Thursday, December 31, 2020

Varieties of Just Intonation


When a contemporary composer proclaims to work with Just Intonation (JI), this statement can refer to widely different approaches to working with pitches. On the one hand, one composer might approach JI purely as a microtonal system with a primary focus on the unique interval sizes that JI generates. Apart from intervals that closely map to those found in Western Classical music, JI can also, even in its simplest low-integer form, generate intervals that deviate from these intervals by tenth-tones, sixth-tones, and quarter-tones. Through using more complex ratios, an almost infinite palette of interval sizes is found. On the other hand, another composer might be primarily concerned with the tunable aspect of JI–using the simple whole number ratios at its basis to fine-tune resting, composite periodicities. The first composer uses ratios without thinking about their tunability. Their practice is more linear and 'microtonal' than the second's, whose practice is more vertical. The second composer is interested in the psycho-acoustic phenomena of how frequencies in simple whole-number ratio relationships fuse and create spectra that are both rich and stable. When 'microtonal' intervals are part of resting, composite periodicities, the sounding result does not necessarily sound 'microtonal' because the focus is shifted away from interval sizes to spectral fusion. For the first composer, this shift would be aesthetically undesirable because they are more interested in the affective qualia of different microtonal interval sizes and their melodic articulation. The relationship these two composers have to JI is very different and generates very different kinds of music. Yet, they are both "JI-composers". 

While both the microtonal as well as the fine-tuned approach to JI can generate successful and captivating pieces of music, I take the standpoint in this text that tunability is the essence of JI. A microtonal practice can easily be achieved by using different types of equal temperaments and by notating cent deviations from their scale steps. There is no need to make things more complicated by evoking the theory of JI if the music does not provide a tunable context. The interval 12/11 is, for example, really nothing other than a neutral second if it arises in an un-tunable context. It can only be enacted as a 12/11 if it appears in a musical context that supports tunability. If the special kind of fine-tuning associated with JI is not an aesthetic value in the music, it is simpler to notate this interval as a neutral second since this is what it will sound like. The difference between an ET-neutral second (150.00 cents) and a 12/11 (150.64 cents) is about half a cent and imperceptible in most musical contexts. While the microtonal practice will have a prominent role in this text, both because it is a common practice among "JI-composers" and because it serves as a useful marker of when JI stops sounding like JI, it will not be considered a species of JI. It will instead be referred to as microtonal music that uses JI-type intervals.

Using JI microtonally or not is not the only way in which JI composers' practices differ. There is a wide range of approaches even among composers who work in a more harmonically fine-tuned idiom. The pioneer who, to my knowledge, first wrote about different ways of using JI was the American composer Lou Harrison. He introduced the strict style and the loose style as a way of distinguishing between different compositional techniques and usages of JI. The strict style refers to music characterized by intonational rigidity; music in which the composer works from a predetermined, small gamut of pitches. The free style refers instead to music characterized by intonational fluidity–where any ratio can be called upon at any moment. In Harrison's strict style pieces, the intonation of each 'focal pitch class'–if we provisionally assume that the theory of focal pitch classes has explanatory power when it comes to describing the perception of pitch (something that will be discussed in greater detail below)–is always the same. If the second scale step is "9/8", it will remain as such throughout the piece or musical section. Harrison's compositions for justly tuned gamelan provide clear examples of a strict style practice. For these pieces, Harrison tuned gamelan sets to original JI interpretations of Javanese scales and then composed pieces for these sets of pre-given pitches. In the free style piece Simfoni in Free Style, Harrison instead locally adjusts the intonation of pitch classes with care to fine-tune vertical harmonies: the focal pitch class of the second scale step can be either "9/8" or "10/9" depending on the context. No intonation of any scale step is pre-given as the intonation of each pitch is sculpted out locally. 

In the present text, I will use Harrison's two terms as a starting point to analyze different kinds of music in JI. But rather than merely focusing on compositional techniques, I am interested in how different compositional strategies lead to enabling different modes of listening. The difference between the strict style and the free style is not only about the composer choosing to work with either a limited gamut or a limitless gamut but about the different ways of listening that are enabled by these approaches–the different musical worlds that the listener can be attuned to. The perspective of this text is primarily one of poetics–how a composer can use JI to achieve certain sounding results. A crucial insight here is that not all pieces in the same style generate the same kind of mode of listening. The strict style–the mere fact of using a small gamut–can, for example, be used to create music that moves in the direction of either approach mentioned in the first paragraph while still being a species of JI: music that is primarily about microtonal interval sizes or music that is primarily about tuning pitches into composite periodicities. Emphasizing either the horizontal or vertical is not only what distinguishes JI proper from microtonal music that uses JI-type intervals, but these different poles are also found within JI music proper. Different modes of listening are enabled depending on where the music is situated on this spectrum. 

The aforementioned justly tuned gamelan pieces by Harrison clearly move in the direction of emphasizing the vertical: the gamelan is tuned justly but then used in a primarily melodic idiom. The simple intervals used in these tunings make the sound distinctly JI–the simple ratios ground the pitches in a tunable context–but the free linear, melodic treatment places the rational origin of pitches in the background as Harrison treats the melodic steps much more like they belonged to a traditional Javanese scale than as if they belonged to a set of harmonically interconnected simple ratios. Rather than being careful to link all pitches through an integrated 'tunable path' that allows the act of rational tuning, Harrison uses the gamelan as an already tuned set of pitches–a 'tuning'. Because of these different ways of using a limited gamut–either treating the gamut as something that stipulates the available pitches to draw upon when constructing tunable links, or treating the pitches as constituting an already-tuned set ready to be used rather freely–the strict style is subdivided into the integrated strict style and the strict style as a 'tuning' to accommodate these tendencies. Within the integrated strict style, the correct intonation of pitches has to be brought forth by a compositional craft that ensures tunability, while the intonation of pitches in the strict style as a ‘tuning’ is treated as if already brought forth. Since within the strict style as a ‘tuning’, a composer can either articulate the pitches as coming from a primarily microtonal and melodic scale, as in Harrison’s gamelan pieces, or as coming from a harmonically interconnected tunable matrix, as in La Monte Young’s The Well-Tuned Piano, the strict style as a tuning must be subdivided further so that these different poles are represented. The sub-categories of the strict style as a ‘tuning’ will in this text be termed rough and informed because the former treats the pitches freely, without considering their tunability deeply, while the latter carefully considers the tunable relations between pitches.     

Just as the mere usage of a strict style can lead to many different modes of listening, the simple feature of using a non-predetermined gamut of pitches that characterize the free style can create vastly different modes of listening. Just as with the strict style, different sub-divisions are used to point to these differences. There is a free style as a 'tuning' and an integrated free style. Within the latter, there is a subdivision that is related to how pitches distanced by commas are perceived categorically: the free style that readily invites separating comma-distanced intervals categorically, and the free style that readily invites a simplified organization of comma-distanced pitches into focal pitch classes.

In addition to further sub-dividing Harrison's two styles, this text also introduces the intermediate loose style–a style that shares features with both the strict and free styles. The most salient characteristic of the loose style is a hierarchical, modal ordering of pitches that dictates the degree of intonational fluidity that different regions in pitch space have; pitches low in the modal hierarchy can be freely replaced by their close neighboring pitches distanced by small commas without sounding out-of-tune and with the replacements being easy to perform–in other words, their intonation is loose. Pitches higher up in the hierarchy, however, can not be replaced without sounding out-of-tune and being difficult to perform–in other words, their intonation is strict. Just as with the free style, there is a subdivision here into music that readily invites separating comma-distanced intervlas categorically, and music that readily invites a simplified organization of comma-distanced pitchs into focal pitch classes.

The musical examples discussed in relation to the three styles will come from pieces by composers such as Marc Sabat, Catherine Lamb, Lou Harrison, and La Monte Young. The majority of examples, however, will come from my own musical practice because this is the music I have the most intimate understanding of. Although many examples come from my musical compositions, the insights drawn from these passages will not only be given a first-person phenomenological explanation but also be continually related to research from the fields of music cognition and tuning theory in order to argue for the intersubjective validity and relevance of my findings. Experimental research from music cognition labs and music theorists' investigation of intonation will help provide an explanatory framework for the analyses of the text. However, because these particular theories tend to assume a cognitivist theory of perception, they will also need to be complemented with insights from enactive, socio-cultural, and ecological theories of perception in order to create an adequate theoretical foundation for the practical, tacit insights gained from composing and performing in JI.

Before delving into the different styles of JI, this text will first outline its basic theoretical foundations. Specifically, it will introduce the concept of tunability and its relationship to JI composition and performance. Other important theoretical topics will be introduced later in the text, such as the idea of a 'categorical' perception of pitches and the concept of 'focal' pitch classes. These will first appear in the section discussing the free style.

This text was first given as a lecture in Tel Aviv on July 12, 2017, at the Tzlil Meudcan festival, and was from the year 2017 to 2019 expanded and reworked into multiple different versions that were shared and circulated informally as "Modality and tunability in strict, free, and loose style Just Intonation". The current version, under a new name, Varieties of Just Intonation, was published in 2020, with revisions and updates being made continuously since. The latest update was on April 10, 2023.

Playing in tune and its prerequisites

From the perspective of a composer, Just Intonation (JI) can be defined as the practice of notating pitches as whole-number ratios, such as 3/2, 5/4, or 25/24. This definition, however, needs to be amended; you need more than notated ratios for the music to actually sound like JI. A belief that it is the use of ratios that leads to JI music is mistaking the means for the end. What defines JI is rather, as Thomas Nicholson and Marc Sabat (2018) state in the very first sentence of their introductory text to JI, the practice of playing in tune.

In the context of notated music, it is the composers who must make sure that the tuning aspect of playing music can be properly activated when performing their pieces, and this requires more than just writing ratios. Composers achieve this by using tunable intervals and connecting these into tunable paths. These tunable intervals and paths are the fundamental elements of justly, actively tuned music, not ratios.

All tunable intervals are possible to express as whole-number frequency ratios, but not all ratios are tunable. There is a limit to how complex patterns musicians can bring to the point of stable rest that characterizes a tuned interval. Ratios can be untunable because one of the pitches falls within the critical band of the other pitch, and other ratios are untunable because they fall within the tolerance range of ratios possible to express with lower integers (Tenney, 1984); the 10935/8192 in isolation, no matter how exact we produce it with computer-generated sounds, can not sound any other way than a very slightly out-of-tune 4/3. It cannot generate an identity of its own as "10935/8192". 

Having said that, it is not the theoretical standpoint of this text that all non-tunable intervals are heard as 'approximations' or out-of-tune versions of just intervals. As will become clear in the discussions below, there are reasons to be skeptical of adopting the cognitivist idea of pitch perception that stems from the work of James Tenney as an all-explaining model. This model perfumes much of the discourse around Just Intonation today. Kyle Gann, for example, in his introductory book to microtonality, states clearly in the opening that the "theoretical standpoint of the book is that ... non-just systems (meantone, equal temperament, and so on) are attempts to incorporate, systematize, and simplify the just systems by approximation" (Gann, 2019, 5). Robert Hasegawa's 2008 doctoral dissertation is another application of this brand of cognitivism; by drawing upon the theoretical arsenal of Tenney and expanding it with more recent research from the field of music psychology and music cognition, Hasegawa analyzes Equal Tempered music by composers such as Scriabin and Schönberg according to the just intervals they are assumed to approximate.

My standpoint is not that the cognitivist explanatory model is wrong. On the contrary, I believe that explaining meantone temperament or the spectrally sounding music of Scriabin as attempting to approximate just intervals can provide us with valuable insights into why this music is successful and beautiful. Rather than dismissing this model, the cognitivist analyses of Tenney and Hasegawa will throughout this text be utilized to illuminate some important features of pitch perception in justly tuned music. But what I want to stress here at the outset is that this text does not take such cognitivism as its foundation or sole theoretical standpoint. It rather sees it as one explanatory model, but one that also needs to be complemented with insights learned from other perspectives, such as socio-cultural, enactive, and ecological theories of perception. We will see examples below of how an intersubjectively conditioned search for enacting meaning will make possible the actualization of intervals that according to strict cognitivist ideas of pitch perception–where pitch perception is analyzed as a kind of 'data-processing' of sounds according to a kind of justly tuned grid–should be impossible. Whereas such a search for meaning might not easily be able to override the process in which a 10935/8192 comes to be perceived as a very slightly out-of-tune 4/3, it helps explain why we can actualize an interval such as the salendro whole step–an interval we will meet again in this text–as a meaning bearing interval in and of itself that does not take its function from being heard as an approximation of either 8/7 or 7/6–its two closest tunable intervals.

To state again what I wrote above, Just Intonation is about the usage of tunable intervals. In other words, a polyphonic piece of music is not written in Just Intonation if it completely lacks tunable intervals. To lack tunable intervals means that the musician never is given an opportunity to tune pitches by ear into the stable, resting composite periodic resonances that are the building blocks of JI. These tunable, distinctive sound patterns are all possible to express as simple whole-number ratios, but even simple ratios might not be enough to ensure tunability if other contextual parameters, such as the speed with which melodies are played, interfere. If tunability is lacking and intervals are articulated slowly enough for the composite pattern to be audible, intervals will typically have a kind of fast-beating, 'indistinctive', and somewhat 'unfocused' quality. 

This fast-beating, 'unfocused' quality is a characteristic feature of Equal Temperament (ET) as well as many other intonation systems, such as the scales used in traditional Sundanese music. Among the first generation of Just Intonation composers, one can sense a strong anti-Western (anti-Classical music) sentiment, and JI was proposed as an alternative to Western imperialism, but very few traditional forms of music in the world use Just Intonation, and the use of 'indistinctive' fast-beating intervals where pitches never fully fuse into periodic signatures is characteristic of far more music than only ET-music. It is also very important to recognize that fast-beating, 'indistinctive' sounds happen even in pieces that are written with Just Intonation accidentals–pieces that use ratios to notate pitch relationships and that are performed on justly tuned instruments. This is because not all music produced in this way only uses tunable intervals. In other words, tunable intervals and rational intervals are not the same.

Similarly, a keyboard tuning might be said to lack tunable intervals if the instrument tuner is unable to tune the pitches to resting, composite, and periodic intervals but must instead rely solely on the BPM of beatings to achieve accurate intervals. This does not mean that tuning by counting BPM is an unreliable practice. In fact, it can often generate greater precision than what tuning to the beatless justly tuned intervals can. The reason for this is that the quality of 'beatlessness' has a slight relativity to it compared to the exactness of counting the BPM of beats.

To complete the picture, we can also say that if tunable intervals appear as completely isolated events at certain points in a piece of music but without tunable connections between them (this is the case in much spectral and post-spectral instrumental music), the music can be said to have tunable intervals but to lack tunable paths

The act of tuning

By compiling a large number of studies and research, Burns (1998) concluded that the most common way for musicians across the globe, including musicians of Western art music, to play musical intervals is by 'reproducing' them from mental representations of how these intervals 'should' sound. (I will leave the discussion of what these 'mental representations' might entail just for now, but will return to a problematization of it below.) In many musical styles, musicians produce highly irrational intervals with great accuracy and consistency even though no tunable references are, or have been, audible. Through cultural learning and repeated exposure, musicians have internalized the quality of the intervals. Burns (1998, 257) concludes in his research that, 

"the intonation performance of a given musician is primarily determined by his or her ability to reproduce these learned categories and is little influenced, in most situations, by any of the psychophysical cues that may underlie sensory consonance or harmony." (Burns, 1998, 257). 

What Burns refers to with “psychophysical cues that … underlie sensory consonance or harmony” is precisely the practice of tuning intervals to low-integer ratios, i.e. Just Intonation. This is a very important point to recognize. Burns is saying that even though Western music’s pitch material—the diatonic scale and the categories of perfect and imperfect consonances—might be derived from the intervals found within the lower partials of the overtone series, this origin is not reflected in a musical practice where musicians tune intervals accordingly. This origin is obscured in the actual performance of Western music as the musicians reproduce pitches based on their stylistic, culturally defined positions in Western music. These positions, even when playing without a piano, come close to the intervals of ET (Burns 1998, 246). This is not to say that these musicians do not play in 'tune' in the conventional meaning of this term, but rather that their idea of 'in tune' does not coincide with the practice of JI. It should, however, be mentioned that there are musical cultures in which the culturally defined pitch classes come close to JI, the most famous example being perhaps Hindustani classical music. 

When a JI composer asks a musician to maneuver through the tunable paths that make up a JI piece, the composer is thus asking something of the musician that, given that she is not a JI specialist, is different from her normal way of producing pitches. It is not only that the actual pitches are different, i.e., JI is not only an expanded palette of new interval sizes to learn, but the actual method for playing pitches is different. She is not only reproducing culturally defined pitches but is also tuning a tunable path. This different way of playing music calls for a compositional craft markedly different from the crafts used in contemporary and traditional Western art music. The failure of many contemporary composers to recognize exactly how vastly different this compositional craft needs to be has led to the production of microtonal pieces that provide musicians with extraordinary difficulties in producing the correct pitches. This in turn has given JI a faulty reputation as being something difficult. 

Learning to tune

Just because a piece of music provides the musician with a tunable path does not by any means imply that the musician no longer needs to have learned the musical intervals to be played. Tunable paths do not make the tuning of a piece automatic. To be able to tune rational intervals as they appear in tunable paths, it is necessary to practice the intervals extensively and to develop a feeling for their size, composite pattern, and character. Embodied representations of intervals are acquired through sensorimotor rehearsal and by engaging the intervals in intersubjectively meaningful contexts, if 'intersubjective' is not taken to only mean 'with other people' but includes the solitary practices that always take place in intersubjective contexts. In this process, we do not simply learn the 'sound signatures’ of intervals but also how they feel, affectively, when we actualize them through our mind-bodies. 

Through such a practice, we become able to audiate the intervals–to sound them to ourselves mentally–and couple this audiation with our rehearsed performance. This process is given a sensitive explanation from a performer's perspective in the violinist Mira Benjamin's doctoral thesis (2019). In it, Benjamin describes a performance scenario in the following way:

"I would observe [...] the surrounding context, make decisions based on my observations, audiate my intended sounding result, and carry out a [rehearsed] set of physical and technical movements which would bring me as close as possible to that intention. I would then listen to what had sounded, respond to it, and add that impression to the overall context going forward." (Benjamin, [2014, 57) 2019, 26)

'Embodied representations' of memorized sounds–learned categories–are crucial for both microtonal music that uses JI-type intervals as well as for JI proper. In both cases is the kind of feedback system between auditing, performing, and adjusting/responding that Benjamin describes present. What distinguishes JI proper from microtonal music that uses JI-type intervals is that these learned categories are not only reproduced as learned sizes but are taken one step further by being subjugated to a process of real-time tuning–they are precisely allowed to be influenced by "the psychophysical cues that may underlie sensory consonance or harmony" (Burns, 1998, 257) that the mere reproduction of learned categories is not.

Passive and non-conceptual

'Learning an interval' (whatever it might mean to 'know' an interval) is something that we do in order to perform it. The way I see it, we do not need to learn anything to actualize the interval as a meaningful aspect of music listening, or if we do, it is an incredibly swift and tacit process that happens without us ever having to make it our explicit object of study. In the vocabulary of ecological perception theory, we can describe it as a type of perceptual learning that is passive. That this learning is passive means that, as Clarke explains, 

"there is no explicit training involved, no human supervisor pointing out distinctive features and appropriate responses. It is "passive" in the sense that it is not under the direct guidance of any external human agency, but it is, of course, profoundly active from the perspective of the organism itself." (Clarke, 2005, 23)

In order to actualize the aural phenomenon that can be represented by the ratio 147/128—a common JI interpretation of the Sundanese salendro whole step—as a meaningful musical aspect, it is not necessary to know that this interval's ratio is "147/128". In order for this phenomenon to be adequately actualized, we do not need to hear it "as" a "147/128" or "as" a salendro whole step—most people who are moved by the beauty of this interval will not actualize the phenomena as mediated by the cultural tool that is the theoretical ratio-label "147/128" or the salendro whole step. Countless music listeners are moved by sounds without knowing their theoretical foundations and without having been 'taught' how to categorize them or respond to them by other humans. This is indeed a trivial point to make but has to be established in the beginning when we clarify our theorizing about JI.

Luntley (2003) makes a similar argument in the article "Non-conceptual Content and the Sound of Music". In this text, it is the dominant V7 chord in tonal music rather than the salendro whole stop that is discussed. According to Luntley, 'novice' listeners without music-theoretical education can perceive the 'tension' and 'incompleteness' of dominant V7 chords without knowing anything about what dominant chords are. Through a series of arguments, Luntley shows that although these 'novice' listeners "have a discrimination of the dominant 7th", they "hear something that they cannot pick up conceptually" (2003, 421). According to Luntley, what marks the presence of conceptuality is the capability of representations to contribute to subjects' rational organization of behavior "by figuring in their inferential reasons for belief/action" (2003, 421). Since the tension and incompleteness of the V7 chord are experienced by the novice and yet "unavailable to the subject's inferential reasons (without cognitive enhancement)" (2003, 421), these experiences represent a kind of pure phenomenality unstained by concepts–"an experience with a nonconceptual content" (Luntley, 2003, 421). Luntley agrees that nonconceptual representations are "extremely difficult to spot because of the reach of concepts into most areas of experience" but finds no surprise in the fact that the examples he gives are "primarily aesthetic" (2003, 421). Aesthetic experiences exist outside of rational organization–an idea that has clear resonances with the aesthetics of Schopenhauer. An aesthetic experience is to Luntley something that "creates a certain impression, but it is not primarily an experience productive of a rational response" (2003, 421).

Although Luntley does not explicitly go as far as Schopenhauer in considering musical engagement to paradigmatically be nonconceptual in nature, the idea of aesthetic relishing as being 'useless' in terms of our ordinary beliefs and actions (i.e., our ordinary willing), and therefore serving as a respite from ordinary modes of being, certainly points in this direction. Based on a recognition of music as something 'direct' and 'immediate', Schopenhauer considered music to not merely be a 'copy of Ideas'. Ideas, Schopenhauer tells us, are the 'objectivity' of the Will, i.e. they are conceptual. Music, however, actualizes the pure nonconceptual movement of the 'Will' directly–which for Schopenhauer was "the reality underlying and bringing into being the appearances of the world" (Cross, 2013, 183). Actualizing this level of reality means forgetting our ordinary willing–the dualistic, conceptual mode of everyday life in which the ego is taken to be an object among other objects. Luntley concludes his text by writing that "[p]ossession of concepts requires possession of will and, for developed human adult experience, you have to look hard, and in normally unexamined corners, to find those areas of experience where we find it hard to exercise will" (2003, 424). For 'novice listeners', musical attunement can be such an area, but for musicians who need to learn how to perform music reliably, conceptual theoretical labels (such as ratios) are essential, but they are not what enables meaningful music experiences for the listener.

Active and conceptual

It follows from these preliminary points that what we as musicians want to learn when we study JI intervals are primarily certain strategies that enable us to perform JI intervals consistently and reliably, not primarily ways of hearing them any differentlyA performer's ear will, however, always be different from a listener's ear. Performing entails a different hearing. Acquiring the skill of performing intervals means starting to hear them in specialized ways, for example when actively listening for the resultant combination- and difference tones in order to tune and stabilize the intervals. When learning to tune a 11/9, for example, actively listening for the arising pitch "2" (the first-degree difference tone) helps the musician to establish the interval's identity. But listening for this "2" is not a "better" way of listening to the interval. It is not revealing the sound in a more "true" and "direct" way; it is simply a practical way of listening that has the purpose of facilitating tuning. It is perhaps similar to a painter's perception of color. She knows exactly how to mix two colors to create a specific third color and knows the unique name for this third color. She will certainly engage this third color in a way that is colored by her practical knowledge and the affective meaning this color has to her practice, but she does not perceive this third color more 'vividly', 'radiant', more 'detailed', or 'better' than someone without this practical knowledge.

Another useful analogy is that of dialects in speech. We can identify a for us novel dialect of a spoken language long before we are told what the name of this dialect is. We can engage it as a novel, and distinct, dialect, and when someone later points out that it is "a Gotlandic dialect", we will appropriate a term that in the future will mediate our experiences, but it is not the case that before we had that term "a Gotlandic dialect" we simply could not perceive the spoken dialect as being different from other dialects that we previously had encountered. We can hear new sounds even when we do not know what they are. This, too, might be a completely trivial point to make but marks a crucial dividing line between the sociocultural theorists—who argue for the primacy of linguistic discourse—and proponents of ecological perception—who argue for the primacy of sense perception and perceptual learning. A sociocultural theorist might argue that the term "Gotlandic dialect" is precisely what enables us to hear the dialect to begin with, while the ecological theorist would argue the reverse, that the term "Gotlandic dialect" only becomes meaningful because there is something to apply it to. According to ecological (Gibson & Gibson 1955) as well as phenomenographic (Marton & Booth 2000) accounts of learning, phenomena arise primarily in relation to prior experience—linguistic categories are secondary. The example used by Gibson and Gibson (1955), is that of a wine connoisseur. This expert taster can distinguish a multitude of aspects of the wine's tastes, and he "can consistently apply nouns to the different fluids of a class and he can apply adjectives to the differences between the fluids" (1955, 35) but the role of language is to describe the nuances of experiences after the experience. Wallerstedt (2010) comments that for ecological theorists, a wine's "dry taste" can never be understood by a person who has never experienced dry taste, while sociocultural theorists would say that the term dry taste is what enables us to experience it as dry in the first place. According to the sociocultural perspective, the concept of dry exists prior to us, in the collective social knowledge, and we make it our own by appropriating it and by engaging the world in a way that is attuned to the way others in our community engage it.

Learning from the sounds

The sociocultural perspective emphasizes that it is difficult to discover for oneself what is valuable to discern in each situation; we need to have access to linguistic descriptions and analyses (discourses) and through communication become involved in communities of action and meaning (Säljö 2000, 62). Taken to its extreme, this socio-cultural vision leads to a heavily anthropocentric vision of social constructionism. While I agree with a vision of the world as radically relational and non-realistthere are no discrete selves and no material reality 'out there' distinguished from a mental representation 'in here'we also have to overcome any dualism between "human" and "non-human" spheres of existence and influence. If social constructionism argues that the illusion of 'reality' is created only through human social interactions, and where human language is seen as the primary agent in shaping this reality, it misses out on emphasizing that also mountains, the spring breeze, and bronze are agents (equally empty, illusory, and relational as humans). They take an equal part in constructing this illusory reality. 

It is here that the ecological theory again becomes relevant because of their theory of affordances (Gibson, 2015). Sounds have certain affordances in that they offer relationally constituted possibilities for interaction. When we practice JI-intervals, we are not simply learning culturally defined pitches or some kind of 'social convention' dictated by other humans. We are learning the intervals from the sounds themselves. By tuning a pitch to another pitch, and searching for common resonances with it, we learn about which intervals are tunable and how their composite periodicities feel directly by engaging the sounds themselves. The sounds are the agents that point out to us what is important to discern, not other humans. We enter into a community of praxis with sound; we are moved and directed by the sounds. As Bashō said, we "enter into the object, perceive its delicate life, and feel its feeling (sono bi no awarete jōkanzuru)" (Thornhill, 1998, 351). We become involved and initiated by sounds' nonverbal communication in a community of action and meaning that is more direct than language. 

It is for this reason that Timothy Morton saw JI as being an important part of the 'non-human' turn in aesthetics. According to Morton, La Monte Young made the piano "as open to its nonhumanness as is possible for humans to facilitate" (Morton, 2013) by the act of tuning the piano to JI. In this way is JI to Morton "a deliberate attunement to a nonhuman" (Morton, 2013). But precisely because reality arises dependently–where neither subject and object can be said to exist by themselves–this also means that sounds and their affordances are not "invariant" or, as Gibson (2015) had it, "always there the be perceived" (130) as if they existed independently from the perceiver. We learn JI from the sounds themselves, but these sounds do not exist 'out there' in some kind of reality that exists in a dualistic relationship to us. They are rather, as Varela, Thompson, and Rosch (2016) have argued, enacted

When Bashō famously told his students to "[l]earn about a pine tree from a pine tree, and about a bamboo plant from a bamboo plant" (Thornhill 1998, 351), he did not have in mind that they were going to write scientific treatises about the pines and bamboos. The result of these poems were not objective representations of material reality, but rather an account of reality where the poets' emotions fused with the natural world nondually. These poems had the aesthetic quality of zōka (造化)what can be described as a kind of 'naturalness' but only if we define this naturalness as something that is not separate and detached from the artist. It is in this way that we can say that the intervals of Just Intonation are 'natural'. They are, on the one hand, not created by musicians since the "psychophysical cues" that Burns (1998) speaks of are not solely made up by social conventions or artists' imagination. On the other hand, the JI intervals have no existence independently–they only come to be because they are enacted by musicians. They are not solely 'discovered' as they can not be separated from the creative act of, as cultural and social beings, enacting these. In other words, Just Intonation intervals arise interdependently–and recognizing this interdependence is what it means to be open to the nonhumanness that Morton speaks of.

In his important 1984 text "John Cage and the Theory of Harmony", James Tenney drew a link between John Cage and Just Intonation, despite Cage himself never working with JI. It is precisely in this act of focusing on sounds themselves that Tenney found a connection. Cage–similarly to Bashō's focus on a naturalism without human ego–envisioned an art that moves "from being a selfish human activity to being [...] fluent with nature". It is difficult to think of an approach to working with pitches that is more "fluent with nature", according to Cage's intention with this phrase, than Just Intonation. In a letter from 1956, Cage succinctly expressed his vision of radical non-foundationalism and relativity–a vision in which the artist is attuned to the agency of non-humans: 

We live in a world where there are things as well as people. Trees, stones, water, everything is expressive. I see this situation in which I impermanently live as a complex interpenetration of centers moving out in all directions without impasse. (2016, 188)

With a Just Intonation practice, we are allowed to precisely be in touch with the interpenetration of phenomena–the nonduality between perceived and perceiver. The practice of JI invites us to be attuned to what can be learned from the act of hearing itself–a hearing in which neither the faculty of hearing nor the object of hearing exist independently but arise interdependently.

Expert listening

The fact that professional musicians hear music differently from non-trained listeners should come as no surprise, and here again, we must emphasize that the listening tools that musicians bring to the engagement of music have been developed with the purpose of performing some particular kind of music reliably. For this reason, these tools can sometimes be maladaptive if applied to the 'wrong' music. Many learning theorists have been inspired by the concept of adaptation in describing the process of learning. van Glasersfeld (1989) argued that Piaget actually shared much of Dewey and the pragmatists' view of knowledge, as both camps described learning in Darwinian terms of adaptation. Knowledge is not a 'representation' of the world based upon how it is beside our practical interests but is precisely shaped by our motives. Unlike some cognitivist colleagues, Piaget did not mean (pace Philips & Soltis 2014) that our cognition is "a mirror of the outside world" but that it is a practical construction out of the need to adapt. Organisms, through their active action, construct the 'laws' and 'structures' of the phenomenal world from recurring patterns. These recurring patterns return to man as 'external forces', but must be seen as the product of a relational and dialectical process. Among these patterns must be included phenomena such as tunability: the phenomena how many low-integer JI ratios are perceived to be still and resting while more complex ratios cause beating patterns. Many of these constructed 'nature's recurring patterns' reach most people in similar ways, but this does not necessarily make the theory universalist (pace Säljö 2000), although it is often interpreted as such by its critics. 

The sociocultural theory, on the other hand, does not describe learning as the adaptive process of building up mental maps that process information (as cognitivists such as Piaget did), but prefer to speak of our experience as being mediated by tools that have been appropriated in praxis. The keyword is appropriation rather than adaptation. According to the theoretical arsenal of this theory, it is not entirely clear whether JI-ratios should be referred to primarily as signs (psychological, mental aids) or intellectual (tertiary) artifacts. The discussion can be set aside by simply describing them as mediating resources (Jakobsson, 2012, 155). Learning the interval 147/128 means that the musician appropriates the mediating potential of the 147/128. When we have learned it, we hear and think in a new way through it. The 147/128 stands in a non-dualistic relationship to our perception; it is not just a cognitive category that filters "the great blooming, buzzing confusion" (James, 1890, 488) but is what mediates the world. It is something we act through and what enables thought (Jakobsson, 2012, 153). To emphasize this non-dualism between person, action, and mediating resources, the influential socio-cultural theorist Wertsch introduced the heavily hyphenated term "individual-operating-with-mediational-means" (1998, 26) as the single analytical term for the individual agent. 

Research has demonstrated that for people who have been trained in Western music, the perception of frequencies is 'sorted' into twelve categories that correspond to the twelve chromatic pitches of Western music. Anything between 60 cents and 140 cents is 'categorized' by the musician as the same interval—a 'minor second'. These categories are built up through sensorimotor rehearsal by playing music—it is not enough to just passively listen to music. People who despite having listened to Western music their whole lives but have not had the practical training, "show no evidence of a category boundary effect" (Burns 1998, 229). In the research by Burns, this insight is expressed as endorsing a cognitivistic view of interval processing: aural training leads to the construction of a 'model' in the student's long-term memory that 'sorts' and categorizes incoming sound frequencies, but I want to make the point that the results can also be explained by sociocultural theories: the subject actualizes pitches as mediated by the pitches in the twelve-tone scale. The conclusion I want to suggest is that from both a cognitivist as well as a sociocultural perspective, it seems like the response of musicians to music can be limited by the 'wrong' kind of prior expertise. Musical training can be maladaptive. This rings a familiar bell from our everyday experiences; we know how certain pieces in 11-limit JI can sound completely natural to a non-musician while a Western musician might not get past the experience of hearing the intervals as 'out-of-tune'. The research cited by Burns seems to suggest that musical training leads to expertise in listening to some kind of music, not any kind. The famous slogan of the Zen master Shunryu Suzuki easily comes to mind: “In the beginner’s mind there are many possibilities. In the expert's mind there are few.” 

There are, however, reasons to be skeptical of the validity of experimental research of the kind studied and compiled by Burns. That many Western musicians only have access to vocabularies and discourses pertaining to twelve discrete tones per octave is no proof that there is a corresponding cognitive twelve-tone structure that filters perception. I have met many Western musicians who, upon closer questioning, perfectly well hear that a "147/128" is different from both a major second and a minor third, but simply do not have any other way of talking about it other than using the terms provided by 12TET. Answering that the sound heard is a major second does not necessarily mean that she hears it as a Western major second, only that it is her only way of talking about it. When the same musicians try to copy the 147/128 with their voice, they often sing either a major second or a minor third. While this might corroborate the idea that they "hear" it as a major second or minor third (i.e. through the categories of Western music), we also have to remember that singing is a practical motor skill that relies not only on hearing but also on the memory and habituation of the body. These musicians have not had the chance to practice singing this interval yet. Often, they can hear that what they sing comes out wrong, but they can not adjust it. My point here is that neither singing nor verbal accounts can be viewed as some kind of neutral medium that gives 'access' to some postulated 'inner' experience. To bring back our example of the 'Gotlandic dialect', we can hear it perfectly well even if we cannot reproduce it ourselves. Most people can identify dialects of their mother tongue, but only people with the right training, such as actors who have worked with dialect coaches, can reproduce a truly convincing reproduction of any dialect other than their own. It is similar to JI; we can all hear the particular beauty and meaningfulness that results from using Just Intonation, but learning to play it is a separate skill.

In this text so far, I have been trying to gesticulate toward what learning such a skill consists of. It is about through a direct engagement with sound, tune stable, resting composite periodic resonances. The JI-intervals are as such not primarily learned as culturally defined 'interval sizes' with certain 'microtonal profiles' that we learn from entering a cultural convention, but are sounds that present themselves as unique composite sound signatures resulting from the fusion of two pitches into a periodic pattern. A musician without prior JI experience can expect a sharpening of the percept as the listening will be geared toward stabilizing the exact periodicity of intervals. The 'dissonances' of Western music in particular will demand increasing attention; small differences will reveal themselves to carry great importance. Between the perfect fourth and perfect fifth, there are five tunable intervals (11/8, 7/5, 10/7, 13/9, and 16/11). These consonances require more attention to distinguish and comprehend compared to Western music where everything between a fourth and a fifth simply is considered 'dissonant'. Learning new JI intervals is greatly expedited if these intervals are meaningful to the performer. Simply sitting with a drone and learning the periodic signatures is not necessarily meaningful enough for everyone. It is better if the 11/9, for example, arises in a musical context where it serves a clear musical purpose and where no other third (such as the neighboring tunable intervals 6/5 and 5/4) would suffice.

Having until now established the fundamental topics of what it means to hear and perform JI intervals, I will now slowly move toward the main topic of this text: the three styles of Just Intonation composition: strictfree, and loose. Before beginning an exploration of these styles proper, I will first introduce some more specific topics in JI composition that will be important to draw upon in our study of the strictfree, and loose styles. These topics concern microtonality, modality, and tunability.   

Microtonal music that uses JI-type intervals

As mentioned at the beginning of this text, ratios notate not only specific 'composite patterns' but also signify certain interval sizes when put into a scalar system. JI is not only about using ratios to create periodic resonances but is also about creating melodic steps that can be analyzed as derived from a musical scale. "7/6" is both a way of tuning sounds as specific periodic resonances with particular sets of combination- and difference tones, as well as being a specific interval size (with a particular microtonal profile) between the two frequencies. The interval size is usually measured in cents. "7/6" is around 267 cents, and "6/5" is around 316 cents. If we adopt the language of Western music, we can say that both of these intervals are types of minor thirds. The first one has a microtonal profile of being narrow, while the other one has a microtonal profile of being wide. "7/6" does, therefore, imply both a harmonic relationship (a periodicity where one repeating soundwave repeats seven times in the same duration as another soundwave repeats six times), as well as an interval size with a characteristic microtonal profile. Ratios have these dual properties (harmonic and intervallic) and the composer can choose to give more emphasis to one or the other through the craft of JI-composition.    

Music that is notated with ratios but lacks tunable intervals and paths will have to rely solely on the musician’s ability to memorize the size–the microtonal profile–of intervals. In these cases, only the ratios' intervallic properties are utilized. The musician might practice these intervals with an electric tuner and reproduce them as closely as possible in performance. The result of this will, however, not be JI music according to the strict definition that has tunability as a requirement. The resulting music can instead best be described as microtonal music that uses JI-type-intervals. This music will be performed just like the common music described by Burns where intervals are reproduced based on memory and approximation. Some composers appreciate the size of rational intervals, such as the intervals that can be found between the first thirteen partials of the overtone series, without caring much if these intervals actually fuse into stable, tuned sounds; it might simply be the case that many of these interval sizes are deemed beautiful and considered to have a powerful affect regardless of whether they are tuned precisely or not. Many contemporary composers of the post-spectral inclination seem to use JI in this fashion. One such composer, Taylor Brook, even writes in the preface to the piece Amalgam (2015) that "the microtones can be understood within a system of just intonation or as written-out bends and slides." Brook recognizes that in this piece, even if the musicians conceptualize the pitches as JI, the effect, due to the musical context that never calls upon tunable paths, will not be different than if pitches were conceptualized as microtonal deviations ("bends and slides") to the conventional pitches used in Western music. A brief look at the score confirms this as the music begins with a line that under a D drone plays a microtonal sequence of 1/1, 81/80, 64/63, and 33/32, none of which are tunable but that nonetheless creates a particular sequence of narrow fast-beating microtonal variations. In this music, JI becomes only a method for generating particular interval sizes and microtonal inflections, and the whole aspect of tuning intervals to resting, composite patterns is lost. 

Since microtonal music that uses JI-type-intervals is so widespread, musicians encountering JI music proper have to first learn that the purpose of the microtonal inflections of scale steps is to form relational properties with other scale steps, not to draw attention to the inflection itself. A friend told me how a prominent harpist in the German "New Music" community had protested against tuning a string 2 cents lower since this change is imperceptible. The harpist questioned the composer's intention and painted the composer as making impossible and unnecessary requests. What the harpist did not understand was that the point never was to microtonally inflect a certain string, but to form a simple, tunable relation to another string. As a microtonal inflection, 2 cents might not be perceptible, but tuning a 3/2 (2 cents is the difference between an ET perfect fifth and a 3/2) in a musical context that supports tunability is very much perceptible.

There is an interesting difference between the 'traditional' musical practices described by Burns and a piece such as Amalgam. While both practices rely on approximation and memorization, the process in which this internalization comes about is very different. In the first case, it happens because the musician is taking part in an intersubjective culture of meaning. The musical repertoire, the community of performers, and the instruments create a praxis of meaning in which the microtonal profiles of intervals make 'intuitive' sense, despite being irrational (not expressable as ratios). When I studied Sundanese music, I marveled over how easy it was to learn to sing the different Sundanese scales and the microtonal profiles of the scale steps, despite being irrational and different from other music that I had practiced. Were I to write these scales down and give them to a Western musician reading from a score, she would not be able to produce them with the same ease simply because she would not have access to the praxis in which these intervals are meaningful. It would indeed be very difficult and painstaking for this musician to learn the same intervals that I so easily had internalized. The contemporary composer who wants to use irrational intervals and have the pieces be easy to learn has therefore the great—but by no means impossible—task of needing to compose this meaningfulness into the piece. The piece itself is what must provide the entire sphere of meaning. Above, I wrote how the 11/9 is easier to learn if it arises in a musical context where it serves a clear musical purpose and where no other third (such as the neighboring tunable intervals 6/5 and 5/4) would suffice. In microtonal music that uses JI-type intervals, the composer must give every microtonal inflection a similar clear purpose and necessity. If not, the musicians will find themselves practicing the intervals with an electric tuner and reproduce the intervals as closely as possible to this in performance, but never feel that she 'understands' why the microtonal profiles are the way they are. And there are indeed countless examples of such pieces in contemporary music. Such situations become akin to behavioristic conditioning; the musician conditions herself by repetition to reproduce arbitrary intervals accurately. But this is refusing the musicians the opportunity to embody and actualize the intervals as meaningful. Or rather, musicians will have to do this anyway because that is what it means to be a good musician, but only after a painstaking process and perhaps with a different meaning inscribed to the intervals than what the composer intended. 

It is, however, not my intention to be overly critical of the above-described microtonal 'interval-size-generating' approach to JI but rather to argue for the fundamental definition of JI as that of hearing tuned sounds. Since the practice of microtonal music that uses JI-type intervals is widespread, it is important to emphasize how the JI music that will be described in this text is different from it. We will soon see, however, that even music that purports to be an integrated tunable path (including many of my own compositions) often makes use of plenty of intervals that are not, as a matter of fact, tunable. We will see that a definite line between microtonal music that uses JI-type-intervals and what we will call integrated JI is impossible to establish but often represents two idealized extremes. In reality, much music moves back and forth in complex ways between using JI for its tuned resting sounds and using it for the affect associated with different interval sizes—where whether or not these intervals are justly tuned is of little importance.

Tunable intervals

Since it is the composer who is faced with the responsibility to ensure tunability, she must develop a thorough understanding and insight into the real-time tuning processes of playing music and will have to reach an answer to two questions: "What is a tunable interval," and "What is a tunable path?" The contemporary composer who perhaps has contributed most towards answering these questions is Marc Sabat. Apart from his many compositions in Just Intonation, one of his great contributions is the compilation of a list of tunable intervals in the order of difficulty (Sabat and Hayward 2006, see also Sabat 2008/2009). This list, which is based on tunable intervals above and below A4, may not be universally valid in every musical situation as tunability changes with context, register, timbre, and dynamics, but it is a valuable reference and often surprisingly accurate.

Among the easiest tunable intervals within one octave, we find 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, and 9/5. It can be noted that within the first octave’s list of easy intervals, all the intervals involve no higher prime number than 7. Within the second octave, we can easily add to these 9/4, 11/4, 13/4, 11/3, and 15/4. It is thus only with one octave’s distance that we can add two 11-limit intervals (11/4 and 11/3) to our list of easily tunable intervals. Among the more difficult intervals to tune, we find within the first octave also 8/7, 11/9, 13/10, 11/8, 10/7, 13/9, 16/11, 14/9, 11/7, 13/8, 12/7, 11/6, 13/7, 15/8, and 23/12. What is important to take note of here is that the collection of tunable intervals is small (26 notes to an octave in total and 11 easy ones), and that many common intervals are completely lacking from this list, such as the 11-limit neutral second of 12/11. This list, however, only pertains to harmonic dyads, and, as we will see below when more closely examining the craft of JI composition, untunable intervals like 13:11 can be tuned both melodically—by, for example, moving from an “11/8” to a “13/8” that are both tuned to a “1/1”—and harmonically—by, for example, having all three pitches sound at once.

The internal ordering between more and less difficult intervals differs between instruments, timbres, and registers. I find for example that for me, the 9/7 is easier to tune than its inversion 14/9 in mid-and high registers. In low registers, it is the reverse; the 14/9 is easier to tune than the 9/7. It should also be stressed that some people will find certain intervals more difficult to learn due to their unique previous musical experiences. Even though 11/9 is listed as more difficult to tune than 11/6, I remember how, for me, learning 11/9 was less difficult than 11/6. This 11-limit neutral third was very easy to learn; a solid "2" (the first-degree difference tone) arose in my inner ear which helped me stabilize the composite interval. 11/6 was, however, more difficult for me to learn: the first-degree difference tone "5" caused 'distracting' harmonic interference with the pitches and made it more difficult for me to stabilize. In the development of my JI-hearing, it was interesting to observe how all the intervals I had a hard time mastering were all intervals to which the arising first-degree difference tones offered 'rich' harmonic contextualizations. Such was often the case with the non-5-limit intervals that had "5" as their first-degree difference tone, but with the notable exception of 13/8 which was partly so easy to learn to tune because of the clear "5". 14/9, 12/7, 11/6, and 16/11 seemed all to be difficult for the very same reason; they had "5" as their first-degree difference tone. In these cases, it was my difficulty in finding stability in a richer harmonic context and 'listening for the 5' that dictated the internal ordering of difficult intervals. Now, however, I can hear that 11/6 is an easier interval to tune than 11/9, and agree with this ordering.  

Sabat’s list is not only valuable when composing for intonating instruments but should also be taken into consideration when composing for fixed-pitch instruments or electronic sounds as the list of finite tunable intervals is also the list of finite perceivable JI intervals. Tunability must be ensured for the musician and listener alike. It is a logical impossibility to imagine an interval that is perceivable as JI but not tunable because perceiving it as JI means precisely that it has a stable core with a concomitant 'gravitational pull’ that can be tuned to. Later on in this text, we will see examples of complex ratios that even when accurately tuned in JI will fail to sound like JI.

Tuning to short-term memory and modal fields

If there was no short-term memory and only present-moment awareness of musical attunement, the craft of JI would end with the list of tunable intervals. If such was the case, the two categories mentioned above of microtonal music that uses JI-type-intervals and integrated JI would be mutually exclusive; whenever a frequency ratio was untunable, the music would be microtonal music that uses JI-type-intervals, and whenever a frequency ratio was tunable, the music would be in integrated JI. The craft of JI composition would then have nothing more to it than this very list and the investigation of how this list translates to different registers, timbres, and instruments. 

While we can contend that 'true tunability' only is possible to achieve with simultaneous sounds, we frequently encounter the phenomenon of how during the performance of a piece, tunability seems to 'seep out' from simultaneous, harmonic sounds and infuse non-simultaneous pitches that in and of themselves are untunable with a meaningful, albeit much weaker, degree of tunability. It is precisely this subtle quality of a kind of emergent tunability that makes JI a more complicated craft than simply categorizing intervals as tunable or untunable. This point is subtle, there should be no question about it. Nothing can replace the power of simultaneous tuning, and some people are right in questioning my choice of vocabulary in calling this a form of 'tuning'. But what else could it be? What other word could we use to describe this phenomenon? Whatever we prefer to call it, it is some kind of emergent property that helps the performer adjust the intervals in a tuning-like manner. Despite being subtle, we will see through concrete examples in this text how this phenomenon has significant effects on harmonic and melodic writing in JI. 

There are to my present knowledge two different types of phenomena that can be called 'non-simultaneous' tuning.

Firstly, there is the fact that points in harmonic space seem to have what James Tenney (1984) described as a kind of 'persistence': “once activated, a point in harmonic space will remain active for some considerable amount of time after the tonal stimulus has stopped sounding" (30). Tenney attributed these to “a sort of neural 'resonance' in short-term memory.” For example, the 9/8 is only possible to tune with difficulty in high registers (Sabat, 2008/2009). In a musical passage, however, a mid-range 9:8 melodic movement consisting of the scale degrees “4/3” and “3/2” might be perceived 'as if' tunable if an emphatic “1/1” has been audible for some time before it and thereby constitutes an external reference to both parts of the 9:8. Strictly speaking, however, we are not tuning a 9:8 but rather a successive “4/3” followed by a “3/2”, between which there is an interval we can describe as 9:8.

Secondly, there is the phenomenon that happens when as soon as a composition unfolds, pitches and harmonies build up a 'supportive' context for tuning. Previously heard pitches support the intonation of present pitches by reinforcing their position in a modality. Playing the series of pitches that make up a piece becomes slightly less a matter of tuning each pitch to the immediately preceding pitch, like a relay race, and more about relating them to the established emergent modality. In this way, intervals that are not directly tunable can be perceived 'as if' tunable because they will be possible to intone to something other than just the immediately preceding notes. Think of how most musicians would agree that singing a prima vista is easier if the music is tonal rather than atonal since one does not have to create each interval individually 'by itself' but is helped by the existence of a latent tonal scale, a kind of grid, to place it in. This is in a sense similar to the phenomenon described here but the modality of precise points of intonation (rather than approximate pitch classes as in Western tonality) is much more feeble and weak; there is a reason for the careful language I am employing here. This topic of modal intonation will be extensively explored in this text. 

As the time between sounds increases, the margin of error before a note sounds out-of-tune also increases. This can be demonstrated by, for example, playing or singing a pitch (for example an A) for a few seconds, then waiting five seconds in silence before singing or playing a tunable interval (for example a 5:4 down, an F+ [2]). We observe ourselves tuning the second interval to the short-term retention of the first, and we might say, if we are pleased with our performance, that we performed the second pitch in tune with the first. If we make an audio recording of ourselves doing this and then edit the recording in such a way that the two pitches overlap, we might realize that what we perceived to be in tune was, in fact, not quite so. As tunable sounds are separated in time, either with silence or other pitches in between, sounds will be heard as being in tune even though they are quite far from it when measured exactly. It is for this reason that these sounds above have been referred to as 'as if' tunable. This 'as if' is, however, tautological since to human perception, there is never anything other than 'as if'. If JI is defined as hearing tuned sounds, it does not matter if the sound is 100% in tune when scientifically measured—which it never is when listening to music played on acoustic instruments anyway. 'In tune' happens as humans actualize intervals; it is not some property of a postulated scientifically measurable 'reality' simply located 'out there'. If an interval appears 'as if' tunable–if it has gravitational pull toward a 'core’ where the pitch can find 'rest' and where anything either higher or lower sounds out-of-tune–, it is simply tunable.  

It should be added here that when playing a piece in JI, ample opportunities arise for the musician to doubt whether she is producing a certain pitch from a rehearsed embodied mental representation of it or if she is tuning it to a previous pitch that is starting to approach a distant memory. Is she 'tuning' to her silent 'representation', connected to an embodied praxis, or the retained 'sound'? We will in this text see that as soon as we move out of the simultaneity of tuning and into non-simultaneity, things rapidly get unclear and less clean-cut. In this text, fortunately, I will not attempt to answer such complicated questions. What primarily interests me is not to find out exactly how out-of-tune some of these sounds we believe to be in tune really may be, or that what we thought we were doing when 'tuning' actually just was the reproduction of pitch positions from memory, but rather how this type of non-simultaneous tuning practically impacts the musician’s and composer’s task of constructing a piece as a tunable path. If we define tuning as that which has gravitational pull toward a 'core’ where the pitch can find 'rest' and where anything either higher or lower sounds 'indistinct' or out-of-tune, the dualism between 'tuning' and 'approximating a representation' becomes unimportant. I am not so much trying to construct a theory as I am sharing observations from my practice and how I have tried to make sense of it theoretically. By showing the non-simultaneous tuning at work in my compositional process (primarily in Figures 27-29 and 11-13), the reader is hopefully convinced that this process indeed is best described as one concerning tuning rather than simply approximation.

The balance between tunable and untunable intervals

To exemplify what has been discussed so far, Figure 1 and Figure 2 show two brief melodies. Both figures include the same pitches, A, B, C#- D, E, and G#-, but ordered differently. In the first figure, the pitches are ordered as E:B:C#-:A:D:G#-. In the second figure, they are presented in the order of E:D:B:A:C#-:G#-. The different order that the pitches are presented in greatly influences the tunability of the two figures. 

Figure 1

In Figure 1, the first interval is a tunable perfect fourth (4:3) between E and B. The second interval between B and C#- is an untunable Ptolemaic major second (10:9). The thesis that I propose is as follows: the intonation of the C#- is aided by the short-term memory (or rather, retention) of E to which it forms a 6:5, and the musician does not have to solely approximate a 10:9 according to a rehearsed mental image. It is a similar situation with the fifth interval between D and G#- in the fourth measure. While the third interval between C#- and A is a tunable 5:4 and the fourth interval between A and D is another tunable 4:3, the fifth interval between D and G#- forms the untunable interval 45:32. Rather than only approximating this 45:32, I suggest that the musician retains both the A—to which the G#- forms a tunable 15:8—and the C#- —to which the G#- forms a tunable 3:2—and tunes the pitch in some reliance upon both of these previous pitches. The performance of the G#- is positively aided by the retention of pitches previously played but not immediately adjacent. When playing the G#-, it does not feel like I am solely approximating the untunable 45:32 ratio, but that I am bringing it forth through an act of tuning in JI.

In Figure 1, even though 40% of the pitches are untunable to their immediately preceding pitches, they can all still be played by tuning them to varying degrees, and no intervals have to be solely approximated from memory. Of course, when I observe myself audiating the first 10:9 in anticipation of performing it, I am also bringing to the foreground an embodied sense of what that narrow major second feels like in and of itself. This kind of approximation from a rehearsed memory is an important part of playing the C#-. But I also observe how in this particular case, the presence of the retained E allows a 'tuning aspect' to come into play. I am not only approximating a 10:9, but the retained E informs the positioning of the C#- as a 6/5 to it. This tuning aspect does not come into play when I start the passage on the B directly and does thus not have the retained E. What I propose is that this whole passage can be considered tunable; tuning two out of five intervals to their penultimate pitches is not enough of an obstacle to render this passage ‘untunable'.

In Figure 2, the proportional relationship between tunable and untunable intervals is reversed. Only 40% of the pitches are tunable to their immediately preceding tones, and none of these pitches fall within the first three measures. The first interval between E and D is an untunable 9:8. Since no other pitches have been heard before it, it can only be played by approximation from a mental representation. The second interval between D and B is the untunable Pythagorean minor third (32:27). This third pitch, B, might be given some intonational help by the retention of the first pitch, similar to what we saw in Figure 1, but here in a 4:3 relationship. When playing this particular phrase, however, I notice how this requires me to make a very active effort to retain the first pitch in my mind. The reason for this seems to be that the second pitch, D, is distracting. I must resist the temptation to treat this D as a tuning reference for my B. The third pitch should neither be tuned as a syntonic comma lower (to form a 6:5, to the D) nor as a septimal comma higher (to form a 7:6 to the D). The second pitch in Figure 2 is more 'distracting' to the third pitch’s accurate tuning than the second pitch in Figure 1 was to its third pitch. This makes my task of tuning the third pitch to the first pitch more difficult in Figure 2. In Figure 1, retaining the first pitch in memory was more passive and felt 'natural'. In Figure 2, I have to more actively retain this note as the tuning reference for the third pitch.  

Figure 2

This talk of a difference in the musician's active effort of retention points to the topic of audition—the internal hearing and projection of pitch in the mind-body before playing it. In Figure 1, the process of auditation is supported by how the musical context provides an emergent tunability. Because of this, it feels more effortless to perform. In Figure 2, auditation has to rely more on embodied experience, and it feels more like something I have to produce and 'project', and therefore feels more 'active'. 

Moving on in Figure 2, we find the third interval to be another untunable 9:8. Despite this untunable relationship between the B and A, the A becomes possible to 'tune contextually' to the short-term retention of E and D—two recently heard pitches that form perfect consonances to the A. This contextual tuning of A happens relatively effortlessly without having to make the very active effort to retain E and D in the mind (as when tuning the B). Because the A has been preceded by more pitches to establish a modal, tunable context, it is easier to 'contextually' tune than the B, despite both of them being untunable to their immediately preceding neighbors. Lastly, the final intervals of Figure 2 are the tunable 5:4—from A to C#- —and the tunable 3:2—from C#- to G#-. 

The examples above illustrate that music that functions as a tunable path does not have to be confined to only using the tunable intervals from Sabat’s list. There must, however, be a careful balance between tunable and untunable intervals. One might even say that the specialized craft that is JI composition is the very mastery of this balance. We saw how by simply changing the order of pitches in a brief melodic passage, the difficulty of tuning can be significantly altered. When playing the two figures above, I find Figure 2 considerably more difficult to tune precisely than Figure 1. The first figure showed us a fine balance, and the second showed us a less-than-optimal balance. As we will see in further examples below, the construction of a fine 'balance' often entails using pitches in such a way that harmonic spaces, or modalities, are formed in which a single pitch often becomes tunable to multiple other pitches in an 'emergent tunability'.

Examples of 'modal' intonation 

Let us continue to dwell on the music in Figure 2. This musical fragment can be made either more difficult or easier to tune by adding more notes before it. Experimenting with this will show us how a modal context with many 'supportive' pitches facilitates tuning. 

Figure 3

In Figure 3, a G and F#- are added. In my experience, this addition makes the B even more difficult to tune than before. This is because the G and the F#- provide untunable references to the B in the form of a Pythagorean third (81:64) and a wolf-fifth (40:27). Above, it was noted how the musician might be 'tempted' to tune the B as a B- in order to form a 6:5 to the D. In the version with the added G and F#-, this temptation grows stronger as the F#- is a perfect fifth (3/2) to B-, and G is a pure major third (5/4) to it. A version with a B-, seen in Figure 4, will be significantly easier to tune since it fits the suggested modal context implied by the previous pitches. In integrated Just Intonation, there will often be an implied root—a 1/1, or modal center—to a passage. The first four pitches in Figures 3 and 4 clearly indicate a modality centered on D. It is D rather than G that is the 1/1 of this passage because of E’s position as a “9/8” above it (if it had been an E-, then G would have been implied as the root). If we visualize the pitches in question as a ratio lattice (Table 1), we can see that the B is further away from 1/1 than the B-. We see that the B- is just diagonally to the left of the D and has a simple (low-number) ratio to it, while B is three steps to the right and has a more complex (high-number) ratio to it. The B is thus further away from the locally implied modal center.

Figure 4

Table 1

By doing practical experiments such as these where we are adding pitches to brief melodies and experientially assess if the melody becomes easier or more difficult to tune based on what we add, we come to see clearly for ourselves that there is a way to construct melodies that facilitates non-simultaneous tuning by allowing a type of modality to arise. Pitches support the intonation of other pitches by reinforcing their intonational tendencies. A context is created where some pitches will feel 'correct' to introduce—they sound as if they already belong to the mode—and other pitches will feel foreign and incorrect. This latter quality can, of course, be exploited in a composition to create meaningful contrasts and modulations. In the last bar of Figures 3 and 4, there is a modulation to A major as 5/4 (C#-) and 15/8 (G#-) are built upon the A. As the B- is further away from A major than B, the modulation to A major will be more unexpected in Figure 4 than in Figure 3. In Figure 3, the use of the B actually helps to hinder D from becoming established as a very strong modal center, and this makes the subsequent modulation to A more smooth. When discussing the loose style below, we will explore this kind of subtle writing in greater detail, but what we already see here is that even though Figure 3 is more difficult to tune in and of itself, there might be good artistic reasons to not write a B- even though it is easier to tune; we might have A as our modal intention and goal, and a B- would too much imply another root. The modulation to A itself is not really made any easier to perform by changing the B- to B since the C#- is present in both D and A modalities, and the G#- is introduced as a simple 3:2 to it, but the effect of the modulation will be different. The modulation in Figure 3 will sound less surprising and more 'prepared'.

Figure 5

As a final example, Figure 5 shows a context in which the music in Figure 2 is experienced differently to intonate by changing the root to E from the very beginning. By E emphatically being 1/1, there will be less problem with the B even as the 32:27 relationship to the D still exists. But new problems arise; it is now rather the fourth pitch, D, that becomes the main focus of intonational problems. This pitch has no tuning reference and will not have to be solely approximated. 

The three styles

The modal fields that emerge from these kinds of musical procedures can either be strong or weak. A strong mode means that there are precise regions in an octave pitch-continuum where pitches must be placed to feel 'correct'. A strong mode, as we will see in greater detail below, has the benefit of facilitating intonation of untunable and non-simultaneous intervals, but it will also make the introduction of new pitches (modulation) more difficult. If the scale degree "5/4" in a melody like "9/8":"1/1":"5/4" sounds too low and out-of-tune, it is likely because of the existence of a strong ["1/1", "9/8", "81/64"]-modality. Such modality is considered strong because it made a consonant, common interval like "1/1":"5/4" (5:4) sound out-of-tune, which might sound odd and unlikely on paper but is commonly encountered in JI counterpoint. It is the same strong modal force that makes the 5:4 sound out-of-tune that also, elsewhere in a composition that uses that mode, facilitates the intonation of the untunable "1/1":"81/64". Pieces that exclusively use pitches belonging to one strong modal gamut are in this text categorized as being in the strict style.

If the microtonal difference between the "5/4" and the "81/64" in the melody "81/64":"9/8":"1/1":"5/4" does not sound jarring but simply pass by naturally, then there is not a strong modality present of the kind that is present in the strict style. A lack of modal rigidity and the listener’s acceptance of a fluid intonation is characteristic of successful writing in the free styleMusic in the free style does not generally lack modalities or pitch hierarchies, but these are much more temporary and unestablished than in the strict style. The stylistic context of ever-changing and very local modal centers creates an acceptance of pitches constantly being replaced by neighboring variants comma-distances apart. Pieces in the free style typically draw upon a large gamut of pitches; it is not uncommon to in a single piece have more than four comma-distanced variants of a single focal pitch class. The weak modality in the free style has the benefit of freedom but will demand more carefully constructed tunable paths as the musician will be able to rely less on tuning to strong modal fields. She must instead find tunability in the neighboring pitches and the highly locally arising modal hierarchies. The result of this is a tunable path that is more feeble as it becomes more important that each pitch is played accurately. This description of the free style may sound advanced, and in Lou Harrison’s practice, the free style was something very complex and difficult. Later on in this text, however, it will be suggested that the free style is a most 'normal-sounding' style to an ear accustomed to Western classical music; it is the conceptually more simple strict style that to this ear will sound difficult and unusual.

Lastly, there can be situations where we in the same section of a piece hear nothing jarring in the melody "81/64":"9/8":"1/1":"5/4", but the "81/80" in the sequence "5/4":"1/1":"9/8":"81/64":"81/80" might sound like a mistake even though it forms a 5:4 to the "81/64." In such a case, the third scale degree enjoyed 'intonational fluidity' as it could be both "81/64" and "5/4". In other words, to use a slightly awkward formulation, the third scale degree did not have a strong 'modality' attached to it. The root position, however, did; it could be "1/1" but not "81/80". These kinds of situations, where it sounds jarring to change the intonation of some pitch classes but not others, or where some focal pitches have strong modal commitments and others do not, are characteristic of music in the loose style. As we will see below, it is the existence of a modal hierarchy that dictates which pitches that are 'fixed' and which ones that are 'fluid'. 

These three modal strengths give rise to different compositional treatments of tunable intervals, tunable paths, and melodic writing in general. In the upcoming sections of this text, starting with the strict style, these will be discussed separately and in greater depth. 

Strict style

When Lou Harrison introduced the two categories of strict style and free style, he did not have the criteria of perceived modal strength in mind. His perspective was rather one of pre-compositional planning. In the strict style, the composer sets out to work from a limited gamut of frequency ratios and builds a piece from these. Typically, the gamut is small, and the pitches are usually relatively far apart. Although intervals of 35 cents or smaller can occur in a strict style piece, they can not occur in the same way as they often occur in loose or free style pieces: if these close intervals were to be perceived as different intonational 'variants' of the same pitch rather than discrete pitches, the pitches would not have the strong modal commitments that I here define as being characteristic of the strict style. The
 music would instead be variants of the loose or free style

When writing music for instruments with fixed tuning, such as zithers, pianos, or tuned percussion, the strict style is often the most idiomatic and most practically feasible style of JI to employ. For this reason, the strict style is often associated with music for fixed-pitch instruments. For Harrison, the difference between strict style and free style was indeed almost the same as the difference between writing for instruments with fixed or flexible tuning. Strict style is, however, not limited to fixed-pitch instrumentation, and it can be successfully employed when using intonating instruments. 

Figure 6

An example of a strict style piece that includes an intonating instrument is my composition Vårbris, porslinsvas for violin and piano. The gamut used in this piece (repeating in all octaves) is shown in Figure 6. It is a simple 5-limit gamut with only one 'comma-level'; only one kind of comma is used to change pitches from their default Pythagorean definition. By doing an inventory of the types of intervals found in the gamut, we find that there are no tunable minor or major seconds for mid-register use–the 9/8 (found between A/B, F#-/G#-, and D/E) is only tunable is higher registers or as a ninth. There is a total of three pure major thirds (5:4, or minor sixths, 8:5, if we inverse them)–"1/1":"5/4", "4/3":"5/3", and "3/2":"15/8"–and three pure minor thirds (6:5, or major sixths, 5/3, in inversion): "5/4":"3/2", "5/3":"1/1" and "15/8":"9/8". These thirds will all be calm and resting. The only untunable 'third' we can find in this gamut is between "9/8" and "4/3"; this minor third of 32/27 is always narrow and fast beating. There are many pure fifths and fourths; the leap between "9/8" and "5/3" is the only instance of an untunable 'wolf-fifth'. This thorny interval will easily sound out-of-tune and must be handled carefully. There are two tunable major sevenths–"1/1":"15/8" and "4/3":"5/4"–and only one tunable minor seventh–"5/3":"3/2"–while there are three untunable Pythagorean minor sevenths (16:9, between D and E, F#- and G#-, and A and B). The gamut both contains a handful of tunable intervals–especially thirds, fourths, fifths, and sixths–as well as a handful of untunable intervals–primarily major seconds and minor sevenths. Constructing a tunable path out of this gamut means primarily using the former rather than the latter.

Turning our attention to the score, excerpted in Figure 7, we can, by tracing the tunable intervals, see that the melodic writing for the violin is deeply conditioned by the tunable paths available in this particular gamut. The first harmonic interval, formed between the piano and violin, on the downbeat of measure 144 is a tunable 4:3. The violin then plays a melodically descending tunable 6:5 to form a tunable 8:5 to the sustained piano note from the downbeat of the measure. The piano then plays a descending 5:4 to this violin note (or a 2:1 to its previous note), and the violin adds another tunable 5:2 to this. When I composed this piece, I was not 'free' to choose any pitch at any time from the gamut. Since I wanted to ensure a good balance between tunable and untunable intervals, my options for which pitch to choose at any time were heavily restricted. In measure 178-179 (also in Figure 7), in order for the violin to move from a B to an F#-, I added an A in between–an A that itself was tunable to the same sustained E that the B was tuned to–so that the F#- had a tunable reference point and so as to avoid a direct wolf-fifth. 

However, and as will be repeated many times throughout this text, music in JI is not restricted to exclusively employ the still, tunable, consonances as long as there is a balance between these and the untunable, tense intervals. In measure 160 of Vårbris, porslinsvas (Figure 7), the untunable Pythagorean major sixth 27/16 between D and B is held as a point of harmonic tension. This works because the interval has a thorough preparation; it is arrived at by tunable steps. The B is tunable to all pitches previously sounding: 9:4 to A, 10:9 to C#- and 5:4 to G#. The D is a tunable 4:3 to the previously sounding A. The tense, untunable, 27/16 is then resolved to a tunable C#- unison in measure 161, which is a tunable 15:8 to the D. Since the Pythagorean interval is treated in this careful way of being tunable to the immediately neighboring pitches and not approached directly from nowhere, it does not sound jarring but rather reveals itself as a natural tension inherent in and characteristic of the mode; it is part of what gives poetic meaning to the mode. The gamut dictates to the composer not only which pitches and intervals will create a feeling of relaxation but also which ones will create tension

Figure 7

One result of this kind of restricted and highly limited strict style melodic writing is that each scale degree gets endowed with a clear, characteristic function and individual flavor. Each tone becomes imbued with a specific 'psychological feeling' as its relationship to the other scale tones is so particular and so uniqely defined. One should, of course, remember that a similar kind of situation (where each tone in a scale receives an unique quale) also has been noted for diatonic scales in ET. David Huron has written extensively about the different qualia of scale tones in ET. He notes how "each scale tone appears to evoke a different psychological flavor or feeling" (Huron, 2006, 144). According to Huron, this is only true for listeners with sufficient experience of Western music as "several of the most salient qualia evoked by different pitches can be traced to simple statistical relationships" (Huron, 2006, 173). Qualia are not, as Thompson (1995) explains, found 'out there' in the music but arise as "relational properties constituted by the environment and the perceiving-acting subject" (286). Thompson explains that "[o]n the side of the environment, they correspond to various physical, chemical, and biological properties; on the side of the perceiving subject, they correspond to modes of presentation in perceptual experience" (see further discussion about qualia in Thompson 1995, 286). In ET, it is primarily stylistic knowledge that, on the side of the perceiving subject, that colour the qualia.

Although music in ET depends on stylistic knowledge and statistic relationships for the qualia to arise, the argument can be made that the qualia of strict style JI are more immediate. With the unequal scales of strict style JI, each pitch comes with a particular behavior pattern based on the unique relationships it forms with other pitches in the gamut, whereas in the equally tempered scale, all such meaningful relationships have been evened (tempered) out. As a strict style gamut unfolds in a single piece, the affective differences between scale tones become very pronounced in the listening experience. My hypothesis is that if the qualia in Western music have to do with, as Huron says, stylistic knowledge attained through sufficient experience of Western music, the qualia in strict style pieces are accessible through passive perceptual learning from the piece itself. In Vårbris, porslinsvas, the B takes on a special quale because it forms a pure fifth to the E; it can rest with the C#- as a 9/5 minor seventh but not as a 10/9 major second; it will be in unrest with the first degree if played as a 16/9 (B/A) but can be tuned and rest with it if played as a 9/4 (A/B). These relationships as articulated through a piece of music give the pitches their qualia and the mode its rasa. The B is so modally 'correct' that even in measure 172, where a B- would have been more 'in tune' with the simultaneously played D, the Pythagorean minor third B:D sounds right and expected; it is its meaning-making role to be untunable here. It would be an exaggeration to say that occasionally replacing the “9/8” (B) with a “10/9” (B-) would sound 'jarring', but it would dilute the unique affect that perfumes this piece. It would, for this reason, sound peculiar in the context. However, if used consciously, such a simple comma alteration can have dramatic, modulatory effects, similar to that of modulations between key signatures in ET. We will look closer at this phenomenon when discussing the loose style below.

The very distinct pattern of the gamut leads to one of the characteristics of strict style pieces—a clearly defined underlying affective, atmospheric field. A strict style piece will often have a clearly perceivable rasa or Stimmung. It would, however, be misleading to say that this rasa is 'built into' the gamut; every piece written for the same gamut will not have the same rasa. Just as with two other genres of music that are valued for their modal and affective qualities, Renaissance polyphony and Hindustani classical music, affects and rasas evolve through the particular movement of tones. In Renaissance music, the affects were not inherent in the different scales per se (Dorian, Mixolydian, etc), but had to be constructed by, among other things, restricting melodic outlines and leaps to designated scale degrees. Similarly, in North Indian Hindustani music, which is based on rational intervals and thus like JI has distinguishing unequal scales, the scales are not what has rasa. Rather, it is how the scale degrees are combined and the particular path that is taken through a scale that gives rise to a rasa (Mahthur et al., 2015). In strict style JI, the rasa is similarly not inherent in the gamut. It is rather the restricted compositional choices it gives to the composer and the way it forces her to relate to the predetermined tunable paths that give rise to music with a clear rasa.

At this point, it is valuable to acknowledge that any piece, whether it is JI or an irrational system of intonation, will have its intonation facilitated by the two aspects that have been highlighted here as important features of the strict style. Firstly, since the pitches are few in number and repeated, the musician will quite readily learn and remember the position of each pitch. The repetition facilitates the formation of mental representations in the musician’s mind of the gamut in use. This allows the performer to 'tune' in a more contextual manner 'to the gamut' rather than in an exclusively local manner to immediately preceding pitches. Many studies have suggested that the firm establishment of a pattern of interrelationships (Krumhansl, 1979) and the frequent repetition of pitches (Deutsch, 1972, 1982; Dewar, Cuddy, & Mewhort, 1977) have this effect. Secondly, the unique qualia of scale degrees further facilitate intonation; one can even speak about the musician 'tuning' to the affect associated with each pitch. The intervals do not need to be tuned to low-number ratios for these benefits to happen. When I was studying Sundanese vocal music, reliance upon these two aspects was crucial for tuneful singing even though the intervals were far from being in JI. What the addition of JI contributes to the equation is to, on the one hand, introduce intervals that because they are possible to tune into composite, resting wholes, have clearly defined identities that are easier to learn than irrational intervals; asking a musician to learn a 7/6 (267 cents wide) is 'less of an ask' than asking her to learn how to play a narrow third of 275 cents. On the other hand, by being able to bring about the modal field effect, or emergent tunability, described in the previous section—where pitches support the intonation of other pitches by reinforcing their intonational tendencies—tunability seeps out from the harmonic sounds and greatly helps the musicians place the notes in their correct position (in this case within a tunable modality). We saw an example of this in Figure 4. Having the gamut be in JI thus greatly facilitates its accurate performance. This is echoing Ben Johnston’s argument that justly tuned intervals simply are more intelligible than irrational intervals; using just intervals ("ratio-scale thinking") is efficient because of their clarity and role within an interconnected harmonic space: 

Interval scale thinking emphasizes symmetry of design. The harmonic and tonal meaning of symmetrical pitch structures is ambiguity. Chordally they produce either a sense of multiple root possibilities or of no satisfactory root possibility. Tonally they cause either a sense of several possible tonics or of no adequate tonic. Ratio-scale thinking, on the contrary, emphasizes a hierarchical subordination of details to the whole or to common reference points. The harmonic and tonal meaning of proportional pitch structures is clarity and a sense of direction. (1964, 28)

It should now be clear to us why the strict style is the style of JI where the composer can be the most careless when constructing tunable paths. Figure 2, which was described as having an 'imperfect' balance between tunable and untunable intervals, could work moderately well in a strict style piece due to the simple reason that the strong modality makes the experience of playing this music slightly less a practice of tuning each pitch individually and more a practice of relating them to a unified modality and an emergent tunability. This, however, does not by any means mean that an interval like 81/64, the 'Pythagorean major third', automatically becomes tunable in the strict style. The balance between tunable and untunable intervals still has to be good enough for the music not to become completely untunable, and the musicians should not be asked to dwell on untunable intervals for too long as this will cause the memory of the necessary reference points needed to play complex intervals to fade. The extra carelessness possible in the strict style means that the composer can introduce untunable intervals more directly and with less preparation. We do not perhaps always have to be just as cautious as demonstrated in measure 160 of Figure 7 where the performance of a Pythagorean major sixth (27/16) was thoroughly prepared by being arrived at by directly tunable intervals.

The risk we run by not being as cautious as in Figure 7 is that the complex intervals become 'detached' from their harmonic origins and positions within a tunable matrix. Since these intervals have no tunable identity of their own, they rely on their modal context to be comprehensible in JI. If the context is not providing such support, the untunable intervals will no longer be harmonically relatable to any emergent tunability; they will instead function like scale degrees with particular microtonal inflections. In Johnston’s terminology, we would then have moved from ratio-scale listening to interval-scale listening. In other words, the intervals would no longer sound as being in JI but rather sound like microtonal music that uses JI-type intervals. In a strict style piece, because of the frequent repetition of a small amount of non-tempered scale degrees, such microtonal JI-derived intervals will still be relatively easy to play accurately enough, but we can not say that we still hear them as being in JI. Take for example the Pythagorean third of 81/64. Since it is only a syntonic comma wider than the common and simple 5/4, the 81/64 can very easily be heard as a 'too wide’, out-of-tune variant of it. In integrated JI, all styles considered, it is the case that if a composer wants to use an 81/64 and not make the listener hear this as a 'bad 5/4', she must use skillful melodic and contrapuntal writing that carefully considers the tunable paths and the modal strength in place. Basically, the composer must make the musician/listener 'grasp' or 'understand’, through passive perceptual learning, that an aggregate consisting of a Pythagorean third is not supposed to form a major third 5/4 nor a 9/7. The composer must make the listener enact the 81/64 as two notes that due to their unique and individual melodic trajectories and positions within temporary modal frameworks, produce a ditone rather than a 'third.' This is done by clearly establishing the tuning references to which we are to relate the individual pitches. In other words, great attention to the preparation of an 81/64 must be given.

Writing an 81/64 to be heard as a ditone is especially difficult in the free style where a stylistic context of constant micro-adjustments of pitches by commas means that the 5/4 variant always is a possible alternative. In the strict style, however, where such alterations are not stylistically possible, an ill-prepared 81/64 is not necessarily always heard as an out-of-tune 5/4 even as it is not modally articulated as a ditone either; the fast beating (the 'out-of-tune-ness') that occurs between the composite pitches of such an ill-prepared 81/64 can in fact still be recognized as a quality that gives the piece its particular mood or rasa–similar to the 32/27 we met in Figure 7. In the strict style, the 81/64 can be enacted by the listener as two pitches that due to their unique characteristic positions in a strong gamut form a kind of 'third' that has the particular quality of beating fast. That is, even as 81/64 stops sounding like a ditone (ratio-scale) and starts behaving like a 'wide third' (interval scale), it can still sound correct because of the small, characteristic gamuts used in the strict style. At such moments, however, it is important to recognize that we have left integrated JI behind and entered the realm of microtonal music that uses JI-type intervals, despite the fact that we perceive the 81/64 as a 'correct' sound. 

Additional caution should also be noted for music reminiscent of Western tonal music. For this music, the strict style is very limited. The Western tonal style of composition requires far more intonational flexibility to account for all the different possible chord formations. It is no coincidence that the rise of tonal music happened simultaneously with the rise of keyboard temperaments; a modal system based on ratios simply cannot contain all the commas needed to play tonal music. Because of our stylistic expectations of tonal music, transposing 'tonal music' into a strict style gamut of the kind we met in Figure 7 often makes the music sound out-of-tune. If the music is too chordal and reminiscent of tonal functional music, the “81” in a 64/81/96-formation easily starts sounding like an out-of-tune third and is not saved by any strong modality, nor its affect.

Figure 8

An example of a strict style usage of the 81/64 in a piece for intonating instruments can be found in Vid stenmuren blir tanken blomma for violin and viola. The quasi-strict style of the last section of the piece creates a musical context in which the Pythagorean major third between C and E in measure 543, seen in Figure 8, does not sound too wide but precisely right with regards to the mode and its affect. Adjusting this interval to 5/4 sounds contextually incorrect. When musicians accidentally do this in rehearsals, it is immediately recognizable as sounding wrong, despite them adjusting a non-tunable interval to a tunable. Although the pitches in measure 543 in the score look as if they form a conventional major triad in second inversion (G C E), the music, if performed accurately, does not sound chordal enough for this E to sound better a syntonic comma down. The modal context is rather one in which this " 'C major' chord with a 'high' E "sounds appropriate.

It can furthermore be noted how careful orchestration can provide intonational support in passages such as these. Open strings and the harmonics of these are here used extensively to facilitate a predominantly Pythagorean intonation; only pitches found as such and their octave transpositions are used. The entire tonal world of this piece is derived from the lowest five natural harmonics of the open strings. In the violin dyad in measure 543, the E is the second harmonic of the violin's first string, and in m. 541-542 it sounds together with the second harmonic of the violin's second string (A) to form a 3:2. The pitch C is ultimately derived from the viola's fourth string, but appears here as a stopped note to form a 4:3 to the viola's G–performed as the first harmonic on the third string. The aural effect of simultaneously hearing the pitch that is a 3:2 above A and the pitch that is a 4:3 above G is a 'ditone'—not an out-of-tune 5/4. That the E is added last and played as a harmonic facilitates its harmonics. The violinist tunes the C to the preceding and simultaneous G as a 3/2 and then adds the E. Of course, playing the E as a harmonic does not guarantee that it will be in tune as the intonation of harmonics can be adjusted. But harmonics are helpful in rehearsals in establishing these sounds in the musicians’ ears.

Two types of strict styles

As mentioned above, the strict style is most commonly found in pieces for fixed-pitch instruments. The craft of writing for fixed-pitch instruments is different from writing for intonating instruments in several ways. Firstly, because of the clarity and exactness of the intonation on fixed-pitch instruments, we can write passages with high intonational complexity–passages that would simply be too difficult for intonating performers to execute clearly–and hear these with accurate intonation. Secondly, the craft of writing for fixed-pitched instruments is different not only because the composer is guaranteed exact intonation but also because the way we listen to fixed-pitch instruments is different. Part of the reason for this has to do with musical convention: the typical listener is accustomed to hearing out-of-tune intervals and irrational temperaments on keyboard instruments and metallophones, but this is not the entire reason. The listener to music performed on pre-tuned instruments also hears that there is nothing the musician can do to adjust the pitch; the intonation sounds 'automatic' as no tuning indications, such as micro-adjustments or pitch fluctuations, are present in the sound. The intonation is, therefore, not evaluated in the same way as when played by an intonating musician. Unskillful usages of untunable intervals (such as blatant Pythagorean sixths and thirds) that would sound out-of-tune if played on intonating instruments rarely appear as audible 'problems' here; a 128/81 is, for example, much less likely to sound like an out-of-tune 8/5 if played on a fixed-pitch instrument than if performed by intonating musicians.

Because of these two reasons, the strict style pieces that are written for fixed-pitch instruments often have a distinctly different character from the strict style pieces for intonating musicians. To account for this instrumental idiom, the strict style is in this text subdivided into the integrated strict style—paradigmatically written for intonating musicians and that takes the limitations of intonating musicians in mind when constructing the tunable paths (an example of this was Vårbris, porslinsvas)—and strict style as a 'tuning', paradigmatically written for fixed-pitch instruments. 

Strict style as a 'tuning' and its two sub-categories

Simply transcribing the music of the integrated strict style-piece Vårbris, porslinsvas to a fixed-pitch instrument, as done in the fourth movement of the kacapi piece Andra Segel, does not produce a piece that uses strict style as a 'tuning' because the transcription still has all the tunable paths that make it possible to play the pitch sequence of this piece on an intonating instrument. What is intended with this category strict style as a 'tuning' is rather the pieces where a composer tunes an instrument to tunable ratios and writes music for this tuning while not thinking about the tunable intervals and paths in the same way as she would have needed to have done if she were writing a piece for intonating musicians. Since the accurate tuning will happen by itself, there is no need to consider the tunability of the 'tuning' as carefully. It would be a mistake to consider this type of strict style inferior to the integrated strict style. This style opens up many artistic merits that are simply impossible to achieve when having to consider tunable paths and some of the pieces that we today think of as the 'masterpieces' of JI are in this idiom. 

Within strict style as a 'tuning', an enormous spectrum of compositional practices and types of music is accommodated. Many, but not all, pieces written for precisely tuned instruments fit somewhere on the spectrum of this style. At one end of this spectrum, we have pieces like La Monte Young’s The Well-Tuned Piano and Michael Harrison’s Revelation. I call this end of the spectrum the informed style of strict style as a ‘tuning' because the pitches of these pieces arise in tunable contexts. Not all intervals of these pieces may be directly tunable, but the pitches are experienced to arise out of a latent, justly tuned modal field. These pieces therefore embody and reveal a deep understanding of their tunings. 

At the other end of the spectrum, we have pieces like Julia Wolfe’s STEAM, written for an ensemble consisting of Harry Partch’s JI instruments. The pitches of this piece do, contrary to the work of Young and Harrison, not arise in tunable contexts. What we hear is not so much tunable pitches as microtonally flavored pitches. The piece utilizes the microtonality of these instruments' tunings but does not reveal any insight into the tunable origin of the pitches. Rather than having tunable relations, the pitches are treated like arbitrary microtonal pitch classes. The music treats the justly tuned instruments not as providing intricate webs of tunable paths through overlapping overtone series but rather as found objects. This end of the spectrum will be termed as the rough style of strict style as a ‘tuning'. The piece STEAM, due to an insouciant or 'rough' treatment of the tuning–in the sense of not paying attention to the subtle details of the tunability of pitches–, ends up sounding more like undefined microtonal music than JI music. How an ensemble of instruments tuned exactly in JI can end up not even sounding like JI should at this point in this text not be surprising. It takes more than simply tuning instruments to ratios for the music to actually sound like JI. Above, we saw for example how an 81/64 may start sounding like a 'wide third' rather than a JI ditone due to features in melodic and contrapuntal writing. Earlier in this text, the point was made that when writing music in JI, tunability must be ensured for the musician and listener alike—the list of finite tunable intervals is also the list of finite perceivable JI intervals. In STEAM, maybe we could say that tunability has been ensured for the 'executor' (it happens automatically because of the tuning of instruments) but not for the listener.

In defense of Wolfe’s 'hands-off' approach to the tuning of these JI instruments, it should be mentioned that the music by Partch himself frequently sounds like this as well; he builds instruments based on careful tunings but then composes on the instruments as if they were a given big collection of microtones. He is rarely, it can be argued, considering the tunable paths. In pieces like Delusion of the Fury (1966), Partch draws upon a very large collection of pitches and chooses intervals from these that are complex and far from tunable, often with the intent of approaching the subtle shades of intonation and glissando effects of the human speaking voice. 

Important to note here is that all the examples mentioned above include intervals smaller than 35 cents in their tunings and include pitches separated by commas. It can be recalled that I above wrote that it was rather a feature of free style and loose style pieces to freely use comma modulations and alterations. The Well-Tuned Piano does not, to me, sound like free style as a tuning or loose style as a tuning despite having small intervals. When hearing this piece or Harrison’s Revelation, it sounds like I am listening to a single, fixed tuning system; not to an intonation that is 'freely' moving. Maybe it is the fixed nature of the pitches and the limited gamuts that cause this, but it also has to do with compositional treatment. In the case of The Well-Tuned Piano, I believe this has to do with the way the pitches are harmonically articulated in the piece, where 1/1 and 63/32 are clearly separated as distinct nodes in a harmonic net. In pieces like Delusion of the Fury, however, it might make more sense to actually describe this music as something like in the rough style of free style as a 'tuning' due to a large number of pitches and the way they are spread out over multiple instruments. The reason it is rough is because it does not take into consideration tunable paths. If Partch had written music more informed by tunable paths but still using the same amount of pitches, the music would have been in the informed style of free style as a 'tuning'. What this shows is that pieces written for fixed-pitch instruments tuned to JI necessarily need not sound like they are in the strict style.

The rough style of strict style as a 'tuning'

When writing for intonating instruments, the sounding result of this kind of compositional approach that does not consider tunability and merely uses ratios as a collection of pitches frequently ends up sounding like microtonal music that uses JI-type intervals. Writing for fixed-pitch instruments, like keyboards or zithers, however, often produces an audible result slightly different from such microtonal music that uses JI-type intervals. When using the same kind of compositional approach but using instruments that are already tuned, the resulting music often manages to retain the quality of 'being in JI' even when the music only has incidental tunable paths and no directly tunable intervals for long periods of time, although STEAM successfully defies this rule largely due to the compositional focus on clusters, small intervals, and the complete avoidance of tunable intervals. In other pieces less extreme in their denial of tunability than STEAM, the power of the precise, exact tuning, often 'prevails' in a way that in a subtle but important way separates this music from microtonal music that uses JI-type intervals. The exact tuning perfumes the music with the 'sound of JI' in a way that causes the music in Figures 9 and 11 to still sound like JI; something that would not happen–or at least not happen as naturally and effortlessly–if this music were to be performed by intonating musicians. It is for this reason that there is a difference between microtonal music that uses JI-type intervals and the rough style of strict style as a 'tuning'. 

Figure 9

We should not be too quick to dismiss the 'rough' approach of Partch and Wolfe as simply being worse because it is 'less informed' about tunable paths and intervals. I want to argue that both ends of the spectrum outlined above have merits and artistic qualities. In my own compositions, I have explored both ends of the spectrum. In the piece Stenskrift for piano, I used a 5-limit JI tuning in this rougher or 'uninformed' way to achieve a certain effect. Listening to this piece, it is true that we not only hear a complete disregard for tunable paths and a disregard for the tunability of simultaneous sounds, but also a disregard for the modal functions and modal origins of pitches within this tuning (what this means exactly will become more clear below). But I would argue that in the piece (as in many other pieces that uses this approach), this disregard is not audible as a 'problem' because the music is not constituted of modal harmony and contrapuntal melodies but is derived from aggregates and complex, often chromatic, sounding-blocks (see Figure 9). In this sense, Stenskrift uses a very similar approach to STEAM. Due to their mixture of tunable and untunable intervals, the aggregates in Stenskrift express different degrees of stillness and activity. It is the oscillation between 'active' aggregates (which with their predominantly Pythagorean thirds and sixths are fast-beating) and resting aggregates (which with their Ptolemaic thirds and sixths embody stillness) that gives the music its breathing pattern and constitutes its syntax. It almost sounds to me as if the composer found the piano already tuned in this way and then allowed the points of rest and tension discovered within this tuning to dictate the flow of the music. Therefore, the phrasing of the music and the tuning speak in a single voice with the same goal in mind despite the seeming lack of insight into the actual tuning. 

This rough treatment of JI is not without its qualities. Certain kinds of chord relationships, sharp modulations, and chromaticisms that would be impossible to conceive when thinking about tunable paths become possible with this approach. This unpredictable, thorny, and rough style is itself a poetic quality and can be highly essential for many pieces. 

Figure 10 is another example from a strict style piece, Ribuan Matahari for kacapi and violin, that is leaning towards the rough style. In this piece, most intervals completely lose their harmonic, justly tuned functions and start working solely like microtonal intervals. Even though we can trace some kind of tunable path between all of the pitches in the gamut of this piece, the tunable intervals are never clearly enough articulated for the listener. The minor third between the first two notes in the violin is a 6/5, and in measure 7, the kacapi bass line plays a tunable minor ninth (11/5). In practice, however, it is its interval sizes rather than the composite periodicities of these intervals that matter here. In this piece, the pitches lose their function as tunable intervals and start functioning almost solely as scalar pitches.

Figure 10

The informed style of strict style as a 'tuning'

In Stenskrift, many intervals are complex and their pitches are notated far apart when visualized in a ratio lattice. Some of the pitches are so far from each other that we no longer perceive the resulting intervals as being in JI. The interval between Bb and F#- is 405.87 cents–1.95 cents out of tune from 81:64–an untunable interval that we in the discussion above saw needed a very supportive context to not sound out-of-tune. Many of the piece's intervals are so irrational that they, in this way, could be reinterpretable as simpler–yet often untunable–ratios that are slightly out-of-tune, but precisely because Stenskrift, unlike Vid stenmuren blir tanken blomma, does not provide the context to make sense of the simpler untunable ratios, a musical situation is created where many intervals only barely sound like JI. Between F and G#- is, for example, an interval with the size of 296.09 cents, which is just 1.42 lower than the simpler, yet untunable, ratio of 19:16. The interval between G#- and Eb is another example, but this interval can be enacted as a straight-forward tunable interval: as a slightly out-of-tune 3:2. As we will see below, the usage of these kinds of 'enharmonic equivalents' can be the basis for a harmonically ambiguous informed style of JI. Their sporadic and accidental appearances in Stenskrift (such as in m. 40) are, however, not enough for that.

All these examples reveal that in Stenskrift, many pitches do not sound in a way that corresponds with their rational notation; either they sound not like JI at all, or they are so irrational that they become possible to retranslate as other ratios, something that in this piece probably only is true for the false fourths and fifths between G#- and Eb. In contrast to this, a characteristic feature of many informed style pieces is that the pitches' original modal functions are audibly retained even when they are highly complex–even when they are untunable. In this style, the rational notation accords with how it sounds. Robert Hasegawa's (2008) discussion about the complex intervals 256/147, 147/128, 49/32, and 63/32 in La Monte Young’s informed style piece The Well-Tuned Piano is illuminating in this regard. Hasegawa writes that if we hear these intervals outside the context of the piece, we might interpret them as out-of-tune versions of simpler ratios, such as 7/4, 8/7, 3/2, and 2/1. He writes that when we hear the 256/147  

"with no other sonic information, then we’re most likely to understand it as representing the 7/4 septimal minor seventh—or even an out-of-tune minor seventh from a more standard five-limit just tuning (16/9, or 966 cents) for example, even further out-of-tune" (2008, 45). 

This is because there is "a perceptual bias toward simple intervals, since very complex ratios are likely to be heard as mistuned versions of simpler intervals" (Hasegawa, 2008, 74). The 147/128 is indeed only a gamelisma (1029/1024) of 8.4 cents wider than an 8/7. This relatively large comma can be arrived at by stacking the two smaller commas–often used for creating enharmonic equivalents–441/440 (3.93 cents) and 385/384 (4.5 cents). In The Well-Tuned Piano, however, the 147/128 is not heard as an 8/7 but precisely as a 147/128. What is so remarkable about Young’s composition is that 

"the simple intervallic building blocks in Young’s tuning can combine to lead us securely into very distant harmonic territory—we have to negotiate two 7/4 intervals and one 3/2 to reach the F↓ from the D↑; but if these intervals are introduced carefully, it may be possible to follow the path on the harmonic lattice from 1 up to 147 through the intermediate points." (2008, 45)

Hasegawa concludes that "Young offers enough parallels between the new pitches and those we’ve already heard to allow the ear to make these esoteric connections. The complex intervals are convincingly broken down into simpler steps of 3/2 and 7/4" (2008, 48). The informed style of strict style as a 'tuning' is often characterized exactly by this quality that Hasegawa finds in Young's piece; the music allows the ear to follow along with the harmonic steps and reach a comprehension of intervals that we under other circumstances might have re-interpreted as simpler ratios. It is not literally the case that the ear traces steps, but the harmonic field of the piece helps the listener to enact complex intervals that under other conditions would be collapsed to out-of-tune versions of simpler intervals. This line of reasoning should sound familiar to how we discussed the usage of 81/64 in Vid stenmuren blir tanken blomma. There, we saw that the composer must guide the listener’s ear into recognizing this interval as a ditone rather than as a third for it to still be comprehensible as JI. The composer must, as Hasegawa writes of Young, "lead us securely into very distant harmonic territory".

Figure 11

Figure 11 shows an excerpt from the second movement of Andra Segel for the kacapi siter. This music is representative of the informed style of strict style as a 'tuning'. All intervals have only a few steps between them in a ratio lattice. The two intervals furthest apart are "8/7" and "35/32". The easy tunable steps between them are only four in number, 7/4+7/4+5/4+3/2. A compact harmonic space where all pitches are reached by a few tunable steps means that the harmonic relationships of the tones in the gamut are easy to articulate clearly. “35/32”, for example, is related to “8/7” as the seventh’s seventh's major third, and this relationship is (passively and non-conceptually) 'graspable' in the music.

What makes this movement from Andra Segel and The Well-Tuned Piano characteristic of strict style as a 'tuning' in a way that distinguishes them from integrated strict style pieces is the high complexity of the music that makes it too difficult to transcribe for an intonating ensemble. Although the music looks relatively tunable on paper when tracing tunable paths, strict style as a 'tuning' pieces are often impossible to play on anything other than a fixed-pitch instrument due to the complexities that arise from the application of instrumental idioms. For example, the first three notes in Figure 11 are created by stacking two 7/4 intervals. Although the 7/4s themselves are easy to tune, the frame interval between the first and third pitch is the difficult interval 49/16. As this interval is only a sixth tone higher than a perfect fifth (plus an octave), considerable beating and a distracting gravitational pull toward an implicit "3:1" can (if the pitches are sustained) make the performance of it for an intonating musician difficult despite the simplicity of the 7/4 building blocks. Later in this excerpt, the interval between the fourth and fifth pitch is a common minor sixth 8/5, but the interval between the third and fifth pitch will be an untunable tritone of 626 cents, 8 cents off from a 10/7, or 10 cents away from a 13/9, again causing considerable beatings (if the pitches are sustained) that could disturb the performance of the simple 8/5. When performed on a fixed-pitch instrument, however, these kinds of passages are completely clear and unproblematic for both the performer as well as the listener. 

It is important to mention that the music in Figure 11 appears towards the very end of this movement and has been preceded by a slow unfolding where pitches have been gradually added to the harmonic space. Formal features such as these have an important impact on the perception of pitches. A slow, gradual establishment of the modality is what enables aspects of these faster passages to be intelligible. Because of the clearly established modality at this point, the third pitch retains its identity in the harmonic space despite forming an untunable 294/128 (a 147/128 with an octave in between) to the preceding note in the melody. As explored above, this interval is just 8.4 cents lower than the also untunable 16/7. The pitch also forms an untunable 21/16 to the phrase's second pitch. When played in completely accurate tuning and after the modality has been clearly established, however, the ear clearly grasps the fourth pitch as a 4/3 to the phrase’s first pitch and thus as forming a 294/128 with the third pitch. Because we do not hear the fourth pitch either as an out-of-tune 16/7, as an out-of-tune 9/4 to the phrase's third pitch, or as an out-of-tune 4/3 to the phrase's second pitch, we say that the piece is in the informed style of strict style as a 'tuning'.

The tolerance range of each interval is, as Hasegawa explains, "inversely related to the complexity of a ratio" (2008, 74). This means that simpler intervals allow for greater mistuning while still retaining their identity, while more complex intervals need precise tuning to be recognized. In other words, an interval like 147/128 has considerably less margin of error than what an 8/7 has and thus requires more precise tuning. When writing for fixed-pitch instruments, the composer can rely on this precision and therefore take greater liberties in her melodic writing. The strict style as a 'tuning' allows the composer to work with the category of intervals that are very sensitive to mistuning yet are still audible "as themselves" (and not mistuned simpler intervals) if tuned with great precision. The precise tuning is, however, not enough; the interval also needs to arise in a musical context where the composer is guiding the listener’s ear "securely" to comprehend the complex ratios. The musical context must support the perception of a 147/128 as a 147/128 and not as an out-of-tune 8/7. 

What the process of composing such support looks like can be seen by comparing Figures 12 and 14 from the unfinished and finished versions of the same passage from the keyboard piece Nattviol, nattviol. This piece might be described as a piece that has the ambition of being in the informed style of strict style as a 'tuning' but that often lapses into a rough style. One of the primary aspects that cause this lapse is the chromatic idiom that characterizes the piece throughout. It is precisely due to the larger (12-tone chromatic) gamut involved that it becomes difficult for the mind to make sense of the untunable intervals and actualize them as JI-intervals in this piece (in the way that we did with the 294/128 in Andra Segel or the 81/64 in Vid stenmuren blir tanken blomma). Due to the chromatic idiom, no 'strong modality' arises to guide the enaction of untunable intervals as tunable. Twelve chromatic pitches are perhaps too many to form a unified modality. In Figure 12, an early version of the very ending of Nattviol, nattviol is shown.

Figure 12

Figure 13

At the point in the piece from which this excerpt is taken, we have been exposed to this tuning for fifteen minutes and we have heard the tuning articulated through both tunable paths as well as through untunable leaps. All the pitches in the gamut given in Figure 13 have been adequately articulated. Yet, when the "9/7" (the C# raised by a septimal comma) appears in measure 259, it sounds out-of-tune to me: perhaps I was expecting a 9/2 in its place. Even though the ear many times throughout the piece has heard that the pitch-class "9/7" originates as the seventh of the "9/8", we still cannot at this point accept this untunable inversion without it sounding 'out-of-tune' or 'awkward'. This reveals how this music is much more 'sensitive' to the usage of untunable intervals than what Andra Segel is, where we more easily accepted that such inversions resulted in untunable intervals without them sounding 'wrong' to the ear. It also reveals that this passage is best analyzed as being in the informed style. Although other passages of this piece are best analyzed as rough, the mere fact that the ear does not want to hear this interval merely as a microtonally inflected scale step but wants to make harmonic sense of it implies that the conditions for the informed style have been established.

The solution to this problem of the out-of-tune sounding "9/7" was to add another B ("9/8") in the octave above the "9/7", and have this pitch be articulated slightly later than the low B ("9/8") by way of arpeggiation rather than simultaneous attack, and extend this added B pitch over the "9/7". The final score is given in Figure 14. In this final version, the ear has enough help to accept the C# as a "9/7" articulated as a 16/7 to the low B, and not just an out-of-tune ninth (9/4). The difference between Figure 12 and Figure 14 shows how a composer needs to work to create music that sounds in the informed style of strict style as a 'tuning'. Without this intervention, Figure 12 sounds more like the 'rough style'. It sounds more 'microtonal' than it sounds JI, even though the instrument in both cases is tuned completely by using tunable intervals.

Figure 14

It is important here to clarify that saying that the musical context supports the perception of (the pitch-class) "9/7" as "itself" and not an out-of-tune (pitch-class) "81/64" does not by any means entail that "9/7" is articulated as "the 9th partial of a harmonic series"; it simply means that the music affords the sonority in measure 258 to not be 'misinterpreted' as an (out-of-tune) 9/4. This is what the difference between the 'rough' and 'informed' styles means. The "9/7" does not imply a harmonic space in which this interval is the 9th partial of a harmonic series. The harmonic space that this piece opens up is unrelated to the harmonic series. In fact, this is true of most intervals in most pieces. Only very rarely does a rational interval imply the harmonic series in which its ratio seemingly belongs. A "5/4" does not by default sound like the simultaneous performance of a 5th and 4th pitch in a harmonic series and therefore somehow "imply" this harmonic series as a type of "latent" harmony. Take for example the music in Figure 10. To argue that the 11/5 between the kacapi strings ".3" (D) and ".3-" (E flat raised by 5-limit and 11-limit commas) would sound to actualize the harmonic series that can be built around these intervals is incorrect. The music has a 'harmonic space', but it is a different one from the harmonic series in which these two pitches can be analyzed as belonging.

Moving between rough and informed idioms

As is the case with all the sub-categories of ratio-based composition outlined in this text, these rough and informed styles of strict style as a 'tuning' represent two ideal ends of a continuous spectrum. Many pieces move seamlessly back and forth between these ends. Nattviol, nattviol is an example of a piece that sometimes sounds informed and sometimes sounds rough. When composing the clavichord suite I Sommarluft, I wanted to use the differences between the rough and informed styles as a way to give the piece its form. The form was therefore designed so that the movements of the piece would constitute an oscillation between the rough and informed styles.  

The tuning of I Sommarluft is presented in Figure 15. From this gamut of twelve chromatic pitches, each of the six movements focuses on a different traditional Western diatonic subset: C minor, D major, F major, etc. Some of these scales, like A major [“1/1”, “9/8”, “5/4”, “4/3”, “3/2”, “5/3”,“15/8”], are highly tunable (indeed, this A major set is the same gamut showed in Figure 6 from Vårbris, porslinsvas). Other scales end up not sounding much like JI at all. An example of such a scale is the scale used in the opening movement. This scale–[“32/27”,“4/3”, “45/32”, “128/81”, “16/9”, “15/8”, “135/128”]–is in effect a Pythagorean Aeolian scale based on “32/27” but where the minor third, minor sixth, and minor seventh are all a schisma (32805/32768, 1.95 cents) too wide. When hearing this key in the opening of the suite, there is very little information reaching the audience that they are listening to music in Just Intonation. No intervals except for octaves, fifths, and fourths fuse in periodic entities. The tuning sounds like a 'temperament' even though no pitches are tempered.

Similar to in Stenskrift, we might gather from listening to the piece that the composer took into consideration the characteristics of the 'temperament' (i.e. which intervals rest and which are in unrest) when writing the piece as nothing strikes the ear as jarring; the phrases and melodies seem to speak in a single voice together with tuning. Apart from this, however, it is not possible to hear any intimate engagement with JI as most pitches lack tunability. The first movement of I Sommarluft therefore clearly fits the rough style of strict style as a 'tuning'. 

Figure 15

The second movement continues in the same way with another untunable scale, F major. The third movement, In Nomine, also begins with mainly untunable chords and intervals; it gives special prominence to the Pythagorean minor third between D ("4/3") and B ("9/8"). A few minutes into the piece, however, the inclusion of tunable 5-limit pitches ("5/4" and "5/3") transforms the music to start to sound like the informed style of strict style as a 'tuning', or even very close to true integrated JI music in D major ["4/3", "3/2", "5/3", "16/9", "1/1", "9/8", "5/4"]. The D major scale differs from the A major scale of Vårbris, porslinsvas  (Figure 6) in that the sixth degree is a Pythagorean sixth, whereas A major had a just sixth. Many passages from this section of In Nomine could actually be performed by intonating musicians but the carefree usage of the sixth degree, B, still reveals that this music was written for a keyboard. An example of this kind of carefreeness can be seen in the simultaneous attack of the Pythagorean minor third B/D directly following an F#- (wolf-fifth to B) in measure 84 in Figure 16.

Figure 16

One idea behind I Sommarluft was to explore the sounds and affects of the scales that do not sound like JI (e.g. the first movement's C minor). These were juxtaposed with the scales that clearly sound like JI (e.g. the fifth's movement's E major), and the piece was shaped as a process of going in and out of JI by working with the scales in between (e.g., the third movement's D major). In other words, the artistic idea was to shape the piece as a transition between the rough and informed styles of strict style as a 'tuning'. This is an artistic idea that would be impossible to realize with intonating instruments. It is an idea that is utilizing the limitations of the keyboard instruments and the fact that it anyway is impossible to tune all different common Western key signatures to 'sound good' with only twelve keys to an octave.

One of the subtle merits of I Sommarluft is how the music can be said to have two primary tonalities. On the one hand, the tonality is C minor in which the piece begins and ends; when the last movement returns to C minor, it feels like a return to the home tonality. On the other hand, the tuning itself can be said to gravitate towards E major and A major because this is where the instrument is the most resonant and where the most pitches fuse and enforce strong modal centers. We might describe this as the root or tonality inherent in the tuning system itself. The composer can explore these two opposing centers of gravity; on the one hand, there is the gravity set by the musical context (the root of the key established by the composer), and, on the other hand, there is the gravity of the intonation system itself, which leans towards tunability and resonance.

Summary of the strict styles

Before moving on to the free style, we can summarize what has been said about the strict style. We have seen how music in this style is characterized by the usage of small, fixed gamuts. Because of this, the music usually embodies strong affective qualities (like moods, rasas, or Stimmungs). The usage of small gamuts facilitates intonational accuracy, and the composer can therefore be more careless or 'free' in terms of melodic and contrapuntal writing compared to the two other styles we will examine below.

The strict style was subdivided into two sub-categories: strict style as integrated JI, usually written for intonating instruments and where the music is constructed as a tunable path, and strict style as a 'tuning', usually written for fixed-pitch instruments and where the composer does not consider tunable paths to the same degree. Within the latter sub-category, we can find two different approaches. 

First, there is an informed approach to strict style as a 'tuning'. When this approach is taken, the music is composed in a way so that pitches still audibly retain their harmonic, relational articulation within a tight, tunable matrix. This approach differs from integrated JI in the higher levels of complexity and freedom in harmonic and melodic writing that the automatic intonation of fixed-pitch instruments affords. The informed approach manages to still construct the music as a tunable path for the listener even though it might be too difficult for musicians to actually intonate. 

Secondly, there is a rough approach to strict style as a 'tuning'. When taken to the extreme, pitches in the pieces that takes this approach no longer sound as if in JI because the composer neither constructs the series of pitches as tunable paths nor articulates the pitches as belonging to an interconnected, tunable harmonic space. We noted that the rough approach is a similar approach that when applied to intonating instruments leads to microtonal music that uses JI-type intervals. In the rough strict style as a 'tuning', however, the precise, exact tuning facilitates (as the composition unfolds) an emergent tunability that still provides the music with a (sometimes subtle) quality of being in JI. The rough strict style as a 'tuning' still provides the music with the perfume of JI that microtonal music that uses JI-type intervals lacks.  

Pitch classification in free style

Moving on to look at the free style, we can choose to describe its difference from the strict style in a couple of different ways. On the one hand, we can say that the free style is music where the pitches are fluid instead of fixed; a "B" can at any time be either a "16/15", "135/128", "256/243", or "25/24." Such a description is saying that we are changing the nature of the pitches; in the strict style, pitches have fixed intonation, while in the free style, they have flexible or fluid intonation. On the other hand, we could say that we are expanding the gamut of pitches; in strict style, the gamut is small and does usually not include pitches closer to each other than 1/6th of a tone, whereas, in the free style, we are adding a possibly infinite amount of pitches that exists as close to each other as commas, such as 21.5 cents (a syntonic comma).

The second way of conceptualization is in accordance with the compositional styles and methods used by composers such as Partch and Johnston who both drew their pitches from huge ratio lattices. Larry Polansky (2009) described this as a method of multiplicity. But the question we should ask ourselves is if Johnston’s composition method is reflected in the way our minds actualize pitches in his pieces. Do we hear the pitches as drawn from a colossal gamut, or do we rather hear a smaller number of pitches that have fluid intonation (despite this not being the way he conceptualizes pitches in his scores)? This question of how the mind-body 'represents' and 'categorizes' pitches (if we allow ourselves to adapt such a cognitivistic language provisionally) is important to answer if we want to develop a way of talking about how the music sounds regardless of how it is composed. We want our method of analysis to correspond to our aural perception rather than a compositional technique. Our goal should be to employ an analytical model that corresponds as closely as possible with the way our mind actualizes pitches. According to Huron (2006), "how minds represent music has repercussions for what listeners remember, what listeners judge to be similar, and other musically important functions" (73). Describing this kind of representation well is, in other words, crucial if we want to get the phenomenology right. In this context, this means trying to give a phenomenologically correct answer to the question of wether the mind in the listening act judge "9/8" and "10/9" as different pitches that are to be retained separately because they carry very different musical meanings, or if it judges them as intonational variations of the same focal 'pitch class'.

When trying to answer these questions, it is important to remember the point that was already sufficiently argued for above: when listening to music, most of us do not listen actively to pitches per se. Unless doing an exercise in transcribing a melody, we do not strain ourselves to conceptualize and categorize pitches, but we are rather attuned to a musical world that is non-reflective and non-conceptual at the gross level. As soon as we actively start paying attention to the intonation of different pitches, we might start hearing differences where we would not if we were effortlessly listening in an attunemental, 'natural', non-critical mode of listening. Being able to hear certain differences might be important to the musician, but this does not mean that the listener needs to hear them. The musicians' listening is necessarily different from the listeners'. To be able to tell if we, as listeners, hear the shift from "10/9" to "9/8" as a modulatory moment or not requires us to first adopt an easy, effortless, natural way of listening to the music, and from this perspective (and not the perspective of a critical listener focusing on pitches) make an assessment about whether pitches are grouped into focal pitch classes or not. This is not an always easy phenomenological exercise, but a skill that is crucial to master if we are to report back our modes of listening adequately.

An important idea from music cognition to bring into this discussion is Dowling's (1978) suggestion, made by drawing upon Miller's (1956) classic cognitivist idea, that the diatonic scale contains only seven discrete pitch classes because this matches our limited ability to remember and label items reliably along continuous dimensions such as pitch frequency. This type of cognitivist restraint can further be interpreted to imply that the mind’s perceptual and memory systems seek, because of its limited ability, a simplified organization of the sensory information received when listening to, say, the huge collection of pitches in the work of Johnston or Partch. In these piece, the mind would according to this idea categorize the sounds into focal pitch classes. Even though the composer works from a huge gamut of pitches where "10/9" and "9/8" clearly are distinct, they are nonetheless simplified as different tonal shades of the same focal pitch classes in the listening experience. In relation to C as "1/1", "10/9" and "9/8" share a 'D-ness' in the same way as the colors crimson and ruby share a redness. In other words, we do not hear the shift from "10/9" to a "9/8" as a modulation. To hear it as a modulation would mean that we hear it as a shift from one 'scale' step to another, but according to a cognitivistic thesis, these pitches are collapsed into a single scale step.

But what range of pitches would be eligible to share this D-ness? Is also an "8/7" collapsed into the same category as a "10/9"? Sabat has observed in relation to this, and I believe that many musicians will agree with him on this observation, that "intervals smaller than 1/6 of a tone (approximately 35 ¢) begin to take on the character of enharmonic shadings of pitch rather than functioning as distinct tones" (Sabat 2008/2009, 1). Generally speaking, the smaller the interval, the greater the tendency for the pitches that make up the interval to collapse into each other. This rule helps explain the different general usages of the septimal and syntonic commas that we find in the JI repertoire. Pitches separated by septimal commas (64/63, 27.26 cents) have a greater proclivity to not sound like enharmonic equivalents compared to the syntonic comma (81/80, 21.51 cents), even though the syntonic comma is only 5.75 cents more narrow. In my own experience, I would say that to make pitches separated by septimal commas collapse into a single scale step requires quite special compositional circumstances. It is 'easy' to make them actualized as separate pitches; it is easy to use the movement between "9/8" and "8/7" as a melody–albeit very 'microtonal' sounding. Syntonic commas, however, are instead 'easy' to actualize as enharmonic variations and require very special circumstances to separate (we will see examples of such special circumstances below in the music of Sabat and Lamb). It is not as easy to articulate the melodic movement between "9/8" and "10/9" without it sounding like 'the same pitch being re-intonated'. Both the differences between 9/8 and 10/9, and 9/8 and 8/7 imply new harmonic regions, yet one is more distinguished as a melody than the other. Maybe the answer is that 5.75 cents provide enough of a difference for the tendency to 'distinguish' to flip over into 'fusing'. 

If we accept Sabat's threshold of around 35 cents, pitches separated by the comma "49/48" should function as distinct tones. This interval is just above 35 cents at circa 35.7 cents. This is the interval found between the scale steps “7/4” and “12/7” in the opening of Av dagg och fattigdom and the third movement of Andra Segel. From these compositions, I would say that this is the case: the melody moving from "7/4" to "12/7" is a very clear melodic movement involving two separate pitches. In other compositions, such as Mellan bleka stränder (efter Ni Zan), I have used the slightly smaller "56/55" found between a "7/4" and "55/32". As this "56/55" is 31.2 cents wide, it falls below the postulated 35 ¢ threshold. And indeed, it is very interesting to note how in this piece, when compared to Av dagg och fattigdom, the two pitches making up the 56/55 behave slightly more like 'shadings' of each other, creating more of a hazy and fuzzy enharmonic effect rather than clearly functioning like two separate pitches. While the pitches making up the 49/48 in Av dagg och fattigdom were clearly separated tones, the pitches making up the "56/55" in Mellan bleka stränder (efter Ni Zan) has more of a shared 'fused' identity. There seems, like Sabat postulates, to be a very crucial threshold of just around 35 cents.

But to what extent pitches that are separated by a "56/55" fuse is a matter of context. In the harmonic context of the piece Om dagen stilla, the same 56/55 of 31.2 cents is used but sounds in this piece more like separate pitches than was the case in Mellan bleka stränder (efter Ni Zan). My intuition tells me that this has to do with the more fleshed-out harmonic space that operates in Om dagen stilla. While Mellan bleka stränder uses a quite small harmonic material that is very scalar, Om dagen stilla has a more fully articulated harmonic space that is more spectral in orientation. This allows for the two different pitches that make up the 56/55 to more easily be separated since they are clearly associated with different spectral sub-sections–with different harmonic tunable paths. In Om dagen stilla, the 56/55 interval is found between the pitches called "27/22" and "135/112" in the matrix in Figure 17. Both of these pitches have many tunable paths to other pitches. "135/112" clearly functions as a 5th partial in a spectral sub-section. Both the root as well as the fifth, ninth, and seventh related to this 5th partial are present in the scale. "27/22" is in a similar way scaffolded by many tunable intervals–it is the 7th partial in a spectral sub-section where also the 11th, 9th, and 3rd partials are represented. 


Figure 17. Matrix of Om dagen stilla

In Mellan bleka stränder (efter Ni Zan), the 56/55 is found between "55/32" and "7/4". As can be seen by looking at the matrix, while the "7/4" is well connected to many other points in the harmonic space, the "55/32" (as well as the "165/128") seem to float outside of it with only one tunable connection. Because of this, the "7/4" comes across as the main pitch, while "55/32" becomes a microtonal variation of it. The more fleshed-out harmonic space in Om dagen stilla makes the difference between the pitches making up the "56/55" in that piece more harmonically meaningful–the pitches are harmonically important in their own right, and it is this harmonic intelligibility, my thesis here goes, that makes it easier for us to distinguish them as separate pitches in that piece.

Figure 18. Matrix of Mellan bleka stränder (efter Ni Zan)

On the other side of the spectrum, there are pieces that use 'enharmonic equivalents'—intervals below 5 cents, such as 441/440 of 3.93 or the schisma of  1.61 cents. This kind of repertoire illuminates another threshold: where intervals are so narrow that they do not even sound like the kind of 'enharmonic shadings' that Sabat referred to with intervals below approximately 35 ¢. When using these narrow enharmonic equivalents that are smaller than 5 cents and no pitches separated by commas, the music starts to sound like the strict style rather than free or loose. But if the commas become bigger than 5 cents, they start to sound like different 'shades' again rather than collapsing completely. The distance of 7.71 cents between A# - - ("225/128") and Bb ↓ ("7/4") is, for example, too big to sound like an enharmonic equivalent. Using these two pitches rather creates a loose style context where these two pitches appear as different shades. It seems like the narrow band between 5 and 25 cents is especially crucial for enharmonic shadings. Smaller intervals than this are too small to sound like a 'variation' whereas larger intervals can start acquiring too much of an identity of their own.

Combining these observations with the above cognitivistic idea about the difficulty in keeping more than seven pitches 'present' at the same time might lead us to want to at least as a hypothesis say that in many free style styles, the large number of frequencies separated by commas smaller than 35 cents overwhelms the perception and gives rise to a simplified organization of pitches into focal pitch classes; sounds that are less than approximately 35 cents apart become different 'nuances' of the same tone. Unfortunately, it will not be that easy to postulate that kind of rule, and this is because our perceptual orientation toward sound is not only guided by cognitivistic restraints but also by a search for meaning. The 35-cent threshold is fluid and impacted by the harmonic contexts and particular modes of listening that different pieces of music afford. Some harmonic contexts and modes of listening will allow for a proclivity of 'fusion' while other harmonic contexts will easier allow for separation. We can even draw to mind how in the realm of Equal Tempered music, composers such as Takemitsu have even achieved to make intervals as large as quarter tones sound like enharmonic variations of each other.

Before giving some examples where the organization of comma-distanced pitches in focal pitch classes does not happen, I will first give an example of a piece where I think that it happens. This is a piece of mine coincidentally named Marc (Sabat). This piece uses fifteen pitches per octave as shown in Figure 19. Despite clearly being different pitches in the composition process, these fifteen are in performance heard as 9 'fluid' pitches (A, B, C, D, D#, E, F#, G, G#). This does not mean an endorsement of cognitivism, but simply reflects how I perceive the music. The fact that even I, who composed the music and know about all the differences, intuit the music in this way suggests that other people likely hear it this simplified way as well. This certainly does not mean that a piece like Marc (Sabat) could equally well be played in ET, where the proposed simplified organization of sense data is literally translated into a simplified, tempered, system of pitch classes. The different intonations carry important meanings and psychological cues that are very important to play precisely, but this does not translate into perceiving the pitches distanced by commas categorically as different. Below, at Figure 25, I will return to an analysis of this piece.

Figure 19

The 5-limit free style used in a piece like Marc (Sabat) might actually be the style of JI that in sound comes closest to Western Common Practice Period music. In that musical style, performed for example by a string quartet, the pitches are also fluid without each new intonational iteration sounding like a modulation or a completely new category. The musician accustomed to Western music will probably not hear anything odd about a free style piece like Marc (Sabat) as the pitches are neither significantly flat nor sharp from their stylistic boundaries within Western music. Lou Harrison even implied at times that he thought that free style JI is what string quartets who play classical music naturally do. In the chromatic sections of his Suite for Symphonic Strings, Harrison refrained from notating the music in JI despite the other movements being in JI. Instead, he left it up to the musicians to find 'consonant’ and 'harmonious' intonation themselves (i.e., "play it in JI"). The result, he said, was not very different were he to write out the ratios himself (Miller et al. 1998, 121). In my opinion, this is expressive of the unfortunate idealism associated with JI composers of that generation; they believed that JI was the 'natural’ intonation of musicians. Contrary to Harrison's claim, research has shown that performers of Western art music adhere surprisingly close to ET with no seeming preference for JI, even when playing without a tempered instrument (Burns 1998, 246); the 'span' of the pitch classes’ fluidity is not nearly as vast as they typically are in free style JI, even when only in 5-limit with relatively few comma-levels operating.

If one way of perceiving pitches in the free style is by categorically grouping them into focal pitch classes, there are at least some examples of free style music where we intuitively 'separate' comma-distanced pitches in important ways. An excellent example that can be used to investigate this phenomenon is Marc Sabat’s Gioseffo Zarlino excerpted below in Figure 20. When listening, is the A- and G+ heard as different pitches from the A and G? In my own listening experience, this was found to be the case but only after a while. The music works with repetitions of brief, simple phrases, and this allows the listener to adopt the detailed listening that is required to hear these comma differences as categorical. We become attuned to the micro-audial details as the affordances of the music teach us that this is what is important to listen for. This is where the meaning of the piece resides. After a while, I start to perceive G and G+ as different pitches after initially hearing them as the same focal pitch class. These differences take on a freshness that signals something like a new category, similarly fresh as a Bb will sound in C major in Equal Temperament—a modulatory feeling. 

This modulatory freshness and categorical separation when going from a “5/3” to “27/16” does not generally happen in the free style, and it does not happen in a piece like Marc (Sabat). If we are inclined to cognitivistic explanatory models, we might want to follow Dowling and Miller and say that this is due to our limited cognitive capacities and our propensities for categorical perception. As seen in the example by Sabat, however, certain stylistic choices might invite us to a different kind of listening. This means that we cannot accept a naive cognitivist view of perception. As this example by Sabat shows, pitch perception has to do with searching out and as a listener co-create what is meaningful in music, not about 'processing' some kind of postulated 'sense-data' into perceived pitches. 

Figure 20

Another interesting instance of this phenomenon can be found in the reduced 'melodic duo' version of Catherine Lamb’s Prisma Interius VIII. Even though this music uses many small pitches separated only by small commas, it does not sound like being in the free style. Due to the way the music unfolds, it rather sounds to me like something best described (if sticking with the categories used in this text) as a strict style with a very large gamut of pitches, some of which are comma-distances apart. It sounds, in other words, neither as being in the free style nor as in the strict style the way I have generally described it. The music does not seem to fit the categories outlined in this text since strict style pieces were described as having generally small gamuts and very few, if any, pitches separated by commas. The reason for it sounding strict style-esque is perhaps found in the way the pitches are 'spectrally' articulated as part of a single overtone series; all pitches are notated as partials over the low (inaudible) 5 Hz. Of course, we can find a low common root tone to all pieces in JI, from which all pitches can be considered partials, but the difference here is that this relationship is established with unusual clarity for the listener. The slow unfolding of the piece, beginning with a slow oscillation between G and G- (81:80) and slowly adding pitches, insists on the separation of G and G- as different pitches. They indicate that they perhaps are something more than just different shades of the same pitch class; they are the difference between the 80th and 81st partial. A section like the one at rehearsal number 10, seen in Figure 21, which in isolation looks like a 5-limit free style passage appears in something like a 'single-overtone series-strict style' context that makes the analysis of it in the free style dissatisfying.

Figure 21

A parenthetical point to note in connection with this piece is how its mentioned opening section largely consists of small untunable intervals–some of them as small as the prominent 81:80 (Figure 22). This piece provides a clear example of an important point mentioned in the introduction; a definite line between microtonal music that uses JI-type intervals and what we call integrated JI is impossible to establish but rather represents two idealized extremes. In reality, much music moves back and forth in complex ways between these two styles. Categorically, we must say that Prisma Interius VIII begins as microtonal music that uses JI-type-intervals where the musician must rely on her ability to memorize the sound of the intervals, but that these microtones are later clearly articulated as 'always having been' part of a tunable matrix (in this case a single overtone series). But even as the music in the first minutes of the piece goes back and forth between the two performance modes of tuning and approximating pitches, this is not an oscillation that is readily audible to the listener. For the musician as well, this distinction is blurry; at times we can not tell if we are tuning to short-term memory, or if we are habitually approximating pitches in reliance on mental/embodied representations. Between these two modes of performance, there is a feedback system where one supports the other.  

Figure 22

Poetic moods and tunable paths in the free style

If Harrison in his Suite for Symphonic Strings considered free style to be Western musicians’ natural mode of intonation, Harrison at other times deemed free style to be impossible to achieve without instruments specially built for the purpose. For his Simfoni in Free Style, Harrison employed, for example, an array of differently-tuned custom-made flutes (Doty, 1987); the tunings and key placements would guide the musicians through the intonation. Simfoni in Free Style has never been performed live so the only thing we can listen to is a MIDI mockup. The first thing that strikes me when listening to this mockup is the extreme flexibility of intonation. The pitch level rises rapidly through the different comma levels. In just two measures (8-9), five 5-limit comma levels are present from B- (one comma down) to Bbb+++ (three commas up). This is, by all means, radical intonation and very difficult to achieve accurately at such a fast pace. The result of this kind of intonation is the sound of a highly malleable pitch space. It sounds de-centered, non-hierarchical, fluid, and liquid-like. The affective and poetic 'mood' produced by this intonation is the very opposite of the clearly defined and unifying rasa-s of the strict style that we studied above. The 'rasa' of Simfoni in Free Style is one of almost psychedelic liquidity.

Figure 23 (Measure 8-9 of Simfoni in Free Style)

For me, there is a discord between the liquid-like tuning and the actual melodies and motifs used in Simfoni in Free Style. In fact, everything but the intonation sounds like what Miller and Lieberman in their comprehensive study of Harrison have rightfully dubbed "vintage Harrison" (1998, 118). The music sounds just like Harrison’s other music but now with very spectacular intonation—it is vintage Harrison seen through psychedelic glasses. The music is not only swiftly moving between comma-levels but there is even a blatant use of wolf-intervals. A direct 27/20 dyad between D and G is, for example, called for in measure 5. 

The only way to really play the kind of intonation used in this piece convincingly, one could argue, is if it is backed up by an artistic idea and poetic mood that is 'all about' this kind of liquid-like stretching of the pitch space and that speaks in a single voice together with the tuning—it should be psychedelic Harrison seen through psychedelic glasses. Harrison’s Simfoni in Free Style does not embrace that kind of poetic expression and does not invite such a mode of listening. Instead, it is "vintage Harrison" in which many intervals plainly sound out-of-tune.

Simfoni in Free Style thus provides us with another important example of the rule that the intonation system and the composition must speak in a single voice–speak with the same artistic intention–for the intonation not to sound jarring. As a comparison, an almost equally fast movement between comma levels happens in the passage from Prisma Interius VIII shown in Figure 21. Here, the music swiftly moves between the four comma levels from Bb to D#- - - but sounds here completely natural because of the way it articulates a well-defined harmonic territory that has been slowly and gradually built up and established throughout this slowly evolving piece.  

Figure 24

Even in a free style composition, it has to be noted that it does not take much for temporary reference points to arise—reference points that can make certain pitches sound out-of-tune even when they do not conspicuously appear as such from just reading the succession of notes in the score. In Figure 24, modified from Figure 2 earlier on, the A- will likely sound too low even in a free style context as the D and E provides local tuning references. Such local reference points will always arise—weak and temporary as they may be—as the perception of musical pitches always is colored by other pitches previously heard (see further discussion in Krumhansl 1990, 283). This does not change just because one calls the music 'free style' and glorifies it through a rhetoric of having 'freed' intervals from musical gamuts by working out the intonation of each interval locally. The perception of music is not as local as Harrison seem to have thought. Even when writing free style music, the composer must be attentive to local modalities and modal hierarchies as they arise. It is, after all, the very nature of JI to have a 1/1. The composer must be able to feel which notes influence tuning most and then adjust the music accordingly. Such adjustments do not necessarily always entail replacing pitches; by simply clarifying the phrasing in Figure 24, we can diminish the importance that the D and E have on the A- by, for example, grouping the D and E together and clearly starting a new phrase on the B-. Again, pitch perception is about enacting meaningfulness, not about processing data, and shifts in phrasing and articulation will guide us in this endeavor and will therefore change our perception of intonation.

Successful pieces in free style do not by any means have to go as far as to embrace the kind of psychedelic liquidity of Simfoni in Free Style, but they must speak in a single voice with the intonation system. From the perspective of my own practice, the main quality of the free style is the freedom from strong modal contexts. In my free style pieces, I have been inspired to emphasize how the lack of a unified mode and lack of such a mode's co-emergent affective quality puts the listener in a 'present moment' that is less in the grip of a single modal center or modal hierarchy. The music stays 'new' and becomes modally unpredictable. The piece is not unified by one strong affect (such as a Stimmung or rasa), but each new phrase and section can have contrasting affective qualities brought about by the free intonation. In Marc (Sabat), the fact that the music is in free style helps to emphasize the very fragmented nature of this music as there is no unified mood or affect in terms of intonation that connects the fragments.

An excerpt from Marc (Sabat) is shown in Figure 25. By tracing the tunable paths, we can see just how clearly this melodic and harmonic writing differs from the strict style seen in Figure 7. For example, one can read the modulation happening between measures 23 and 26. The excerpt begins in G Ionian [G, A, B-, C, D, E-, F#-] and modulates to the major chords [B-, D#--, F#-] and [E-, G#--, B-] in measure 27 by building upon the emphasized pitches B- (m. 24) and E- (m. 26). The tuning in this passage is fragile; crucial reference points are found in the immediately surrounding notes rather than from a modal framework. In the three systems of Figure 25, four comma levels are used. Swift modulations such as these where each pitch connects to another in a tunable link and where new modalities must be built on pitches only heard once leaves little room for mistakes. The musicians must stay vigilant with regard to performing each aggregate accurately. By skillful use of instrumentation, the composer can help the musicians on the way. When writing for stringed instruments as in Marc (Sabat), she can, for example, make use of the open strings and natural harmonics to aid the tuning and at certain moments provide anchor points. Consider, for example, the stark modulation from a [E- A- C] to a [F#- A D]-harmony in measure 29. The difficult comma movement in the second violin part from A- to A is facilitated by the use of open strings and harmonics; A- is arrived at through a tunable path, but A is arrived at through using an open string. Additionally, by using orchestration to emphasize the top line C to D as the important part of this gesture (the C is doubled by violin I and trumpet, and the D by violin I and II in octaves), the intonation of A- to A becomes easier to grasp because its pitches are clearly in a second voice attributed to and dependent upon the first voice. 

Figure 25 (Trumpet in Bb)

Another piece of mine in the free style is Radii solis, et sternet (sibi aurum) quasi lutum (Figure 26). In this piece, the music uses five different comma levels, which is one more than Marc (Sabat). Yet, the music moves slowly and gradually through the comma levels through mainly tunable paths. The effect of listening to this music is one in which, just like Marc (Sabat), the music retains a fragmented sound and emphasis on the present-moment harmonic constellations rather than establishing a strong modal mood. 

Figure 26

Before moving on to the loose style, the last style to be discussed in this text, we can now summarize the free style. Basically, it is a style that is the total opposite of the strict style. Instead of working with a fixed gamut, it works with an unlimited and unrestrained number of pitches. Instead of having the kind of strong unifying modal affect that is associated with small gamuts, free style pieces can potentially have no global, unifying modal affect of this kind. This does not mean that the music lacks modal affect, but rather that the modal affects are sculpted more locally: the pitches themselves are less related to a unifying mode and more related locally to their immediately surrounding pitches. This makes the craft of composing a tunable path more feeble as the musician can rely less upon contextual 'modal' tuning and scalar familiarity.

The sometimes teeming number of pitches in a free style piece begs the question of how these are cognitively represented by the listener. Do we hear "27/16" and "5/3" as the same pitch in different intonations or as different pitches altogether? In a piece of mine, such distinctions were shown to be categorically blurred in the listening experience. It was argued that while this blurring may be the most normal for pieces in free style—that is, we do not perceive a move from "27/16" to "5/3" as a modulation in the same way as changing a B to Bb in Equal Temperament would—this can change depending on how the composer constructs her piece. Pieces by Sabat and Lamb seem to suggest listening practices in which pitches commas apart can be distinguished categorically. Within the free style, perhaps we can talk of two sub-categories which represent the two ends of a possible continuum: music that readily invites a simplified organization into focal pitch classes, and music that readily invites separating comma-distanced intervals categorically. 

Loose style

Table 2

While the strict and free style, and the importance of properly distinguishing between them, was already introduced by Lou Harrison, the intermittent loose style is introduced here for the first time. Its most salient characteristic is a hierarchical ordering of pitches that dictates intonational fluidity. Some pitches, structural in nature, are fixed, while the remaining pitches are free. Pitches low in the hierarchy can be freely replaced by their neighboring variants (distanced by commas), while the structural pitches higher up in the hierarchy can under regular circumstances not (if not intentionally used for a modulatory effect, they will sound out-of-tune). In my loose style pieces, the structural pitches are often 'typical' structural pitches such as [“1/1”,“3/2”] or [“1/1”,“4/3”, “3/2”]. The [“1/1”,“5/4”,“3/2”,“15/8”]-pattern, in particular, has found its use in many compositions. Three different iterations of this pattern are shown in Table 2. In this table, these four structural pitches are themselves ordered into three hierarchical levels; “1/1” is on the top level, “3/2” is on the second level, and “5/4” and “15/8” are on the third. These three levels reflect how the higher one moves in the hierarchy, the greater the modulatory effect of changing pitches by syntonic commas will be. The fourth level under the structural pitches in the table includes all the other pitches used in the pieces.

These tables have their origin in observations done in my compositional practice prior to formulating the idea of the loose style. For example, when composing the first piece shown in the table, mot våren bortom havet, I noticed that lowering the D by one syntonic comma to D- had jarring effects, often sounding out-of-tune if not prepared with extreme caution, while changing the A to an A- was very smooth and easy; changing the B- to a B had a big modulatory effect, but not as big as changing the G to G- or G+ had—the alteration which most easily sounded jarring. I wrote many pieces making practical observations like these before I started to realize that maybe this was an idiom distinct from the free style. With the eventual formulation of the loose style of JI, these practical observations found a corresponding theoretical explanation. Later on, I noticed that these tables for the [“1/1”,“5/4”,“3/2”,“15/8”]-structure largely corresponds with the basic pitch space tables used by Krumhansl based on empirical measures of tonal hierarchies (see for example Krumhansl and Cuddy, 2010)—a convergence that has to be further studied in the future.      

To illustrate the workings of these hierarchies in practice, let us look at seven measures from att sjunka i doftande klöver for violin, cello, and piano. Figure 27 shows an early version of these bars. When just reading this score for the first time, we might conclude that everything looks very tunable; we might indeed think that it is written by a composer who has taken great care of the tunable paths. Our only objection is possibly the violin line’s third pitch—we can complain that this three-note melody frames a wolf fourth (E to B-), and we might ask ourselves if this is really tunable. Then, however, we see that there is a strong bass line in both the cello and piano that moves from A to D, and that the violin tunes the E as a perfect fifth to the A, and then a B- as a just major sixth to the D. The relationship to the bass line becomes the important factor here and the fact that the violin melody is outlining a wolf-fifth is obscured and hidden in performance and thus not important.

Figure 27

Having concluded that the tunable path is smooth, we will surely be greatly surprised when listening to this music and hearing how the violin’s A- in measure 261 sounds too low—indeed out-of-tune. This is not at all evident from just reading the score. On the contrary, the fact that the A- in Figure 27 is preceded by both a B- and an E- makes it look absolutely correct with the A- there. It is the third interval in a series of 4/3s. What has happened in listening to account for this phenomenon is the arising of a hierarchy of pitches characteristic of the loose style of JI. The emphatic D in a low register in 258 and 259 in both the piano and cello linger in the memory all the way through 261 and makes the ear want to hear the A as unlowered because it forms a 3:2 to it. There is no doubt that it is the D that serves as the basis-pitch of this passage as a 'fundamental' or 'root'. The A, furthermore, is a structural pitch one step down in the hierarchy from D as the 3/2 to the root. Krumhansl noted that “recognition memory for a tone depends on its position in the tonal hierarchy, with more stable tones in the tonal hierarchy more stable in the memory trace” (1990, 148). In other words, because of its position as the basis pitch of this passage, the D remains present for a long time and has, therefore, a strong influence even as E-, not a basis tone, has sounded more recently. The A can be tuned either to the D (as an A) or the E- (as an A-), but not both. It 'had' to be tuned to the D because the mind’s hierarchical categorization of pitches was more important than melodic linearity and short-term tunability.

In order to not hear the pitch as false to the immediately preceding B- and E-, we must hear it as arising within a modal–or tonal, to follow Krumhansl’s usage of this word–context. If the experience of a modal hierarchy is not present when we hear measures 260-261, the A will sound out-of-tune. When composing in the loose style, the composer will often end up writing passages such as these that in notation looks 'bad'. These hierarchies result in compositionally interesting moments where the ear is more willing to hear untunable intervals like wolf-fourths than structural pitches adapting by commas.  

Figure 28

Figure 28 shows the penultimate version of the passage in Figure 27. The A- in 261 is changed to an A, and this change further necessitated a change from E- to E in the same measure (but not the preceding E- in the measure before). The cello’s E in measure 261 now forms a wolf-interval to the following B- in measure 262. Although this does not sound too out-of-tune in the modal context, the whole passage still has a slight perfume of out-of-tune-ness caused by all the direct wolf intervals. On the one hand, this was not a big problem because prior to this moment, the employment of many 'awkward' outlines and skips that exposed Pythagorean thirds and wolf-fourths had already established a rather thorny and angular mood in the piece. The exposed wolves in Figure 28 fit within this mood very well. It was, therefore, decided to keep this passage without re-writing it completely. On the other hand, the passage in Figure 28 had a little bit too much of this quality. To solve this problem, it was enough to just add a D as a pure fifth under the A in measure 261 as a reminder to the ear of the modal center. It also serves the purpose of masking the 10:9 between B- and A by instead drawing attention to the 5:3 between B- and D. By so doing, the perfume of the wolf’s out-of-tune-ness evaporated but still left us with a rather 'thorny' and 'angular' sounding counterpoint. In order to help the musician navigate the fast shifts between, for example, E- and E, I made sure that these kinds of passages were always supported by the use of natural harmonics and open strings. The final version is shown in Figure 29.

Figure 29

The solo double bass piece Väntar där dimma uppstår employs a similar usage of consciously 'imperfect' JI-writing to achieve a particular poetic mood. In both this piece and in att sjunka i doftande klöver studied above, the 'imperfections' in the tunable paths give rise to specific poetic effects that would not be possible with completely 'perfect' JI melodic writing (i.e., a JI that almost exclusively uses tunable intervals). In measure 187 (Figure 30) a major ninth is played, but it is not the tunable major ninth 9/4 but rather the untunable 20/9. The performer is asked to find this by re-voicing the tunable minor seventh 9/5 in the previous measure by keeping the G+ stable while transposing the A up two octaves. Because the pitch classes are already established in the 9/5, the sonority 20/9 in measure 187 does not necessarily sound 'wrong' or out-of-tune; both the pitches involved are in-mode pitch classes, but it has a fascinating, energetic pulsation to it that wants to be resolved. To perform this 20/9, the musician first tunes the 9/5 in 185, then keeps the G+ stable, and adds to high A as a natural harmonic and while doing so, resist the temptation to lower the G+ to a G in order to form the still, just major ninth 9/4. 

Figure 30

While my main purpose in this text is to argue for the need for JI-composers to take tunability more into consideration, it would be a mistake to ask of all music to only use tunable intervals. As these examples from att sjunka i doftande klöver and Väntar där dimma uppstår shows, there is also a place for untunable intervals. 

Implied modal modulations in the loose style

One of the interesting things about the loose style is that the smaller gamut of pitches used in comparison to free style sometimes leads to slightly less of a readiness for categorical perception of comma-distanced ratios as variants of the same focal pitch classes. Quite often in the loose style, we can really hear the difference and properly distinguish "8/5" and "128/81" as different pitches in a way that we are not usually attuned to when listening to pieces in the free style. When analyzing these moments of heightened discrimination of comma alterations, we realize that this is because alterations between comma-distanced pitches often in loose style sections strongly imply a modulation with greater implications than merely a microtonal change in how one scale-degree is intonated. At these moments, the change in the intonation of pitches implies a different purpose than simply adapting in order to form beatless intervals with other surrounding pitches. 

This is not always the case in the loose style; in att sjunka i doftande klöver looked at above, the alteration between E and E- goes by without such notice or sense of modulation. In that piece, the pitches on the lowest, fourth level of the hierarchy are adapting primarily to tune well to the structural pitches. In loose style pieces like Sommarberg, i glömska and Vid stenmuren blir tanken blomma, however, the pitches in the lowest level are replaced by their neighboring comma variants to articulate modulations between closely related modal scales—scales that in ET would be expressed with the same pitches—such as D Ionian and B Aeolian. In these instances, the new pitch stands out as impactful; it is recognized as a new category and has a modulatory freshness. In these pieces, one is almost inclined to describe the music as constantly modulating between different strict style scales closely related rather than as being in a single loose style mode. Thinking about this music as modulations between different strict style scales rather than a single loose style would be another way to interpret Dowling’s suggestion that only around seven pitches can be kept in mind or remembered at a time.

Figure 31 (transposed score; trumpet in Bb)

Examples of these 'modulatory' comma changes can be found in Sommarberg, i glömska for violin, viola, and trumpet. In this piece, modulations back and forth between the D Ionian mode ["1/1", "9/8", "5/4", "4/3", "3/2", "5/3", "15/8"] and B- Aeolian mode ["5/3", "15/8", "1/1", "10/9", "5/4", "4/3", "3/2"] often occurs. The only difference between these scales is the change of the pitch class E from "9/8" to "10/9". An excerpt of a modulatory passage is given in Figure 31. In measure 127, the music is clearly in D Ionian but modulates throughout the phrase to land in B- Aeolian in m. 136. Notice how the E changed to E- (tunable as a 5/3 to G and 4/3 to B-) in m. 135 to establish this. In m. 139, the phrase starts a modulation back to D, and this is achieved precisely by changing the E- back to E (tunable as a 9:4 to D) in measure 140. Throughout this passage, the alterations between E- and E make this modulation between B Aeolian and D Ionian perceivable both in terms of a shift in modal center as well as in a shift in affective quality. If the music had been in ET, the modulation aspect of this passage would not have been registered since it is, after all, possible for diatonic Western music to end the middle of a phrase on the sixth scale degree without necessarily implying full-fledged modulation to the relative minor mode. In JI, however, such modulations arise as impactful. Furthermore, if the passage had appeared in a free style environment, such comma changes would be so common as to dilute the modulatory effect. In this loose style context, however, one can really perceive these subtle modulations. The pitch changes that clarify them (such as E to E-) stand out as meaningful to the listener.

Figure 32

Another loose style piece that uses comma movements to indicate modulations is Vid stenmuren blir tanken blomma for violin and viola. In one section shown in Figure 32, the music goes back and forth between D Ionian and E- Aeolian. The difference between these scales is bigger than the one between D Ionian and B aeolian discussed above as it also requires an interval to change pitch class (C to C#-). Still, however, these modulations are signified and made clear not only by the change of C to C#- but equally well by changing the A- to A. In measure 180-183, the music is in E- Aeolian. As soon as we hear the natural A (rather than the in-mode A-) in 185, we are already given a premonition that the C will be replaced by a C#- in the following bar, as this change in the intonation of A already carries a modulation with it. The following C#- sounds thus very smooth and predicted since some of the modulation already happened with the A, which by being a Pythagorean sixth (untunable and therefore here played as an open string) to the low C in 184 clearly stands out as being modulatory. In ET, the modulation would only have happened with the introduction of the C#. Here, however, the weight of the modulation is distributed between the C#- and A rather than just falling on the C#-. Later in this section, the modality lands more fully in E- Aeolian, but between measures 184 and 208 (not excerpted here) the A- keeps shifting back and forth from A- to A, suggesting subtle modulations between the E- Aeolian and D Mixolydian modes. 

In an earlier section of the same piece, from measure 84 shown in Figure 33, a modulation that could be jarring, from A Ionian to C Ionian is made smooth by already changing the intonation of E and B to E- and B- while "still being" in A Ionian. Much of the modulation already happens when lowering these pitches by commas, and the introduction of the C in measure 90 does not sound jarring, as it might have had in ET, but smooth. 

Figure 33

Another example can be found in Livets eget bleka flöde on page 21 (Figure 34). In a melodic sequence between a D and a G, we find a E. Why an E and not an E-? In the octave positions given in this music, a E- would be tunable to both the D as a 9/5 and to the G as a 12/5, but instead, the music is written using an E. I made this choice because the E- would have implied a undesirable modulation. E- would have suggested harmonic hierarchies and hinted at harmonic regions to which the piece does not strive. Just as with the example above, the only way to answer why an E would make more sense than the E- is found by looking at the modal context and tonal hierarchies. 

As we have seen with all of these examples, there is a wide range of approaches to modulation in the loose style, and all modulations do not have to be as smooth as possible. While pieces like Sommarberg, i glömska and Vid stenmuren blir tanken blomma attempts for a very smooth effect, other pieces such as att sjunka i doftande klöver and Väntar där dimma uppstår that we looked at above have a rougher treatment of modulations. Just as in non-JI music, modulations can be prepared and smooth, or unprepared and jarring, sometimes noticeable and sometimes unnoticeable. Making poetic use of the kinds of subtle modulations discussed here is, however, according to me, one of the great strengths of the loose style.

Figure 34

Before moving on to a comprehensive conclusion of this text, we can summarize our findings of the loose style. As a style of integrated JI, we saw that one of its defining characteristics is the presence of modal hierarchies. This is not to say that modal hierarchies are lacking in the strict and free styles, but only that they operate in a way here that has a very specific impact on the fluidity of pitches and the construction of tunable paths. This leads us to categorize it as a distinct style. A loose style gamut is comprised of structural pitches and non-structural pitches. The structural pitches are at the top of the modal hierarchy, while the non-structural pitches are at the bottom. We saw that the structural pitches had two primary characteristics. Firstly, they are retained more vividly in short-term memory and can, therefore, be relied upon as tuning references even after other pitches that potentially provide conflicting tuning references have been played. Secondly, the modulatory effect of replacing these pitches with their neighboring comma variants will be very drastic. The structural pitches, therefore, tend to be 'fixed' in their intonation and rarely replaced by pitches comma-distances apart. Pitches lower in the hierarchy have two inverse characteristics: they are not as easily retained, and they are easily replaced by their neighboring comma variants (e.g. they are 'fluid'). Depending on the piece, the neighboring comma variants of the non-structural pitches on the lowest level can be treated in a merely interchangeable fashion—used freely to fit the local tuning situations—or, they can be given a more clear audible distinction. The latter quality can, for example, be achieved by having them articulate subtle modulations between closely related modes (such as C Ionian and D Dorian). Excerpts from my compositions provided examples of this. 

Modulation as new harmonic relationships or as new microtonal inflections

Before concluding this text, there is one final concern that arises from the above discussion that should be to addressed. This has to do with the question of whether modulatory 'effects' are primarily achieved by establishing new harmonic relationships, or by the introduction of new microtonal profiles. Is the modulation from C Ionian to D Dorian primarily achieved by suddenly making the fifth between D and A a perfect fifth (in a basic D Dorian scale) and therefore melodically and harmonically possible (tunable), or is it primarily raising the A (from A- in the C Ionian to A in D Dorian) microtonally that achieves the 'effect' of modulation?Since the changes in cent are quite slim, it might be argued that the modulation sounds like a modulation not primarily because the microtonal profile is different. Rather, what makes it stands out is the new harmonic relationships that can be formed between pitches that previously were not possible. An example of what I mean by this can be found in the brass nonet Tusen tysta skogar. The first part of that piece is almost exclusively in the G Hypoaeolian mode ["3/2", "8/5", "9/5", "1/1", "9/8", "6/5", "4/3"], but at a few crucial moments in cadences, the "4/3" is replaced by a "27/20" in order to form a 9/5 to the "3/2" or a 4/3 to the "9/5". We might argue that this modulation stands out as 'fresh' not because the "F" suddenly is intonated a little bit higher, but because it is now possible to form entirely new harmonic relations between the "F" and the other pitches in the scale. We might argue that it is this that is important, and not the microtonal change in cents. 

But if new harmonic relationships were the only thing that was important in order to achieve the effect of modulation, then these modulations would stand out just as much in Equal Temperament, because even in an equally tempered version, the harmony between G and F would have been avoided equally long before being introduced. The statistical relationship would be the same. And indeed, some of this quality is retained in an ET version, but I also observe that it is not nearly as strong. This implies that the microtonal shifts are also important; the effect of modulation does not merely come about because the just intervals dictate certain 'rules' of melodic and harmonic writing, but also because of the microtonal profiles of the pitches that clarify these rules. It is only in the justly tuned context where ratios are related as simple ratios as this effect truly becomes clear. I am reminded here of a statement by Terry Riley: 

Resonant vibration that is perfectly in tune has a very powerful effect. If it's out of tune, the analogy would be like looking at an image that is out of focus. That can be interesting too, but when you bring it into focus you suddenly see details that you hadn't seen before. What happens when a note is correctly tuned is that it has a detail and a landscape that is very vibrant (in Duckworth, 1995, 283)

It is the details of JI that are not available in ET that make it possible to experience the modulation as a 'powerful effect'.

The enharmonically flexible strict style and JI-temperaments

The pieces in which I have used 'enharmonic equivalents' provide good examples when discussing this question of whether modulation is primarily due to a re-configuring of harmonic space, or due to new microtonal placements of 'scale-steps'. The term 'enharmonic equivalent' is used here to refer to pitches that are less than five cents apart, When pitches caused by very different tunable paths (such as "315/176" and "25/14") land this close in frequency to each other, the music can seamlessly modulate through these enharmonic equivalents to very distant harmonic regions, yet without audibly changing the microtonal profile of the 'scale-steps'. In for example Ljusomflutna, sakta vindar, the viola pitch in Figure 35's measure 56 (a "12/11") is enharmonically equivalent to the violin pitch in measure 60 (a "35/32"). If the theory is that it is primarily the new harmonic context that makes the feeling of modulation appear, then we would expect the 35/32 (as a 5-limit interval to the viola's pitch in m. 60 instead of as an 11-limit interval, 11/4, to G) to feel very modulatory. Yet, it does not. The feeling of this piece throughout is that of strict style, although it is a strict style that is  more complex and ambiguous-sounding than the one used in Vårbris, porslinsvas 

Figure 35

Another example can be seen in the excerpt from I luftens svala dunkel in Figure 36. Here, there are two types of E flats, C's and F's. Even though these different enharmonic equivalents imply vastly different harmonic relationships, I would argue that we do not feel any particular modulatory effects from their enharmonic replacaments. It rather sounds like a strict style-piece. 

Figure 36

Experiences from these pieces, where enharmonic equivalents make possible new harmonic implications, yet still sound like they are just in a strict style-context, implies to me that the implied modal modulations in the loose style are to an important extent a product of the changes in microtonal scale positionings.

This kind of enharmonically flexible strict style constitutes the majority of my pieces in non-5 limit JI (e.g. pieces in 7, 11, and 13 limit). This style of music achieves a kind of polysemic and ambiguous 'mode' with an inherently 'modulatory' feeling to it despite sounding like a strict style piece that exploits only 'one scale'. In a sense, the enharmonically flexible strict style is thus very similar to the loose style in that both styles posit to sit in between the strict style and the free style. In the loose style, the fluidity of certain pitch classes that allowed them to drift by syntonic commas created a style of music that always changed its implied harmonic center; the music continually oscillated between, say, D Dorian, C Ioanian, and A Aeolian. In the enharmonically flexible strict style, similar kinds of subtle modulations happen, but here without the pitches drifting in intonation any more than 5 cents. The result is a highly stimulating balance between a fixed scale and an ambiguous and polysemic pitch space.

Figure 37

The piece in which I took this idea perhaps the furthest was the viola or violin solo piece Rosor och så liljor. This piece uses a simple 7-tone strict style scale, but by allowing each pitch to move within no more than a 5 cent radius, a very rich gamut of other harmonic possibilities is opened up. Because of the complexity of this particular scale, I even decided to not notate this piece with conventional JI-accidentals, but rather to introduce a combination of cent-deviations and ratios. A brief excerpt of how the notation would look in normal JI notation is given in Figure 37; it looks like a very untrammeled free style piece. The chosen notation in Figure 38 instead reveals this free style to be contained within a very strict 'grid' that is the 7-tone basic scale with only a 5 cent flexibility to each scale step.

Figure 38

A similar effect happens in my big keyboard piece Som regn. In that piece, the pitches obviously cannot move because of the fixed tuning, but I decided to anyway notate the implied harmonies in a similar way as in Rosor och så liljor. Here, the cent deviations show pitches' out-of-tuneness rather than how much they have to be adjusted. Som regn operates in many ways like an enharmonically flexible strict style piece but the pitches have to be adjusted 'in our minds' rather than on the instruments. In that sense, this music functions a lot like a kind of temperament in the sense that pitches can have multiple spellings. Figure 39 shows a brief excerpt from Som regn; note how the A and E are notated in different ways depending on the harmonies. The A is both an 11/9 to C as well as a slightly out-of-tune 8/7 to the B-, and it is often difficult to choose which interpretation should be favored. The A is in other words polysemic and multi-stable; both interpretations are correct. This is why it makes sense to call these kinds of tunings temperaments even though no pitch technically has been 'tempered'.

Because the piece is written for a tuned instrument, there are also instrumental idioms such as the use of untunable melodic intervals that suggests that the music should be classifiable as being in the informed style of strict style as a 'tuning'. In Table 3 where many of the different styles are summarized, this kind of 'JI-temperament' is listed as a sub-category to both the informed style of strict style as a 'tuning' and the enharmonically flexible strict style.

Figure 39


My main purpose with this text has been to give the reader an insight into the craft of writing tunable music in an idiomatic and integrated justly tuned idiom. The text began with a basic description of what JI is. It was emphasized that JI is about tuning sounds to resting, fused, periodic entities. As a microtonal system, JI is therefore not only introducing to musicians new interval sizes and microtones and asking them to approximate these as closely as possible in performance–although it does that too. Since rational notation frequently is used primarily for such a solely microtonal purpose, that kind of practice needed to be clearly distinguished from the integrated JI focused on in this text because it uses JI to generate very different kinds of music. This practice was bracketed as microtonal music that uses JI-type intervals.

Research has shown that most music of the world is performed by primarily reproducing pitches in correspondence with cultural and stylistic norms, but the musician performing integrated JI is not only reproducing pitches (although they too must rely on mental representations) but is also tuning them to each other in an interconnected, tunable harmonic space. The very method of playing pitches is, therefore, different in integrated JI compared to most other types of music. The compositional craft needed to facilitate this is, likewise, very different from most other compositional crafts. It is the composer who ultimately has to ensure tunability by connecting tunable intervals into a tunable path

But if all intervals in a JI piece had to link into each other by directly tunable intervals, like a domino show or a relay race, it would not even be possible to play a simple ascending 5-limit diatonic scale in JI since both of the major seconds 9:8 and 10:9, as well as the minor second 16:15, are untunable. It was thus established that a piece in JI does not have to be restricted to only use immediately tunable intervals. The reason for this was found, on the one hand, in our capacities to tune to the short-term retention of previously heard pitches, and, on the other hand, in the possibilities of contextual, modal tuning—the possibility to rely on an emergent tunability of sorts. Although these two types of non-simultaneous tunings are impossible to achieve with the same accuracy as simultaneous tuning—they are weaker and considerably more unreliable—, the effects they have on performing and composing JI are still significant.

Given these basic premises, the claim was made that a big part of the craft of JI consists in providing each piece with something we might call a proper balance between tunable and untunable intervals. What constitutes a proper balance is, however, not yet easily defined by any rules, and the composer is left largely to rely on her own developed sensitivity for tunability. After giving an initial example of the workings of short-term retention by showing how the same pitches in a melody could be shifted around to constitute either a favorable or unfavorable balance, our attention turned to this text’s main topic of modality. It was noted that the proper balance between tunable and untunable intervals was affected by whether the modality in the piece was strong or weak. The simple claim made was that strong modalities afford worse ratios between tunable and untunable ratios, and weak modalities necessitate a higher proportion of immediately tunable intervals. A continuum of modal strength was stratified into three different basic styles of integrated JI composition called strict, loose, and free. The main bulk of this text then discussed each of these styles separately. 

In the integrated strict style, a strong modality affords a balance between tunable and untunable intervals that is quite generous on the untunable side; untunable intervals that are part of this mode, such as Pythagorean thirds and sixths, can sound more correct than their unavailable, tunable 5-limit variants as long as these are properly prepared. Integrated strict style pieces achieve this strong modality by the exclusive use of a small, fixed, gamut of pitches.

Pieces in the free style are generally characterized by a huge—theoretically unlimited—gamut of pitches. In this style, the composer is not helped to facilitate the performance of correct intonation by any unifying modes or tonal hierarchies other than the ones temporarily arising; intonation is based mainly on the immediately surrounding pitches and temporary reference points. The free style is, therefore, the style that is the most fragile and that demands the most careful writing. Here, the balance between tunable and untunable intervals has to generally be such that the majority of pitches are directly tunable.

In the loose style, a hierarchical ordering of pitches influences the intonational 'fluidity' of pitch classes. This hierarchy can create a preference for some untunable intervals over changing these intervals to their tunable comma-variants if the pitches in question are high up in the modal hierarchy and structural in nature. These pitches high up in the hierarchy are more vividly retained by short-term memory and can be relied upon as tuning references even after other pitches that might provide conflicting tuning references have been played. Pitches lower in the hierarchy have inverse characteristics. Here, tunable Ptolemaic intervals are generally preferred over untunable Pythagorean intervals, and the short-term retention of these pitches is weaker. In practice, it means that the structural pitches high up in the hierarchy are immovable and behave as in the strict style; changing the intonation of these pitches would imply a strong modulation. The pitches lower in the hierarchy are behaving in a free style manner; changing these pitches does not necessarily imply a modulation but can either go by completely unnoticed or, if articulated so in a composition, imply a subtler modulation. This loose style is thus the middle ground between the strict and free style, including elements of both.

An important theme in this text has been the search for a way of theorizing that corresponds to how we perceive pitches when listening to JI music rather than only reflecting the compositional techniques used to write them. The impetus for this theme came directly from issues arising primarily in the free style, loose style, and strict style as a 'tuning'

In the free and loose styles, the question was posed if we hear different pitches comma-distances apart as different categories or if we hear them, as in Western classical music, as different intonations of the same focal pitch classes. We saw that it was not possible to answer this question with a universal affirmation or negation but rather that our answer changed depending on which particular piece we discussed. Some (or, as was suggested above, most) pieces give rise to a grouping of ratios into focal pitch classes, but stylistic choices, often involving minimalistic aesthetic choices, can give rise to a perception where such grouping does not happen as easily and where comma variations are possible to hear as separate categories. Within the free style and loose styles, two possible sub-categories were therefore gesticulated toward: a style that readily invites separating comma-distanced intervals categorically, and a style that readily invites a simplified organization of comma-distanced pitches into focal pitch classes.

When discussing the strict style as a 'tuning'–which refers to strict style written for fixed-pitch instruments–, the possibility to construct rational tunings, by way of using primarily complex ratios, in such a way that they completely obscure the harmonic origins of rational intervals was mentioned. This led to a discussion of how to best analyze such music. The question was posed if this kind of music adequately can be said to be in Just Intonation. When the ratios between pitches that are separated by many tunable steps become too complex, we no longer perceive them as being in JI. In my clavichord piece I Sommarluft, for example, the common minor third between “128/81” and “15/8” generates the ratio 1215/1024. This interval is a schisma higher than the Pythagorean minor third of 32/27. The interval is thus an out-of-tune variant of an already untunable interval. In the first movement of I Sommarluft, where such intervals abound, there is very little information reaching the listener that the piece she is listening to even is in JI. Within the strict style as a 'tuning' category, the sub-category rough strict style as a 'tuning' was therefore introduced to account for this phenomenon. It contrasts with the informed strict style as a 'tuning' where the ratios still have their tunable relationships retained due to the closer proximities in ratio lattices and more compact and clearly articulated harmonic spaces used. 

Table 3

Many of the categories and sub-categories mentioned in this text are summarized in Table 3. These categories should by no means be thought of as rigid 'genres' of JI music; there are many instances of JI music that defy the application of these categories altogether. Instead, they are best seen as describing points on a highly malleable continuum. On his continuum, many pieces move back and forth between at certain times strongly embodying the characteristics associated with certain styles, and at other times embodying the characteristics associated with other styles. Among the excerpts used in the text, a piece like Vid stenmuren blir tanken blomma starts in the loose style, at times even intimates a freer approach indicative of the free style, and finally coalesces into something that in the end sounds and behaves like strict style—but that still has the presence of two pitches only a syntonic comma apart, which technically would define it as being in the loose style. Another piece discussed that moves between different styles is I Sommarluft. The music of this piece moves back and forth between the rough and informed styles of strict style as a 'tuning'.

Other pieces do not so much move back and forth between different styles as much as their general style is difficult to determine. An example of this was Sommarberg, i glömska. It is hard to know if we should analyze this music as expressing a loose style collection of pitches or rather as different closely related strict style collections that the piece frequently modulates between. At certain times, it embodies the qualities of both. Two other pieces not discussed in this text, Densamma tystnad and Framför stillheten use scales in which certain pitches have enharmonic equivalents while other pitches also have syntonic variants, generating scales that have the characteristics of both the enharmonically flexible strict style and the loose style, meaning that their proper category is the enharmonically flexible loose style.

The different styles outlined here are thus not to be thought of as rigid labels put on musical passages but rather describe behavioral tendencies. Used in this way, one can say of a passage that it inhabits strict style-type behavior or loose style-type behavior even if the passage does not neatly fit into either category. When we looked at Prisma Interius VII by Lamb, describing the music as inhabiting both free style and strict style characteristics as well as moving between integrated JI and microtonal music that uses JI-type intervals provided us with a way of talking about the use of rational intervals and intonation in this piece even as the categories did not, as they rarely do, fit perfectly.

The styles should furthermore not be thought of as compositional techniques; this would run the risk of justifying bad writing by referencing the theory of the styles. One could then, for example, write an untunable interval and argue that this is acceptable by simply referencing the loose style and a table of structural intervals without relying on one’s ears. One could also, as another example, let a passage quickly rise in comma levels, which often sounds jarring, and then justify this by pointing to the score where everything looks good because the music is comprised exclusively of directly connected easy and tunable intervals.

The idea of the different styles can, however, give some confidence to our subjective interpretations and as such be useful as compositional tools. They can, for example, help us understand why a particular passage that looks tunable, like the one in Figure 27 (the early version of the piano trio att sjunka i doftande klöver), is in reality not, or why a chord that looks like it should sound jarring, like the one in Figure 8 (the 'major triad' from the violin and viola duo Vid stenmuren blir tanken blomma), does not. The styles can also be useful conceptual tools in collaboration and communication between composers and musicians. 

Looking at Table 3, it also dawns upon us that when a composer describes herself as 'working with JI', her practice can entail very different things. One composer might be involved with working only with re-tuning instruments or building new instruments and composing for these sonorities as if they provided a unique collection of microtonal pitches—not thinking at all about concepts like tunable paths and the balance between tunable and untunable intervals. Some composers, also when writing for intonating instruments, approach JI only as a micro-tonal system, not interested in if certain intervals interlock to create the fused identities of justly tuned intervals. Other composers might be primarily concerned with creating tunable paths and tunable harmonies by working with a highly specialized mode of counterpoint and melodic writing in JI to ensure this. The craft of these different approaches is extremely different, but by building upon Lou Harrison’s principal idea of a free style and strict style, I have expanded and sub-divided these categories with the hope of finding a common language and framework for all of these different approaches.

Besides the themes of tunability and pitch category perception, the third theme throughout this text has been the poetic and affective implications of the different styles. We saw that strict style pieces can have strong affects, or rasas, where pitches have distinct affective flavors. Loose style can make subtle yet affectively charged modulations by changing the intonation of certain pitch classes by commas. Free style can give rise to a fluid, liquid sensation of pitch, always in the 'present moment'—unpredictable and lacking in the affective glue that binds the music together into the strong modal rasa-s of the strict style. Throughout the text, it has been emphasized how crucial it is for the composition and the intonation system to speak in a single voice and strive toward the same artistic goal. A piece like Wolfe’s STEAM makes use of the Partch instruments as 'found objects' without any subtle consideration of their intonation, but this is not a problem because the artistic effect of that piece is that of masses moving with a rough, loud, and intentionally inelegant attitude. Harrison’s Simfoni in Free Style was using melodies and motifs from Harrison’s usual arsenal but then overlayed them with a radical, freely floating intonation that made the music sound out-of-tune; here, the elements of the composition were not speaking in a single voice.

Just Intonation composition is still in its early days despite the fact that Harry Partch's  seminal book Genesis of a Music was published in 1949. Few musicians master the skills necessary, and few pieces get adequate performances. More practical research through composition and performance is needed in order to be able to answer the two major questions underlying this text: "What is a reasonable balance to have between tunable and untunable intervals in order to provide the musician with a tunable path?" and "to what extent can we tune to non-simultaneous sounds?" It seems to me that any future formulation of the 'rules' of JI composition is dependent upon the answers we get from these questions. 

In this text, I have, through excerpts from my own compositions, been trying to advance such an understanding by discussing how the balance between tunable and untunable intervals impact pitch perception. I must admit, however, that my compositions are filled with flawed tunable paths. There are multiple reasons for this. The first one is that intonation is, after all, only one parameter of music and some people would even call it a subtle one; it is not uncommon to let other parameters sometimes take the upper hand in an artistic decision that results in less than optimal tunability. The second reason is that if you have a piece where all intervals are always harmonically tunable, this will create a certain mode of listening that is constantly focused on fused harmonic sounds. In pieces such as I den gyllne luften, Offerblommor, and Minnets svala flod, almost every sonority is tunable, and this creates little margin of error on the side of the musician since it then becomes so obvious when the musician plays out-of-tune. In these pieces, it can feel like everything has to be completely in tune, as the music creates a rasa of pristine clarity. This kind of situation can become frustrating for both the performer and the listener alike. A much more leisurely listening experience is achieved in pieces like mot våren bortom havet or Väntar där dimma uppstår, where the prominence of untunable intervals does not create this rasa of pristine clarity. In pieces such as I den gyllne luften, we can at worst stop listening to the piece and start listening to musicians tuning. To interpolate the music with a few untunable intervals here and there, can actually function to block this rasa of pristine clarity to become prevalent, and invites to a more relaxed listening.

I would therefore say that any piece does not have to be only a series of directly tunable paths, but that in the places where pitches are performed 'microtonally', they should be able to be derived from a tunable context, where they are introduced and established as having a tunable origin. A principle that I have tried to follow in my own pieces is that the piece should "teach" the microtonal pitches by itself, not by having the musician learn them elsewhere. A powerful way to achieve a balance between melodic flexibility and derived tunability is to work modally. The free style in its most extreme articulation is not practical, and neither is a strictly 'microtonal' style. Working with clear modes, where pitches not only have a derived tunable relationship to each other—where each mode can be worked out as small tunable units—but also unique microtonal profiles that come to be associated with affective qualia, is the method that I in my own work have found most compelling.

In order to advance the craft of JI, however, it is at the present moment perhaps more important to write tunable pieces and work towards more tuned performances rather than trying to formulate any rules and guidelines about melodic writing and counterpoint in JI. Furthermore, a fuller understanding needs more solid evidence than the experience of just one composer. My hope, however, is that the experiences and findings shared in this theoretical text will contribute to the intersubjuctive understanding and inspire the countless other composers and musicians working with JI to similarly share their findings from their own practices. Eventually, we might move closer to have an idea of what the craft and 'rules' of JI composition could entail. At such a point, perhaps a true master of JI counterpoint will come about and write smooth, unflawed, tunable lines that perfectly balances tunable intervals with interpollated untunable intervals. Hopefully, then, my rough attempts in these early days might still carry a primitive, naive charm, similar to the one that we might perceive from the earliest tonal composers working at the cusp of the transition from modal to tonal music in the last days of the 16th century, long before the codification of that system and the advent of its masters a century and a half later. 


[1] In this text, harmonic ratios are notated with a “/” and melodic ratios with a “:”. When ratios are used to signify scale degrees in a scale or gamut, double quotes “ ” are used. Sets of scale degrees to form a mode or scale are notated in brackets “ [ ] ”. For example: In the mode [“9/8”, “5/4”, “3/2”, “5/3”], the melodic leap between “9/8” and “3/2” is a 4:3 and the harmonic sound of playing them together is 4/3.

[2] Because this text will mostly discuss music in 5-limit Just Intonation, the minus and plus signs “- / +” are used in the body of this text to indicate syntonic commas (21.5  cents). 

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