Introduction
When a contemporary composer claims to work with Just Intonation (JI), this statement can refer to widely different ways of working with pitch. One composer may approach JI primarily as a microtonal system, focusing on the distinctive interval sizes it produces. Even in its simplest low-integer form, JI generates intervals that differ from those of Western Classical music by tenth-tones, sixth-tones, or quarter-tones. With more complex ratios, the palette of interval sizes expands almost infinitely. Another composer, by contrast, may be less concerned with interval size in itself and more concerned with the phenomenon of tunability: using JI’s simple whole-number ratios to fine-tune resting, composite periodicities.
For the first composer, ratios are treated as a resource of intervallic nuance rather than as components of a tuned sonority. Their practice is more linear and 'microtonal', and whether or not the intervals are tunable may not concern them. The second composer, by contrast, works more vertically, attending to the psychoacoustic phenomena that emerge when frequencies in simple ratio relationships fuse into spectra that are both rich and stable. When unusual 'microtonal' intervals are part of such resting, composite periodicities, they do not necessarily register as microtonal, because the listener’s attention shifts from interval size to spectral fusion. For the first composer, however, this very shift may be aesthetically undesirable, since their focus lies in the affective qualia of microtonal interval sizes and their melodic articulation.
These two orientations toward JI are fundamentally different, and they generate very different kinds of music. Yet both are called "JI composers."
While both the microtonal and the fine-tuned approach to JI can generate compelling music, I take the standpoint in this text that tunability is the essence of JI. A microtonal practice can easily be realized using different types of equal temperaments, or by notating cent deviations from their scale steps. There is no need to invoke the theory of JI if the music does not provide a tunable context. For example, the interval 12/11 is functionally nothing more than a neutral second if it arises in an untunable setting. 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, then it is simpler to notate such an interval as a neutral second, since that is how it will be heard. The difference between an equal-tempered neutral second (150.00 cents) and a 12/11 (150.64 cents) is only about half a cent—imperceptible in most contexts. While microtonal practice will still have a prominent role in this text—both because it is common among "JI-composers" and because it provides a useful marker for when JI stops sounding like JI—it will not be treated here as a species of JI. Instead, I will refer to it as microtonal music that uses JI-type intervals.
Using JI microtonally or not is not the only way in which JI composers’ practices differ. Even among composers who work in a more harmonically fine-tuned idiom, there exists a wide range of approaches. The pioneer who, to my knowledge, first articulated such distinctions was the American composer Lou Harrison. He introduced the terms strict style and free style to distinguish between different compositional techniques and uses of JI.
The strict style refers to music characterized by intonational rigidity, in which the composer works from a predetermined, limited gamut of pitches. 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 for describing the perception of pitch (a point I will return to below)—remains consistent. If the second scale step is "9/8" (where the ratio designates a scale degree relative to a fixed "1/1"), it will retain that value throughout the piece or musical section. Harrison’s compositions for justly tuned gamelan provide clear examples of strict-style practice: he tuned gamelan sets to original JI interpretations of Javanese scales and then composed works for these sets of pre-given pitches.
The free style, by contrast, refers to music characterized by intonational fluidity, where any ratio can be called upon at any moment. In his free-style piece Simfoni in Free Style, Harrison locally adjusts the intonation of pitch classes in order to fine-tune vertical harmonies. The focal pitch class of the second scale step might be either "9/8" or "10/9", depending on the context. No intonation of any scale step is pre-given; instead, 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 enable different modes of listening. The difference between the strict style and the free style is not simply a matter of the composer choosing to work with a limited or limitless gamut, but of the different musical worlds that the listener, through these approaches, can be attuned to. My perspective 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 mode of listening with respect to how pitches are enacted. The strict style—the mere fact of using a small gamut—can, for example, be used to create music that moves toward either of the two poles mentioned in the first paragraph: music that highlights microtonal interval sizes (a horizontal emphasis) or music that tunes pitches into composite periodicities (a vertical emphasis). This distinction not only separates JI proper from microtonal music that uses JI-type intervals, but also marks out contrasting tendencies within JI music itself. Different modes of listening are enabled depending on where a piece situates itself on this spectrum.
Harrison’s justly tuned gamelan pieces clearly move toward emphasizing the vertical: the gamelan is tuned justly but then used in a primarily melodic idiom. The simple intervals in these tunings make the sound distinctly JI—the simple ratios ground the pitches in a tunable context—yet the freely linear, melodic treatment places the rational origin of pitches in the background. Harrison treats the melodic steps more as if they belonged to a traditional Javanese scale than to a set of harmonically interconnected ratios. Rather than linking all pitches through an integrated tunable path that enables rational tuning, Harrison uses the gamelan as an already tuned set of pitches—a tuning.
Because of these different ways of working with a limited gamut—either treating the gamut as a framework for constructing tunable links, or treating it as an already tuned set to be used more freely—the strict style can be subdivided into the integrated strict style and the strict style as a 'tuning' to accommodate these tendencies. In the integrated strict style, the correct intonation of pitches is brought forth by compositional craft that ensures tunability, while in the strict style as a 'tuning', the intonation of pitches is compositionally treated as if already achieved.
While the strict style as a 'tuning' marks a style in which the tuning is treated as if already brought forth, important distinctions exist within this category itself. Consider, for example, the difference between Harrison’s gamelan pieces and La Monte Young’s piece for justly tuned piano, The Well-Tuned Piano. Compared to Young’s piece, the gamelan pieces treat JI less rigidly. Although the simple tuning and small gamut of pitches clearly establish the sound of JI in the gamelan pieces, the treatment of the tuning is, as mentioned above, free and primarily melodic. By contrast, Young’s piece is more engaged with articulating the tuning itself: the pitches are heard as emerging from a harmonically interconnected, tunable matrix. Yet Young still relies heavily on the instrument being pre-tuned, as performing most of The Well-Tuned Piano would be impossible on anything other than a pre-tuned piano. It is therefore not a piece in the integrated strict style, but it intimates that style within an instrumental idiom far closer than Harrison’s gamelan pieces do.
As these two examples make clear, the strict style as a 'tuning' must be subdivided further to capture these contrasting tendencies. In this text, the sub-categories of the strict style as a 'tuning' will be termed rough and informed: the former treats pitches freely, without deep consideration of their tunability, while the latter carefully considers the tunable relations between pitches. The term rough is not intended as a negative judgment—indeed, I primarily use it here to describe compositions of my own. The metaphor of roughness points instead to something uneven and irregular, like a rough texture.
The informed style of strict style as a 'tuning' carefully links tunable intervals to articulate a justly tuned harmonic space. Consider the 'pedagogically' clear way in which pitch space is presented in La Monte Young’s The Well-Tuned Piano (analyzed in detail below), compared to the melodic freedom of Harrison’s gamelan pieces. In the informed style, there is a balanced alternation of tunable and untunable intervals to achieve the kind of clarity of harmonic space that Young achieves. In the rough style, this balance is not pursued, resulting in a more uneven and irregular—that is, rough—distribution of tunable intervals.
Within both the strict style as a 'tuning' and the integrated strict style, this text identifies a subcategory that emerges when composers use enharmonic equivalents—ratios separated by less than five cents. Because these intervals fall below the threshold of perceptual differentiation, they are effectively heard as the 'same pitch'. When enharmonic equivalents appear in a mode characterized by tunability, a striking perceptual phenomenon arises: pieces may be experienced not as single nodes in the matrix, but as polysemic and multi-stable. On fixed-pitch instruments, this results in qualities reminiscent of Western classical temperaments—an appealing ambiguity—which I term 'JI-temperaments', even though no pitch is actually tempered. Such use of enharmonic equivalents can be found both in Harrison-type gamelan pieces and in Young-style harmonic explorations, thus creating a further layer of differentiation within these subcategories. When enharmonic equivalents occur in the integrated strict style, such as in pieces for intonating instruments, the resulting subcategory may simply be called the enharmonically flexible strict style. In this text, I will analyze my pieces Ljusomflutna, sakta vindar and Som regn as examples: the former written for intonating instruments, the latter a keyboard piece tuned to a 'JI-temperament'.
Just as the mere use of the strict style can generate many different modes of listening, the intonational flexibility of non-strict style pieces can also lead to multiple ways of enacting pitches. In particular, two basic variables shape how pitches are perceived. First, as in strict style, the presence or absence of enharmonic equivalents influences how harmonic space is heard. Second, there is a subtle distinction in how comma-distanced pitches are perceived—either as distinct modulations or as subtle variations of 'the same pitch'.
We might assume that the large and non-predetermined pitch collection of the free style requires careful integration of tunable intervals into tunable paths to ensure playability. Without such attention, one might expect the result to collapse into microtonal music that merely employs JI-type intervals: in the absence of a tunable context, distinctions such as between "9/8" and "10/9" reduce to microtonal inflections. While this line of reasoning is sensible, it is also possible—at least theoretically—to imagine subdividing the free style in the same manner as the strict style, with an integrated free style and a free style as a 'tuning', the latter which in turn might be either informed or rough. One might even picture a hypothetical work for three re-tuned pianos, each mapped to a different JI lattice. In such a piece, pitches between the instruments could diverge not only by commas but also by enharmonic equivalents, producing overlapping networks of near-identity and difference. Performers might alternate or layer material across the keyboards, such that the act of modulating from one piano to another enacts the perception of comma shifts or enharmonic substitutions. Such a scenario could, in principle, yield an 'informed' version of free style as a 'tuning', complete with enharmonic equivalents and categorically distinct comma-pitches. To my knowledge, however, no such work exists, and an exhaustive taxonomy that attempted to chart every logical possibility would quickly become unwieldy, populated more by theoretical speculation than by actual music.
The one body of work that comes closest to a genuine free style as a 'tuning' is Harry Partch’s ensemble music. His custom-built percussion and string instruments produced a pitch collection too large to be grasped as a closed mode, giving the music a distinctly free quality. That few other composers have pursued this path is unsurprising: such a style demands entire families of re-tuned instruments, and even the re-tuning of a single piano is already a considerable challenge. The produced pitches in Partch’s music are often handled in a rough manner, embedded within instrumental idioms that presuppose the instruments’ fixed tuning rather than integrating tunability into the compositional process. For this reason, the music is more precisely classified within the rough subcategory of free style as a 'tuning'. To classify it as JI requires that the sound of JI—through the exactness of the instruments’ tuning—shine through this roughness to establish a tunable context. This does occur at times, but equally often Partch’s ensemble music comes across as microtonal music employing JI-type intervals.
Furthermore, I know of no free style pieces that incorporate enharmonic equivalents as stable elements of a modal scale. They may occur in the course of modulation—serving, for instance, as pivot points that return a passage to its starting region—but within the constantly shifting context of free style their polysemic effect is inevitably diluted. By contrast, in strict or loose styles, enharmonic equivalents derive their power precisely from appearing within a limited modal framework, where a single pitch can sustain two conflicting harmonic interpretations.
In practice, then, with the exception of Partch’s music, the free style can be treated as shorthand for the integrated free style—the only form I have otherwise encountered. The same principle also applies to the loose style, an intermediate approach that shares features of both the strict and free styles. While theoretically we could image a loose style as a 'tuning', I know of no actual pieces of music that fits that categorization. To avoid proliferating speculative categories, my discussion of both the free and loose style will therefore be limited to their default, integrated versions.
The most salient characteristic of the loose style is a hierarchical, modal ordering of pitches, which governs the degree of intonational fluidity across pitch space. Pitches low in the hierarchy can be freely replaced by neighboring pitches separated by small commas without sounding out of tune and with minimal performance difficulty; in other words, their intonation is free. Pitches higher in the hierarchy, by contrast, cannot be replaced without disrupting the tuning and posing performance challenges; their intonation is therefore strict. For the distinction between 'strict' and 'free' pitch classes to be meaningful, tunable intervals must be embedded within a coherent tunable context. In this respect, the loose style resembles the free style in depending on the composer’s attention to tunability.
Within the loose style, differences primarily concern the perception of comma-distanced intervals, as in the free style. In one variant, these intervals are perceived categorically, while in another, they are simplified into focal pitch classes. However, because enharmonic equivalents do occur in loose style contexts, it is useful to define an additional subcategory: the enharmonically flexible loose style, which can arise in contexts where commas are either perceptually separated or fused. In this variant, the enharmonicity of certain pitch classes imparts a polysemic quality to specific notes, paralleling the enharmonically flexible strict style. An example is my own piece Densamma tystnad.
To summarize and situate the distinctions I’ve outlined, the following overview maps the varieties of Just Intonation (JI) composition discussed in this text:
1. Microtonal music that uses JI-type intervals (e.g. microtonal works employing rational notation but without concern for tunability)
2. JI music proper
2.1. Strict style
2.1.1 Strict style as a 'tuning'
2.1.1.1 Rough strict style (e.g., Lou Harrison’s gamelan works)
2.1.1.1.1 JI-temperament, rough style
2.1.1.2. Informed strict style (e.g., La Monte Young’s The Well-Tuned Piano)
2.1.1.2.1 JI-temperament, informed style (e.g., Som regn)
2.1.2 Integrated strict style
2.1.2.1 Enharmonically flexible strict style (e.g., Ljusomflutna, sakta vindar)
2.2. Free style
2.2.1. Integrated free style (default)
2.2.1.1. Variant 1: categorical separation of comma-distanced intervals
2.2.1.2. Variant 2: simplification of comma-distanced intervals into focal pitch classes
2.3. Loose style
2.3.1 Integrated loose style (default)
2.3.1.1 Variant 1: categorical separation of comma-distanced intervals
2.3.1.1.1 Enharmonically flexible loose style without focal pitch classes
2.3.1.2 Variant 2: simplification of comma-distanced intervals into focal pitch classes
2.3.1.2.1 Enharmonically flexible loose style with focal pitch classes
The musical examples discussed in relation to these three styles include works by composers such as Marc Sabat, Catherine Lamb, Lou Harrison, and La Monte Young. The majority, however, come from my own compositions, as this is the music I understand most intimately. While many examples originate in my own work, the insights they provide are not limited to a first-person phenomenological perspective; they are continually related to research in music cognition and tuning theory to establish the intersubjective validity and broader relevance of my findings. Experimental research from music cognition laboratories, alongside music-theoretical investigations of intonation, provides a foundational explanatory framework for the analyses presented here. Because these theories often assume a cognitivist model of perception, they are complemented by insights from enactive, socio-cultural, and ecological approaches, creating an adequate theoretical foundation for the practical, tacit knowledge gained from composing and performing in JI.
Before delving into the different styles of JI, this text will first outline its basic theoretical foundations, beginning with the concept of tunability and its relationship to JI composition and performance. Other important theoretical topics, such as the categorical perception of pitches and the concept of focal pitch classes, will be introduced later, first appearing in the section on the free style. These foundational concepts provide the framework for understanding how different compositional strategies shape the ways in which pitches are enacted and perceived.
Playing in tune and its prerequisites
From a compositional perspective, Just Intonation (JI) can be defined as the practice of notating pitches using whole-number frequency ratios such as 3/2, 5/4, or 25/24. This definition, however, needs to be amended: more than simply notating ratios is required for the music to actually sound like JI. To assume that the use of ratios alone produces JI music is to mistake the means for the end. What ultimately defines JI is, as Thomas Nicholson and Marc Sabat (2018) write in the opening sentence of their introduction to the subject, the practice of playing in tune.
In the context of notated music, it is the composer’s responsibility to ensure that the tuning dimension of performance can be meaningfully activated—something that requires more than simply writing ratios. This is achieved by employing tunable intervals and linking them into tunable paths. These intervals and paths, rather than the ratios themselves, form the fundamental building blocks of justly and actively tuned music.
Understanding and achieving tunability is a central concern for all three styles—strict, free, and loose—as well as their respective sub-divisions. Whether a composer works within a tightly pre-given gamut, draws freely from an open set of pitches, or lets a hierarchical, modal perception guide the composition, the perceptual reality of tunable intervals underpins the music’s ability to register as JI. Without this, even the most carefully notated ratios remain abstract numbers, and the music risks ceasing to sound like JI.
All tunable intervals can be expressed as whole-number frequency ratios, but not all ratios are tunable. When the composite wave pattern of an interval becomes too complex, musicians cannot stabilize it—i.e., they cannot tune it. Some ratios are untunable because one of the pitches falls within the critical band of the other; others are untunable because they lie within the tolerance range of ratios expressible with lower integers (Tenney 1984). For instance, the ratio 10935/8192, no matter how precisely it is produced with computer-generated sound, can only be heard as a very slightly out-of-tune 4/3. It cannot generate an identity of its own as 10935/8192.
That said, 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 model of pitch perception that derives from the work of James Tenney as a comprehensive explanation. This model permeates much of the current discourse on Just Intonation. Kyle Gann (2019), for instance, states in the opening of his introductory book on microtonality 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" (5). Robert Hasegawa’s 2008 doctoral dissertation provides another example of this cognitivist orientation: drawing on Tenney’s theoretical framework and expanding it with more recent research in music psychology and cognition, Hasegawa analyzes equal-tempered music by composers such as Scriabin and Schönberg in terms of the just intervals it is assumed to approximate.
My standpoint is not that the cognitivist explanatory model is mistaken. On the contrary, explaining meantone temperament or the spectrally inflected music of Scriabin as attempts to approximate just intervals can yield valuable insights into why such music is both successful and beautiful. Rather than rejecting this model, the cognitivist analyses of Tenney and Hasegawa will be used throughout this text to illuminate important features of pitch perception in justly tuned music. What I want to emphasize at the outset, however, is that this text does not take cognitivism as its foundation or sole theoretical standpoint. It regards it instead as one explanatory framework—one that must be complemented by insights from other perspectives, including socio-cultural, enactive, and ecological theories of perception. As will be shown, an intersubjectively conditioned search for enacted meaning can make possible the actualization of intervals that, according to a strictly cognitivist model—where pitch perception is conceived as a kind of data processing of sounds along a justly tuned grid—should be impossible. While such a search for meaning may not override the process by which a 10935/8192 is perceived as a slightly out-of-tune 4/3, it helps explain how an interval such as the salendro whole step—an interval we will return to later—can be enacted as meaningful in and of itself, rather than as an approximation of its two nearest tunable ratios, 8/7 and 7/6.
To restate the central point: Just Intonation concerns the use of tunable intervals. In other words, a polyphonic piece cannot be considered to employ JI if it entirely lacks tunable intervals. When tunable intervals are absent, the performer is never given the opportunity to tune pitches by ear into the stable, resting composite periodic resonances that form the building blocks of JI. These distinctive sound patterns can all be expressed as simple whole-number ratios, yet even simple ratios may not guarantee tunability if other contextual factors—such as the speed of melodic motion—interfere. When tunability is lacking and intervals are articulated slowly enough for the composite pattern to be audible, they typically exhibit a fast-beating, indistinct, or slightly unfocused quality.
This fast-beating, unfocused quality is characteristic not only of Equal Temperament (ET) but also of many other intonation systems, including the scales used in traditional Sundanese music. Among the first generation of Just Intonation composers, one can sense a pronounced anti-Western—and specifically anti-Classical—sentiment: JI was often proposed as an alternative to Western imperialism. Yet very few traditional musical cultures employ Just Intonation, and the use of fast-beating, indistinct intervals—where pitches never fully fuse into periodic signatures—is far more widespread than in ET alone. It is also crucial to recognize that such indistinct sounds occur even in pieces written with JI accidentals—works that use ratios to notate pitch relationships and are performed on justly tuned instruments. This is because not all music written or performed in this way relies exclusively on tunable intervals. In other words, tunable intervals and rational intervals are not synonymous.
Similarly, a keyboard tuning might be said to lack tunable intervals if the tuner is unable to adjust the pitches to resting, composite, periodic relationships and must instead rely solely on the beats-per-minute (BPM) of interference patterns to achieve accurate intervals. This does not imply that tuning by counting BPM is unreliable—on the contrary, it can often produce greater precision than tuning purely by ear to beatless, justly tuned intervals. The reason is that the quality of 'beatlessness' is itself slightly relative when compared to the numerical exactness of counting beats per second.
To complete the picture, we can also say that when tunable intervals appear only as isolated events within a piece—without tunable connections linking them, as is often the case in much spectral and post-spectral instrumental music—the music may contain tunable intervals but lacks tunable paths. The occasional tunable interval in an otherwise untunable work is not sufficient for the music to be considered in JI.
The act of tuning
Drawing on a wide range of studies, Burns (1998) concluded that the most common way musicians across the world—including those in Western art music—produce musical intervals is by reproducing them from mental representations of how these intervals should sound. (I will leave aside for now the question of what such "mental representations" entail, returning to problematize the notion below.) In many musical traditions, performers produce highly irrational intervals with great accuracy and consistency, even when no tunable references are or have been audible. Through cultural learning and repeated exposure, musicians internalize the characteristic quality of these intervals. Burns summarizes his findings as follows:
"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—that is, Just Intonation. This is a crucial point to recognize. Burns argues that although Western music’s pitch materials—the diatonic scale and the categories of perfect and imperfect consonance—may historically derive from the intervals found among the lower partials of the overtone series, this origin is not reflected in actual performance. In practice, Western musicians do not tune intervals according to these low-integer relationships. Instead, they reproduce pitches according to their stylistically and culturally defined positions within Western tonal music. Even when playing without a piano, these positions tend to approximate the intervals of Equal Temperament (Burns 1998, 246). This is not to say that such musicians play out of tune in the conventional sense, but rather that their conception of what it means to be in tune does not align with the practice of JI. It should, however, be noted that some musical cultures possess culturally defined pitch classes that do approximate JI—the most frequently cited example being Hindustani classical music.
When a JI composer asks a musician to navigate the tunable paths that structure a JI piece, the composer is asking for something fundamentally different from the performer’s habitual way of producing pitches—at least if she is not a JI specialist. The difference is not merely that the pitches themselves are new, i.e., JI is not only an expanded palette of new interval sizes to learn; rather, the method of producing them changes. The performer is no longer simply reproducing culturally defined pitches but actively tuning a tunable path. This alternative way of playing demands a compositional craft markedly distinct from those employed in both contemporary and traditional Western art music. Many contemporary composers have failed to recognize how profoundly this craft must differ, resulting in microtonal works that present performers with extreme difficulty in producing accurate pitches. This, in turn, has given JI an undeserved reputation for being inherently difficult.
Learning to tune
The mere fact that a piece of music provides the performer with a tunable path does not mean that she no longer needs to have learned the musical intervals involved. Tunable paths do not make tuning automatic. To tune rational intervals as they arise within such paths, the performer must practice them extensively and cultivate a sensitivity to their size, composite pattern, and character. Embodied representations of intervals are developed through sensorimotor rehearsal and by engaging the intervals in intersubjectively meaningful contexts—if intersubjective is understood not only as "with other people", but as including the solitary practices that always unfold within intersubjective worlds. In this process, we do not simply learn the auditory 'signatures' of intervals but also how they feel, affectively, as we enact them through our mind–bodies.
Through such practice, we become able to audiate the intervals—to sound them internally—and to couple this audiation with our embodied performance. This process is sensitively described from a performer’s perspective in violinist Mira Benjamin’s doctoral thesis (2019). She depicts a performance scenario as follows:
"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 essential both for microtonal music that uses JI-type intervals and for JI proper. In both cases, the feedback system between audiation, performance, and adjustment or response that Benjamin describes is at play. What distinguishes JI proper from microtonal music that merely uses JI-type intervals is that these learned categories are not simply reproduced as fixed interval sizes; they are taken a step further by being subjected to a process of real-time tuning. In JI, they are actively influenced by "the psychophysical cues that may underlie sensory consonance or harmony" (Burns 1998, 257)—cues that the mere reproduction of learned categories leaves untouched.
Passive and non-conceptual
"Learning an interval"—whatever it might mean to know an interval—is something we do in order to perform it. However, we do not need to learn anything to actualize an interval as a meaningful aspect of music listening—or if we do, it is an extremely rapid and tacit process that occurs without ever becoming an explicit object of attention. In the vocabulary of ecological perception theory, this can be described as a form of passive perceptual learning. 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)
To actualize the aural phenomenon represented by the ratio 147/128—a common JI interpretation of the Sundanese salendro whole step—as a meaningful musical experience, it is not necessary to know that the interval’s ratio is "147/128". For this phenomenon to be adequately actualized, we do not need to hear it as "147/128" or as a salendro whole step. Most listeners who are moved by the beauty of this interval do not experience it as mediated by the cultural tools of theoretical ratio labels or modal terminology. Countless listeners are moved by sounds without knowing their theoretical foundations or having been taught how to categorize or respond to them by others. This may seem a trivial point, yet it must be established early on as we clarify the theoretical foundations of JI.
Luntley (2003) makes a related argument in his article “Non-conceptual Content and the Sound of Music.” Here, it is the dominant V7 chord in tonal music—rather than the salendro whole step—that serves as the example. Luntley observes that novice listeners without theoretical training can perceive the tension and incompleteness of dominant V7 chords without knowing anything about what a dominant chord is. Through a series of arguments, he shows that although these listeners "have a discrimination of the dominant 7th", they "hear something that they cannot pick up conceptually" (Luntley 2003, 421). For Luntley, conceptuality is marked by the capacity of representations to contribute to a subject’s rational organization of behavior—"by figuring in their inferential reasons for belief/action" (421). Since the tension and incompleteness of the V7 chord are experienced by the novice listener yet remain "unavailable to the subject’s inferential reasons (without cognitive enhancement)" (421), such experiences constitute a form of pure phenomenality unmediated by concepts—"an experience with a nonconceptual content" (421). Luntley acknowledges that nonconceptual representations are "extremely difficult to spot because of the reach of concepts into most areas of experience", yet he finds it unsurprising that his examples are "primarily aesthetic" (421). Aesthetic experiences, for Luntley as for Schopenhauer, lie outside the domain of rational organization: they "create a certain impression, but [are] not primarily an experience productive of a rational response" (421).
Although Luntley does not go as far as Schopenhauer in treating musical engagement as paradigmatically nonconceptual, the idea of aesthetic relishing as useless with respect to our ordinary, rational beliefs and actions—that is, to our ordinary willing—certainly points in this direction. Recognizing music as something direct and immediate, Schopenhauer regarded it not merely as a "copy of Ideas". Ideas, he tells us, are the "objectivity" of the Will—that is, they are conceptual. Music, however, actualizes the pure, nonconceptual movement of the Will itself, "the reality underlying and bringing into being the appearances of the world" (Cross 2013, 183). To actualize this level of reality is to suspend ordinary willing: the dualistic, conceptual mode of everyday life in which the ego is experienced as an object among other objects. This view resonates with Luntley’s own conclusion, where he writes 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" (Luntley 2003, 424). For novice listeners, musical attunement may be one such area. For performers, however, who must learn to realize music reliably, conceptual tools—such as theoretical ratio labels or names for intervals like the salendro whole step—remain indispensable, even though they are not what make musical experience meaningful.
Active and conceptual
From these preliminary points, it follows that what musicians seek to learn when studying JI intervals are primarily strategies that enable them to perform those intervals consistently and reliably—not necessarily new ways of hearing them. A performer’s ear, however, always differs from a listener’s ear: performing entails a different mode of hearing. Acquiring the skill of performing intervals means beginning to hear them in specialized ways—for instance, by actively listening for the resultant combination and difference tones that help tune and stabilize the interval. When learning to tune an 11/9, for example, listening for the emerging pitch "2" (the first-degree difference tone) helps the performer establish the interval’s identity. Yet listening for this "2" is not a better way of hearing the interval. It does not reveal the sound more truly or directly; it is simply a practical listening strategy that facilitates tuning. This may be compared to a painter’s perception of color: she knows precisely how to mix two pigments to produce a specific third, and she knows the unique name for that color. Her perception of that color is shaped by practical knowledge and the affective meanings it carries in her craft, but she does not perceive it more vividly, radiantly, in detail, or better than someone without this expertise.
Another useful analogy is that of dialects in speech. We can recognize a previously unfamiliar dialect long before we are told what it is called. We can engage with it as something novel and distinct, and when someone later identifies it as "a Gotlandic dialect", we simply appropriate a label that will henceforth mediate our experience. It is not the case, however, that before acquiring the term Gotlandic dialect we were unable to perceive the speech as distinct from other dialects we had previously encountered. We can hear new sounds even when we do not yet know what they are. This may seem a trivial point, yet it marks a crucial dividing line between sociocultural theorists—who argue for the primacy of linguistic discourse—and proponents of ecological perception, who emphasize the primacy of sensory experience and perceptual learning. A sociocultural theorist might argue that the term Gotlandic dialect is precisely what enables us to hear the dialect in the first place, while an ecological theorist would argue the reverse: that the term becomes meaningful only because there is already something to apply it to. According to both ecological (Gibson & Gibson 1955) and phenomenographic (Marton & Booth 2000) accounts of learning, phenomena arise primarily in relation to prior experience—linguistic categories come later. Language catalogues and categorizes what is already perceived.
The example used by Gibson and Gibson (1955) is that of a wine connoisseur. This expert taster can distinguish a wide range of aspects of a wine’s flavor, and "can consistently apply nouns to the different fluids of a class and […] adjectives to the differences between the fluids" (1955, 35). Yet for the ecological theorist, language merely describes the nuances of experience after the fact. As Wallerstedt (2010) observes, from an ecological perspective a wine’s dry taste can never be understood by someone who has never experienced dryness, whereas sociocultural theorists would argue that the very term dry taste is what enables us to experience it as dry in the first place. From the sociocultural standpoint, the concept of dry exists prior to individual experience, within collective social knowledge, and we come to appropriate it by engaging with the world in ways attuned to how others in our community do so.
The example of the wine expert also appears in McGilchrist (2021), who likewise draws a parallel between the wine connoisseur and the musician. In McGilchrist’s work, the debate between the primacy of language and that of sense perception receives a clear resolution: sense perception is primary. The belief that language could precede perception, he argues, is symptomatic of the modern world’s dominance of the brain’s left hemisphere—its mode of attending through concepts, categories, and symbols. As McGilchrist famously puts it, the emissary has become the master: the subordinate left hemisphere has usurped the authority of the right. Surveying scientific research that supports this distinction between hemispheric modes of attention, he notes that
"[i]n a test of wine recognition, control subjects showed activations predominantly in the right hemisphere. Wine experts, however, used both hemispheres, and at times relied more heavily on the left hemisphere. The investigators put this down to the fact that they 'work simultaneously on sensory quality assessment and on label recognition of wine'. Or at any rate, naming." (McGilchrist 2021, 112).
The right hemisphere, he explains, perceives novelty and direct experience, whereas the left represents the world through language. For non-professional musicians, almost all musical processing—except for basic rhythm—is mediated by the right hemisphere, while "professional musicians have a more bilateral representation of music" (110). "Professional musicians", McGilchrist writes, "perhaps like sommeliers, are more dependent on the left hemisphere than amateurs, because amateurs are encountering newness, where experts are categorising familiarities" (113).
Learning from the sounds
The sociocultural perspective emphasizes that it is difficult to discover on one’s own what is valuable to discern in any given situation; we depend on linguistic descriptions and analyses—on discourses—and on participation in communities of shared action and meaning (Säljö 2000, 62). Taken to its extreme, however, this perspective leads to a strongly anthropocentric form of social constructionism. While I share its view of the world as radically relational and non-realist—where there are no discrete selves and no material reality 'out there' distinct from a mental representation 'in here'—we must also move beyond any dualism that separates 'human' from 'non-human' spheres of existence and influence. If social constructionism claims that the illusion of 'reality' arises only through human social interaction, and that human language is the primary agent in shaping this illusion, it fails to recognize that mountains, the spring breeze, and bronze are also agents—equally empty, illusory, and relational as humans. They, too, participate in the construction of this shared illusion of reality.
It is here that ecological theory again becomes relevant, particularly through its concept of affordances (Gibson 2015). Sounds possess affordances in that they offer relationally constituted possibilities for interaction. When we practice JI intervals, we are not merely learning culturally defined pitches or adopting a social convention dictated by others; we are learning from the sounds themselves. By tuning one pitch to another and searching for their shared resonances, we discover which intervals are tunable and how their composite periodicities feel—directly, through engagement with the sounds. The sounds, not other humans, are the agents that show us what is important to discern. We enter into a community of praxis with sound, moved and directed by it. As Bashō said, we "enter into the object, perceive its delicate life, and feel its feeling (sono bi no awarete jōkanzuru)" (Thornhill 1998, 351). Through such attunement, we become initiated by sound’s nonverbal communication—participating in a community of action and meaning that is more direct than language.
It is for this reason that Timothy Morton considers JI an important aspect 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" by tuning it to JI (Morton 2013). In this sense, JI becomes for Morton "a deliberate attunement to a nonhuman". Yet precisely because reality arises dependently—where neither subject nor object can be said to exist independently—sounds and their affordances cannot be regarded as "invariant", or, as Gibson (2015) phrased it, "always there to be perceived" (130), as though they existed apart from the perceiver. We learn JI from the sounds themselves, but these sounds do not exist 'out there' in a reality standing in dualistic opposition to us. Rather, as Varela, Thompson, and Rosch (2016) have argued, they are 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 mean that they should write scientific treatises on pines and bamboos. The poems that resulted were not objective representations of material reality but expressions of a world in which the poet’s emotions and the natural world fused nondually. These poems embody the aesthetic quality of zōka (造化)—a kind of 'naturalness', but only if we understand naturalness as inseparable from the artist.
In this sense, the intervals of Just Intonation can also be called 'natural'. On one hand, they are not created by musicians, since the "psychophysical cues" that Burns (1998) refers to are not merely social conventions or products of imagination. On the other hand, JI intervals have no independent existence: they arise only through being enacted by musicians. They are not simply discovered, because they cannot be separated from the creative act through which cultural and social beings enact these. In other words, the intervals of Just Intonation arise interdependently—and recognizing this interdependence is precisely what it means to be open to the nonhumanness that Morton describes.
In his seminal 1984 essay John Cage and the Theory of Harmony, James Tenney drew a connection between John Cage and Just Intonation, even though Cage himself never worked directly with it. Tenney located this connection in Cage’s focus on sounds themselves. Like Bashō’s naturalism—free from human ego—Cage envisioned an art that would move "from being a selfish human activity to being […] fluent with nature". Few approaches to working with pitch could be said to be more "fluent with nature", in Cage’s sense of the phrase, than Just Intonation.
In a letter from 1956, Cage succinctly articulated his vision of radical non-foundationalism and relationality—an artist’s attunement to the agency of the nonhuman:
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)
Through the practice of Just Intonation, we come into contact with precisely this interpenetration of phenomena—the nonduality of perceiver and perceived. JI invites us to learn from the act of hearing itself: a mode of listening in which neither the faculty of hearing nor the object of hearing exists independently, but both arise interdependently.
Expert listening
That professional musicians hear music differently from non-trained listeners should come as no surprise. Yet it is important to recognize that the listening tools musicians bring to their engagement with music have been developed for the purpose of performing particular kinds of music reliably. For this reason, such tools can sometimes become maladaptive when applied to the 'wrong' kind of music.
Many learning theorists have drawn on the concept of adaptation to describe the process of learning. von Glasersfeld (1989) argued that Piaget shared much with Dewey and the pragmatists, as both understood knowledge in Darwinian terms of adaptation. Knowledge is not a 'representation' of the world that exists apart from our practical interests; it is shaped precisely by those interests. Unlike some of his cognitivist interpreters, Piaget did not claim (pace Phillips & Soltis 2014) that cognition mirrors the outside world. Rather, cognition is a practical construction born from the need to adapt. Through active engagement, organisms construct the 'laws' and 'structures' of the phenomenal world from recurring patterns of experience. These patterns return to us as apparent external forces, yet they are the product of a relational and dialectical process. Among these constructed patterns must be included the phenomenon of tunability: how many low-integer JI ratios are perceived as still and resting, while more complex ratios produce beating patterns. Many such 'natural' patterns of perception recur across human experience, but this does not make Piaget’s theory universalist (pace Säljö 2000), even though it is often read that way by its critics.
The sociocultural theory, by contrast, does not describe learning as an adaptive process of building internal maps for processing information, as cognitivists such as Piaget did. Instead, it understands experience as mediated by tools that are appropriated in praxis. The key term is appropriation rather than adaptation. Within this theoretical framework, it is not entirely clear whether JI ratios should be regarded primarily as signs (psychological or mental aids) or as intellectual artifacts (tertiary cultural tools). This discussion can, however, be sidestepped by describing them simply as mediating resources (Jakobsson 2012, 155).
Learning an interval such as 147/128 means that the musician appropriates its mediating potential. Once we have learned it, we both hear and think through it in new ways. The 147/128 stands in a non-dualistic relationship to our perception: it is not merely a cognitive category that filters "the great blooming, buzzing confusion" (James 1890, 488) but something through which the world is mediated—something we act through and that, in turn, enables thought (Jakobsson 2012, 153). To emphasize this non-dualism between person, action, and mediating resource, the sociocultural theorist Wertsch proposed the heavily hyphenated expression "individual-operating-with-mediational-means" (1998, 26) as a single analytical unit for describing the agent in action.
Research has shown that individuals trained in Western music perceive frequencies by sorting them into twelve categories corresponding to the twelve chromatic pitches of Western tonality. For instance, frequencies between 60 and 140 cents are typically grouped together as the same interval—a "minor second". These categories are formed through active sensorimotor rehearsal, such as playing an instrument, rather than through passive listening alone. Individuals who have only listened to Western music without performing it show no evidence of this categorical boundary effect (Burns 1998, 229).
Burns' research supports the cognitivist view that aural training leads to the construction of a mental model in long-term memory for sorting and categorizing incoming sound frequencies. Yet the results can also be interpreted through a sociocultural lens: musicians actualize pitches as mediated by the conceptual and embodied structures of the twelve-tone scale. Both frameworks lead to a similar conclusion, though by different routes—namely, that musicians' responses to sound are conditioned and limited by their prior expertise.
Musical training, in this sense, can become maladaptive. Everyday experience offers ample evidence: certain pieces in 11-limit JI may sound completely natural to non-musicians, while Western-trained musicians struggle to hear beyond the impression of "out-of-tune" intervals. Burns’ findings thus suggest that musical expertise cultivates sensitivity to particular kinds of music—but not to all kinds. This recalls the famous saying of Zen master Shunryu Suzuki: "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 experimental research reviewed by Burns. Many Western musicians have access only to vocabularies and discourses that describe twelve discrete tones per octave, but this does not prove the existence of a corresponding cognitive twelve-tone structure that filters perception. I have met many Western musicians who, when asked more closely, clearly hear that a "147/128" differs from both a major second and a minor third, yet they lack any language for describing it other than the terms provided by twelve-tone equal temperament. When such musicians identify the sound as a "major second", this does not necessarily mean they hear it as a Western major second—it may simply be the only available linguistic label.
When these same musicians attempt to reproduce the 147/128 by singing, they often produce either a major second or a minor third. While this might appear to support the idea that they "hear" it through Western categories, we must remember that singing is a motor skill grounded in embodied memory and habituation, not merely in hearing. These musicians have simply not practiced producing this interval; often, they can tell that what they sing is incorrect, but they are unable to adjust it. My point here is that neither singing nor verbal description can be treated as a neutral medium that grants 'access' to an inner experience.
Returning to the example of the Gotlandic dialect: we can hear it perfectly well even if we cannot reproduce it ourselves. Most people can recognize dialects of their native language, but only those with specialized training—such as actors working with dialect coaches—can convincingly imitate one outside their own. It is similar with Just Intonation: we can all hear its distinctive beauty and meaningfulness, but performing it requires a separate, cultivated skill.
In this text so far, I have been gesturing toward what learning such a skill entails. Through direct engagement with sound, it involves tuning stable, resting composite periodic resonances. JI intervals are therefore not primarily learned as culturally defined 'interval sizes' with specific 'microtonal profiles' acquired through convention, but as unique composite sound signatures—phenomena that arise from the fusion of two pitches into a periodic pattern.
When first learning Just Intonation, a musician without prior experience can expect a sharpening of perception, as attention becomes focused on stabilizing the precise periodicities of intervals. The 'dissonances' of Western music, in particular, will demand heightened awareness: small differences suddenly prove to be of great importance. Between the perfect fourth and the perfect fifth, for example, lie five tunable intervals—11/8, 7/5, 10/7, 13/9, and 16/11. Distinguishing and understanding these consonances requires a finer degree of attention than in Western music, where everything between a fourth and a fifth is typically categorized as 'dissonant'.
As I will emphasize throughout this text, however, learning new JI intervals is greatly facilitated when these intervals carry musical meaning for the performer. Simply sitting with a drone and memorizing periodic signatures may not, by itself, be musically meaningful enough for everyone to internalize the sounds. It is far more effective when, for instance, an 11/9 arises in a musical context that gives it a clear expressive function—one in which no other third, such as the neighboring 6/5 or 5/4, would suffice.
Microtonal music that uses JI-type intervals
As mentioned earlier, ratios notate not only specific composite resonance patterns but also indicate particular interval sizes when incorporated into a scalar system. Just Intonation is therefore not only concerned with using ratios to create periodic resonances, but also with forming melodic steps that can be analyzed as belonging to a scale.
For example, the ratio 7/6 can be understood both as a way of tuning sounds into a specific periodic resonance—producing distinctive combination and difference tones—and as a specific interval size with a characteristic microtonal profile. This size is typically measured in cents: 7:6 equals roughly 267 cents, while 6:5 is about 316 cents. Using the language of Western music, both may be described as types of minor thirds, the first being narrow and the second wide.
The ratio 7/6 thus implies both a harmonic relationship (a periodicity in which one sound wave repeats seven times in the same duration that another repeats six times) and an intervallic identity defined by its size and character. Ratios therefore possess a dual nature—harmonic and intervallic—and the JI composer can choose to emphasize one or the other in the compositional process.
Music that is notated with ratios but lacks tunable intervals and tunable paths must rely entirely on the performer's ability to memorize interval sizes—their microtonal profiles. In such cases, only the ratios' intervallic properties are engaged. The performer might practice these intervals with an electronic tuner and reproduce them as accurately as possible in performance. The resulting music, however, does not constitute JI according to the strict definition that has tunability as a requirement. It is more accurately described as microtonal music that uses JI-type intervals. Such music is performed in the same manner as the ordinary music described by Burns, where intervals are reproduced from memory and approximation rather than through tuning a tunable path.
Some composers value the distinctive sizes of rational intervals—such as those found among the first thirteen partials of the overtone series—without concern for whether these intervals fuse into stable, tuned sounds. Many of these interval sizes may simply be regarded as beautiful or affectively powerful, regardless of their exact tuning. Numerous contemporary composers of post-spectral inclination appear to use JI in this way. One such composer, Taylor Brook, writes in the preface to his piece Amalgam (2015) that "the microtones can be understood within a system of just intonation or as written-out bends and slides." Brook acknowledges that even if performers were to conceptualize the pitches as JI, the audible result—given a musical context that never engages tunable paths—would not differ from hearing the pitches as microtonal deviations ('bends and slides') from conventional Western tones. A brief look at the score confirms this: over a D drone, the opening line unfolds as a microtonal sequence of 1/1, 81/80, 64/63, and 33/32—none of which are tunable—yet together they form a distinctive chain of narrow, fast-beating microtonal variations. In such music, JI functions primarily as a method for generating specific interval sizes and microtonal inflections, while the practice of tuning intervals into resting, composite patterns plays little or no role.
Since microtonal music that uses JI-type intervals is so widespread, musicians encountering JI music proper must first learn that the purpose of microtonal inflections of scale steps is to establish relational properties with other scale steps—not to draw attention to the inflection itself. A friend once told me how a prominent harpist in the German 'New Music' community protested against tuning a string two cents lower, insisting that such a change would be imperceptible. The harpist questioned the composer’s intention and dismissed the request as impossibly subtle and unnecessary. What the harpist failed to understand was that the goal was never to create a perceptible microtonal inflection of a single string, but to form a simple, tunable relationship with another. As a microtonal deviation, two cents may indeed be imperceptible, but tuning a 3/2—two cents narrower than the equal-tempered perfect fifth—within a musical context that supports tunability is entirely perceptible.
There is an important difference between the 'traditional' musical practices described by Burns and a piece such as Amalgam. While both rely on approximation and memorization, the processes through which this internalization arises are entirely different. In the first case, it occurs because the musician participates in an intersubjective culture of meaning. The repertoire, the community of performers, and the instruments together create a praxis in which the microtonal profiles of intervals make 'intuitive' sense, even when they are irrational—that is, not expressible as ratios.
When I studied Sundanese music, I was struck by how easy it was to learn to sing the various Sundanese scales and to internalize the microtonal profiles of their scale steps, despite their irrationality and their difference from any other music I had practiced. Were I to notate these scales and hand them to a Western musician reading from a score, she would be unable to reproduce them with the same ease, simply because she would lack access to the praxis in which these intervals are meaningful. For her, learning the same intervals that I so easily absorbed would be painstaking and difficult.
A contemporary composer who wishes to use irrational intervals and still make the music easy to learn therefore faces a considerable—but not impossible—task: the meaningfulness of the intervals must be composed into the piece itself. The work must provide the entire sphere of meaning. Earlier, I wrote that the 11/9 is easier to learn when it appears in a musical context that gives it a clear purpose—where no other third, such as the neighboring 6/5 or 5/4, would suffice. In microtonal music that uses JI-type intervals, the composer must provide every microtonal inflection with a similarly clear purpose and necessity.
Otherwise, musicians may end up practicing the intervals with an electronic tuner, reproducing them as accurately as possible in performance, yet never feeling that they 'understand' why the microtonal profiles are as they are. There are countless examples of such situations in contemporary music. They verge on behavioristic conditioning: the musician trains herself through repetition to reproduce arbitrary intervals accurately, but without being given the opportunity to embody and actualize them as meaningful. Or rather, musicians will inevitably seek to do so anyway—because that is what it means to be a good musician—but only after a painstaking process, and perhaps with meanings quite different from those the composer intended.
It is not my intention to be overly critical of the microtonal 'interval-size-generating' approach to JI described above, but rather to argue for a fundamental definition of JI as that of hearing of tuned sounds. Given how widespread the practice of microtonal music that uses JI-type intervals has become, it is important to emphasize how the JI music discussed in this text differs from it.
As we shall see, however, even music that aspires to form an integrated tunable path—including many of my own compositions—often employs numerous intervals that are, in fact, not tunable. A clear boundary between microtonal music that uses JI-type intervals and what I will call integrated JI is therefore impossible to draw. These represent two idealized poles. In practice, much music moves fluidly between them, alternating between using JI for its tuned, resting sounds and using it for the affective qualities associated with distinct interval sizes—where the precise justness of tuning may be of little significance.
Tunable intervals
Since it is the composer's responsibility to ensure tunability, she must develop a thorough understanding of the real-time tuning processes involved in performance and address two fundamental questions: 'What is a tunable interval?' and 'What is a tunable path?' Among contemporary composers, Marc Sabat has perhaps contributed most significantly toward answering these questions. In addition to his many compositions in Just Intonation, one of his major contributions is the compilation of a list of tunable intervals ordered by difficulty (Sabat and Hayward 2006; see also Sabat 2008/2009). This list, based on tunable intervals above and below A4, may not be universally applicable to every musical context—since tunability varies with register, timbre, and dynamics—but it remains a valuable reference and is 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. Notably, within this first octave’s list of easy intervals, none involve a prime number higher than 7. In the second octave, we can readily add 9/4, 11/4, 13/4, 11/3, and 15/4. It is thus only with the expansion of one octave that two 11-limit intervals—11/4 and 11/3—enter the category of easily tunable intervals.
Among the more difficult intervals to tune within the first octave are 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 note here is that the collection of tunable intervals is small—twenty-six notes per octave in total, of which eleven are considered easy—and that many common intervals are entirely absent from this list, such as the 11-limit neutral second 12/11.
It should also be emphasized that Sabat’s list pertains only to harmonic dyads. As we will see below, when examining the compositional craft of JI more closely, untunable intervals such as 13:11 can nonetheless be made tunable both melodically—by, for instance, moving from an "11/8" to a "13/8" both tuned to a common "1/1"—and harmonically—by having all three pitches sound simultaneously.
The ordering of more and less difficult intervals varies depending on instrument, timbre, and register. For instance, I find that in mid- and high registers, 9/7 is easier to tune than its inversion 14/9, whereas in low registers the opposite holds true: 14/9 becomes easier to tune than 9/7. Individual differences also play a role, as certain intervals may be more or less challenging to learn depending on one’s musical background and perceptual habits.
Despite the general belief that 11/9 is harder to tune than 11/6, I personally found learning 11/9 to be less difficult than 11/6. The 11-limit neutral third (11/9) was relatively easy for me to learn, since a clear "2"—the first-degree difference tone—arose in my inner ear and helped me stabilize the composite interval. By contrast, 11/6 proved more difficult: its first-degree difference tone "5" produced harmonic interference that made it harder to stabilize.
When I first began learning JI, I noticed that the intervals I struggled to master often shared a common feature: their first-degree difference tones created complex harmonic contextualizations. This was especially true of non-5-limit intervals whose first-degree difference tone was "5", with the exception of 13/8, which was easy to tune precisely because of the clarity of its "5". Intervals such as 14/9, 12/7, 11/6, and 16/11 were all difficult for the same reason—their difference tone "5" complicated the task of finding stability within a denser harmonic context and of ‘listening for the 5’. Over time, however, my perception shifted, and 11/6 eventually became easier for me to tune than 11/9.
Sabat’s list is valuable not only when composing for intonating instruments but also when writing for fixed-pitch instruments or electronic sounds, since the set of finitely tunable intervals also constitutes the set of finitely perceivable JI intervals. Tunability must be ensured for both musician and listener. It is logically impossible to conceive of an interval that is perceivable as JI yet untunable, because to perceive an interval as JI is precisely to perceive it as having a stable core—a concomitant 'gravitational pull' that guides and informs tuning. Later in this text, we will encounter examples of complex ratios that, even when tuned with perfect mathematical accuracy, fail to sound like JI.
Tuning to short-term memory and modal fields
If musical attunement consisted solely of present-moment awareness—without the support of retention and short-term memory—the craft of JI would end with the list of tunable intervals. In such a world, the two categories discussed above—microtonal music that uses JI-type intervals and integrated JI—would be mutually exclusive. Whenever a frequency ratio was untunable, the resulting music would fall under microtonal music that uses JI-type intervals; whenever a ratio was tunable, it would belong to integrated JI. The composer’s craft would then extend no further than learning Sabat’s list and examining how it translates across registers, timbres, and instruments.
While we can contend that 'true tunability' is only achievable through simultaneous sounds, we frequently encounter the phenomenon whereby, during performance, tunability seems to seep out from harmonic simultaneities and infuse non-simultaneous pitches—pitches that are, in themselves, untunable—with a weaker yet meaningful sense of tunability. It is precisely this subtle quality, this emergent tunability, that makes the craft of JI more complex than a simple categorization of intervals as tunable or untunable.
This is a subtle point, to be sure. Nothing can replace the immediacy and power of simultaneous tuning, and those who question my use of the term 'tuning' in this context are right to do so. Yet what else could we call it? What other term could describe this phenomenon? Whatever name we choose, it denotes a kind of emergent property—one that enables performers to adjust intervals in a tuning-like manner through the lingering traces of harmonic memory. Despite its subtlety, we will see through concrete examples in this text that this phenomenon has significant implications for both harmonic and melodic writing in JI.
To my present knowledge, there are two distinct phenomena that may be described as forms of 'non-simultaneous' tuning. The first concerns the way certain points in harmonic space appear to exhibit what James Tenney (1984) called 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 this to "a sort of neural resonance in short-term memory". For example, the 9/8 interval is usually only possible to tune in high registers (Sabat 2008/2009). Yet in a musical passage, a mid-range 9:8 melodic motion—say, between the scale degrees "4/3" and "3/2"—may be perceived as if tunable if an emphatic "1/1" has been sustained for some time beforehand, serving as a continuing external reference for both tones. Strictly speaking, however, what we are tuning in this situation is not a 9:8 itself, but a succession of "4/3" followed by "3/2"—two independently tunable intervals whose temporal proximity allows the intervening distance to be apprehended as 9:8.
Secondly, there is the phenomenon that arises once a composition begins to unfold: pitches and harmonies gradually establish a supportive context for tuning. Previously heard pitches support the intonation of present pitches by reinforcing their position in a modality. Performing the sequence of pitches that constitutes a piece thus becomes less a matter of tuning each tone to its immediate predecessor—as in a relay race—and more about relating each tone to the developing modal field. In this way, intervals that are not directly tunable may be perceived as if tunable, since they can be intoned in relation to something other than the immediately preceding notes.
Most musicians would agree that singing prima vista is easier in tonal rather than atonal music, since one need not construct every interval de novo but is guided by an implicit tonal grid. The phenomenon described here is analogous, though the modality that governs precise points of intonation—rather than the approximate pitch classes of Western tonality—is far more fragile and subtle. There is a reason for the cautious language I employ here. The topic of modal intonation will be explored in depth later in this text.
As the temporal distance between sounds increases, so does the margin of error before a note begins to sound out of tune. This can be demonstrated by, for example, playing or singing a pitch—say an A—for a few seconds, then waiting five seconds in silence before sounding a tunable interval below it, such as a 5:4 down (an F+ [2]). We may observe ourselves tuning the second pitch to the short-term retention of the first and, if satisfied with the result, declare that we performed the second note in tune with the first. Yet if we record ourselves and later edit the two sounds to overlap, we may discover that what we perceived as in tune was, in fact, not quite so.
As tunable sounds become separated in time—whether by silence or by intervening tones—pitches will still be heard as being in tune, even when, by measurement, they deviate considerably. It is for this reason that such sounds are described as as if tunable. This as if, however, is ultimately tautological, for in human perception there is never anything other than as if. If JI is defined as the hearing of tuned sounds, it makes no difference whether the sound is '100% in tune' when measured scientifically—which it, in any case, never is in live acoustic music. 'In tune' is not an external property of a measurable acoustic reality, but something enacted in perception itself. If an interval appears as if tunable—if it exerts a gravitational pull toward a core of rest, where pitches above or below sound out of tune—then it is simply tunable.
It should be added that when performing a piece in JI, ample opportunities arise for the musician to doubt whether she is producing a given pitch from a rehearsed, embodied mental representation or tuning it to a previous sound that is beginning to fade into distant memory. Is she tuning to a silent internal image connected to embodied praxis, or to a retained auditory trace? As soon as we move beyond simultaneity and into non-simultaneity, such distinctions quickly become blurred.
Fortunately, it is not the aim of this text to resolve these complexities. My concern is not to determine precisely how out of tune some of these sounds we believe to be in tune may in fact be, nor to decide whether what we call tuning is merely the reproduction of memorized pitch positions. What interests me, rather, is how this kind of non-simultaneous tuning affects the practical work of musicians and composers in constructing a tunable path. If we define tuning as that which exerts a gravitational pull toward a core—a point of rest where any deviation above or below feels indistinct or out of tune—then the distinction between tuning and approximating a representation becomes largely irrelevant.
My purpose here is not to construct a formal theory, but to share observations from practice and the theoretical reflections they have prompted. By demonstrating how non-simultaneous tuning operates in my compositional process (primarily in Figures 27–29 and 11–13), I hope to show that this process is indeed best understood as one of tuning rather than mere approximation.
The balance between tunable and untunable intervals
To exemplify the discussion so far, Figures 1 and 2 present two short melodic examples. Both figures contain the same set of pitches—A, B, C#-, D, E, and G#-—but in different orders. In Figure 1, the sequence is E:B:C#-:A:D:G#-. In Figure 2, the pitches appear as E:D:B:A:C#-:G#-. Despite comprising identical pitch material, the two melodies differ markedly in tunability as the order in which the pitches are presented profoundly influences how tunable relationships emerge.
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 I propose is that the intonation of C#- is aided by the short-term retention of E, to which it forms a 6:5. The musician therefore does not have to rely solely on approximating a 10:9 according to a rehearsed mental image.
A similar situation occurs with the fifth interval, between D and G#- in the fourth measure. While the third interval (C#-:A) is a tunable 5:4 and the fourth (A:D) another tunable 4:3, the fifth interval (D:G#-) forms the untunable 45:32. Rather than merely approximating this 45:32, the performer may retain both A —to which G#- forms a tunable 15:8 —and C#- —to which G#- forms a tunable 3:2 — and tune the G#- in partial reliance upon both of these preceding pitches.
In this way, the performance of G#- is aided by the retention of pitches not immediately adjacent in time. When sounding G#-, it does not feel as if I am simply approximating an untunable 45:32; rather, it feels like I am bringing it forth through an act of tuning in JI.
In Figure 1, even though 40 percent of the intervals are untunable with respect to their immediately preceding pitches, all of them can still be performed through tuning to varying degrees; none must be rendered solely through approximation from memory. Of course, when I observe myself audiating the first 10:9 in anticipation of performing it, I am not only situating it within a nascent modal field but also bringing to the foreground an embodied sense of what that narrow major second feels like in itself. Such approximation from a rehearsed memory remains an important part of producing pitches in JI. In this case, the C#- is performed both by approximating a 10:9 and by relying on the retained E as a tunable reference—a 6/5 to E.
This tuning aspect disappears if the passage begins directly on B, without the retained E. What I propose, therefore, is that the entire passage may still be considered tunable: the necessity of tuning only two out of five intervals to their penultimate pitches is not sufficient to render the line untunable as a whole.
In Figure 2, the proportional relationship between tunable and untunable intervals is reversed. Only 40 percent of the pitches are tunable with respect to their immediately preceding tones, and none of these occur within the first three measures. The opening interval between E and D is an untunable 9:8. Since no other pitches have yet been heard, it can only be performed by approximation from a mental representation. The following interval, between D and B, is an untunable Pythagorean minor third (32:27). The third pitch, B, might receive some intonational support from the retention of the initial E, forming a 4:3 relationship, as in Figure 1. When playing this passage, however, I notice that such retention requires a markedly more active effort: the intervening D acts as a distraction, offering a competing harmonic anchor. I must resist the temptation to treat D as the reference for tuning B. The B should neither be tuned a syntonic comma lower (6:5 to D) nor a septimal comma higher (7:6 to D).
In this way, the second pitch in Figure 2 is more distracting to the accurate tuning of the third than the second pitch in Figure 1 was to its corresponding third. The task of tuning the third tone to the first therefore becomes considerably more difficult. In Figure 1, retaining the first pitch as reference felt passive and almost natural; in Figure 2, it demands conscious, active effort.
Figure 2
This difference in the performer’s degree of active effort in retention points toward the topic of audiation—the internal hearing and projection of pitch within the mind–body before it is sounded. In Figure 1, the process of audiation is supported by the musical context itself: emergent tunability provides an external framework that assists the inner hearing, making performance feel comparatively effortless. In Figure 2, by contrast, audiation must rely more heavily on embodied memory and deliberate projection; it feels less like being supported by the context and more like something that must be actively generated.
Continuing through Figure 2, the third interval presents another untunable 9:8. Despite this untunable relationship between B and A, the A can nevertheless be contextually tuned to the short-term retention of E and D—two recently heard pitches that form perfect consonances with it. This contextual tuning of A occurs with relative ease, without the need for the deliberate mental effort required to retain E when tuning the earlier B. Because the A is preceded by more pitches that have begun to establish a modal, tunable context, it can be tuned more naturally, even though both B and A are untunable to their immediately adjacent tones. Finally, the closing intervals of Figure 2—5:4 from A to C#- and 3:2 from C#- to G#-—are both tunable.
The examples above illustrate that music functioning as a tunable path need not be confined to the strictly tunable intervals listed by Sabat. What is required, however, is a careful balance between tunable and untunable intervals. One might even say that the specialized craft of JI composition lies precisely in mastering this balance. As we have seen, simply changing the order of pitches within a short melodic passage can drastically alter the difficulty of tuning. When performing the two figures above, I find Figure 2 markedly more difficult to tune precisely than Figure 1: the first demonstrates a well-proportioned balance, while the second represents a less optimal one. As we will see in later examples, constructing such a fine balance often entails organizing pitches so that harmonic spaces—or modalities—emerge in which a single tone can become tunable to multiple others through 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 additional notes before it. Experimenting with such additions reveals how the establishment of a larger modal context—a field of supportive pitches—facilitates tuning.
Figure 3
In Figure 3, a G and F#- are added before the E, so that the entire sequence of pitches now reads G:F#-:E:D:B:A:C#-:G#-. In my experience, this addition makes the B even more difficult to tune than before. The reason is that G and F#- provide untunable references to B in the form of a Pythagorean third (81:64) and a wolf fifth (40:27). As noted earlier, the musician may be tempted to tune the B as a B- in order to form a 6:5 with the D. In the version with the added G and F#-, this temptation becomes stronger, since F#- forms a pure 3/2 with B-, and G forms a pure 5/4 with it.
A version using B-, shown in Figure 4, will therefore be significantly easier to tune, as it aligns with the modal context implied by the preceding pitches. In integrated Just Intonation, there is often an implied root—a 1/1 or modal center—underlying a passage. The first four pitches in Figures 3 and 4 (G:F#-:E:D) clearly suggest a modality centered on D. It is D rather than G that serves as the 1/1 of this passage because E lies a "9/8" above it (if the E had been E-, G would have functioned as the implied root).
When visualized in a ratio lattice (Table 1), we can see that B lies farther from 1/1 (D) than B-. The B- sits diagonally adjacent to D and forms a simple, tunable low-number ratio (6/5) with it, whereas B lies three steps to the right and forms the more complex, untunable high-number ratio 27/16. The B is thus positioned farther away from the implied modal center.
Figure 4
Table 1
By conducting practical experiments such as these—adding pitches to short melodic fragments and assessing whether the result becomes easier or more difficult to tune—we come to see for ourselves that it is possible to construct melodies that facilitate non-simultaneous tuning by allowing a form of modality to arise. Pitches support one another’s intonation by reinforcing their mutual tendencies. A context is created in which some tones feel 'correct' to introduce—they sound as if they already belong to the mode—while others feel foreign or unstable. The latter quality can, of course, be exploited compositionally to create meaningful contrasts and modulations.
In the final bar of Figures 3 and 4, a modulation to A major occurs as 5/4 (C#-) and 15/8 (G#-) are built upon A. Because B- lies farther from A major than B, the modulation to A will sound more unexpected in Figure 4 than in Figure 3. In Figure 3, the use of B in place of B- actually helps prevent D from being too strongly established as a modal center, thereby smoothing the subsequent modulation to A.
When discussing the loose style below, we will return to this kind of subtle compositional decision. What we can already see here, however, is that even though Figure 3 is more difficult to tune in itself, there may be good artistic reasons to avoid writing B- simply because it is easier to tune. If our modal intention and goal are oriented toward A, introducing B- would imply another root too strongly. The modulation to A is not made easier to perform by changing B- to B—since C#- belongs to both D and A modalities, and G#- enters as a simple 3:2 above it—but the musical effect of the modulation changes. In Figure 3, the transition to A sounds less surprising and more prepared.
Figure 5
As a final example, Figure 5 presents a context in which the last four measures of Figures 3 and 4 are experienced differently in terms of intonation. Here, the first two pitches are changed to E and G#-, so that the full sequence reads E:G#-:E:D:B:A:C#-:G#-. With E now functioning emphatically as 1/1, the tuning of B presents less of a problem, even though its 32:27 relationship to D remains. The B can now be tuned to the retained E and G#-. New difficulties arise, however: the fourth pitch, D, becomes the principal intonational challenge. This tone lacks any immediate tuning reference and must therefore be produced solely through approximation.
The three styles
The modal fields that emerge from the musical procedures explored above can vary in strength. A strong mode implies that there are precise regions within the octave continuum where pitches must be placed to sound 'correct'. Such a mode creates a powerful sense of tonal gravity: small deviations from the established points of intonation are heard as incorrect or unstable. A strong mode, as we will see in greater detail below, facilitates the intonation of untunable or non-simultaneous intervals by providing a clear reference field, but it also makes the introduction of new pitches—that is, modulation—more difficult.
For instance, if the scale degree "5/4" in a melody such as "9/8":"1/1":"5/4" sounds too low and out of tune, it is likely because of the presence of a strong ["1/1", "9/8", "81/64"] modality. Such a mode is considered strong precisely because it renders an ordinarily consonant interval like "1/1":"5/4" (a 5:4) perceptually unstable. This may seem paradoxical on paper but is a common experience in JI counterpoint. The same strong modal force that causes the 5:4 to sound 'wrong' is also what, elsewhere in a composition, enables the performer to intone an otherwise untunable interval such as "1/1":"81/64".
Compositions that exclusively employ pitches belonging to one strong modal gamut are, in this text, classified as belonging to the strict style.
If the microtonal difference between "5/4" and "81/64" in the melody "81/64":"9/8":"1/1":"5/4" passes by naturally—without sounding jarring or out of place—then no strong modality of the kind associated with the strict style is present. The absence of such modal rigidity, and the listener’s acceptance of fluid intonation, characterizes successful writing in the free style.
Music in the free style does not necessarily lack modalities or pitch hierarchies, but these are far more temporary and unestablished than in the strict style. The stylistic context of ever-shifting, local modal centers fosters an acceptance of pitches continually being replaced by neighboring variants a comma apart. Pieces in the free style typically draw upon a large gamut of pitches; it is not uncommon for a single work to contain more than four comma-distanced variants of one focal pitch class.
The weak modality of the free style offers freedom but requires more carefully constructed tunable paths. Because the performer can rely less on the gravitational stability of strong modal fields, she must instead find tunability among neighboring pitches and within the small, transient hierarchies that arise locally. The result is a tunable path that is more delicate, as it becomes more important that each pitch is played accurately.
This description of the free style may seem complex, and in Lou Harrison’s practice it indeed represented a demanding compositional discipline. Later in this text, however, it will be suggested that the free style, to ears accustomed to Western classical music, may sound the most 'normal': it is the conceptually simpler strict style that to such ears will seem the more unusual and difficult.
Lastly, there are situations where, within the same section of a piece, the melody "81/64":"9/8":"1/1":"5/4" may pass by smoothly without anything sounding jarring, yet in the sequence "5/4":"1/1":"9/8":"81/64":"81/80," the final "81/80" might sound like a mistake—even though it forms a 5:4 with "81/64." In such a case, the third scale degree possesses intonational fluidity: it can function interchangeably as "81/64" or "5/4." To put it in slightly awkward but precise terms, this degree lacks a strong modal anchoring. The root, however, does; it can be heard as "1/1" but not as "81/80."
These kinds of situations—where the reintonation of certain pitch classes feels natural while that of others sounds erroneous, or where some focal pitches carry strong modal commitments and others remain fluid—are characteristic of music in the loose style. As we will see below, it is the structure of modal hierarchy itself that determines which pitches are 'fixed' and which are 'fluid.'
These three modal strengths give rise to distinct compositional treatments of tunable intervals, tunable paths, and melodic writing more broadly. In the following sections, beginning with the strict style, each of these approaches will be examined separately and in greater depth.
Strict style
When Lou Harrison introduced the categories of strict style and free style, his distinction was not based on perceived modal strength but on pre-compositional planning. In the strict style, the composer deliberately works within a limited gamut of frequency ratios, constructing the entire piece from this confined set. The gamut is typically small, and the chosen pitches are relatively far apart.
Although intervals of 35 cents or smaller may occasionally appear in a strict-style work, they do not function as they would in loose or free styles. In those latter contexts, such close intervals may be heard as intonational 'variants' of the same pitch. In the strict style, however, each pitch must retain its identity as a distinct scale degree with a strong modal commitment. Were these small intervals to be perceived merely as variants rather than as discrete pitches, the music would no longer belong to the strict style but would instead align with the loose or free styles.
When writing for instruments with fixed tuning—such as zithers, pianos, or tuned percussion—the strict style is often the most idiomatic and practically feasible approach to Just Intonation. For this reason, it is frequently associated with music for fixed-pitch instruments. For Harrison, the distinction between strict and free styles largely coincided with the distinction between composing for instruments of fixed versus flexible tuning. The strict style, however, is by no means limited to fixed-pitch instrumentation; it can also be employed effectively in works for intonating instruments.
Figure 6
An example of a strict style composition that includes an intonating instrument is my Vårbris, porslinsvas for violin and piano. The gamut used in this piece—repeating in all octaves—is small and consists of A,B,C#-,D,E,F#-, and G#- (1/1, 9/8, 5/4, 4/3, 3/2, 5/3 and 15/8), as shown in Figure 6. It is a simple 5-limit gamut with only one 'comma-level'; that is, only one type of comma is used to alter pitches from their default Pythagorean definitions.
An inventory of the intervals contained in the gamut reveals that there are no tunable minor or major seconds suitable for mid-register use: the 9/8 (found between A/B, F#-/G#-, and D/E) is tunable only in higher registers or when used as a ninth. There are three pure major thirds (5:4) and their inversions (8:5): "1/1":"5/4", "4/3":"5/3", and "3/2":"15/8". Likewise, there are three pure minor thirds (6:5) and their inversions (5:3): "5/4":"3/2", "5/3":"1/1", and "15/8":"9/8". All of these thirds are calm and resting. The only untunable 'third' in this gamut occurs between "9/8" and "4/3"; this minor third of 32/27 is narrow and fast-beating.
The gamut includes numerous pure fourths and fifths, with one exception: the leap between "9/8" (B) and "5/3" (F#-), which forms the untunable 'wolf-fifth' 40:27. This interval easily sounds rough or out-of-tune and must be treated carefully. There are two tunable major sevenths ("1/1":"15/8" and "4/3":"5/4") and one tunable minor seventh ("5/3":"3/2"), alongside three untunable Pythagorean minor sevenths (16:9, between D/E, F#-/G#-, and A/B).
Overall, this gamut contains a balanced mix of tunable and untunable intervals—most notably tunable thirds, fourths, fifths, and sixths, contrasted with untunable major seconds and minor sevenths. Constructing a tunable path from this gamut therefore involves privileging the former over the latter.
Turning to the score, excerpted in Figure 7, we can trace how the melodic writing for the violin is shaped by the tunable paths available within this particular gamut. The first harmonic interval—formed between piano and violin on the downbeat of measure 144—is a tunable 4:3. The violin then descends by a tunable 6:5, forming a tunable 8:5 to the piano’s sustained note from the measure’s downbeat. The piano follows with a descending 5:4 to this violin tone (or a 2:1 to its previous note), after which the violin adds yet another tunable 5:2.
When composing this passage, I was not 'free' to select any pitch from the gamut at will. Because I aimed to maintain a balanced alternation between tunable and untunable intervals, my available options at any given moment were heavily constrained. For instance, in measures 178–179 (also shown in Figure 7), the violin needed to move from B to F#-. To facilitate this, I inserted an A between them—an A that was itself tunable to the same sustained E to which the B had been tuned—so that the F#- would have a tunable reference point and the passage would avoid a direct wolf-fifth.
However—and as will be reiterated throughout this text—music in JI is not confined to the exclusive use of still, tunable consonances, provided 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 sustained as a point of harmonic tension. This works because the interval is thoroughly prepared; it is reached through tunable steps. The B is tunable to all previously sounding pitches—9:4 to A, 10:9 to C#-, and 5:4 to G#—while the D forms a tunable 4:3 to the preceding A. The tense, untunable 27/16 is then resolved to a tunable unison on C#- in measure 161, which itself forms a tunable 15:8 to the D.
Because the Pythagorean interval is approached through tunable relationships rather than apporached directly from nowhere, it does not sound jarring but instead reveals itself as a natural tension inherent in, and characteristic of, the mode. It becomes part of what lends poetic meaning to the mode. The gamut thus dictates not only which pitches and intervals create a sense of repose, but also which generate tension.
Figure 7
Qualia in strict style
One consequence of this kind of restricted and highly delimited melodic writing in the strict style is that each scale degree acquires a distinct, characteristic function and affective quality. Each tone becomes imbued with its own specific qualia—a unique color or flavor—because its relationships to the other tones of the scale are so particular and narrowly defined.
One should, of course, remember that a similar situation—where each tone in a scale possesses its own distinctive quale—has also been observed in diatonic scales within ET. David Huron (2006) writes extensively about the differing qualia of scale tones in ET, noting that "each scale tone appears to evoke a different psychological flavor or feeling" (144). Interestingly, Huron argues that this effect is present only in listeners with sufficient exposure to Western music, since "several of the most salient qualia evoked by different pitches can be traced to simple statistical relationships" (Huron 2006, 173).
That qualia are not found 'out there' in the intervals themselves but arise as "relational properties constituted by the environment and the perceiving-acting subject" (Thompson 1995, 286) is a widely accepted view. As Thompson explains, "[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" (Thompson 1995, 286).
While this holds true for all qualia, my hypothesis is that the qualia of strict style JI are more immediate than those of ET.
With the unequal scales of strict style JI, each pitch exhibits a distinct behavioral pattern based on the unique relationships it forms with the other tones in the gamut, whereas in the equally tempered scale, such meaningful differences have been leveled (tempered) out. Because of this, the differences between the qualia of ET pitches depend solely on exposure and stylistic conditioning—on the listener’s internalized statistical associations—while the qualia of JI also arise from acoustic distinctions.
For example, in ET, the interval between the first and second degrees of a major scale is identical in size to that between the fifth and sixth (200 cents), whereas in a typical 5-limit scale, such as the one used in Vårbris, porslinsvas, these intervals differ (203.91 and 182.40 cents, respectively). In ET, one can move melodically between these four tones (scale degrees 1,2,5,6) with equal consonance, but in JI, the same motion between the second and sixth cannot be made consonantly. As a strict style gamut unfolds within a piece, the affective differences among scale degrees become vividly pronounced precisely because of such inequalities and constraints.
Whereas the qualia of Western tonal music, as Huron suggests, depend on stylistic knowledge acquired through long exposure, the qualia of strict style JI are available through passive perceptual learning from the piece itself. In Vårbris, porslinsvas, for instance, the B acquires a particular quale because it forms a pure fifth with the E; it can rest with C#- as a 9/5 minor seventh but not as a 10/9 major second; and it remains unsettled in relation to the tonic if played as 16/9 (B/A) but can rest with it when intoned as 9/4 (A/B). These relationships, articulated through the unfolding of the piece, endow each pitch with its quale and give the mode its rasa.
The B is modally 'correct' to the extent that—even in measure 172, where a B- would have been more 'in tune' with the simultaneously sounded D—the Pythagorean minor third B:D feels right and expected; its meaning-making role is precisely to be untunable here. It would be an exaggeration to claim that occasionally substituting "9/8" (B) with "10/9" (B-) would sound jarring, but it would nonetheless dilute the unique affect that perfumes this piece. It would, in this context, sound peculiar. Yet if used deliberately, such a simple comma alteration can exert a pronounced modulatory effect—comparable, in expressive force, to a modulation between key signatures in ET. This phenomenon will be examined further when we turn to the loose style below.
The distinct intervallic pattern of the gamut gives rise to one of the characteristic features of strict style pieces—a clearly defined underlying affective or atmospheric field. Such works often possess a perceptible rasa or Stimmung. It would, however, be misleading to suggest that this rasa is somehow built into the gamut itself: not every piece written within the same gamut will share the same affective quality.
As in other musical traditions that prize modal and affective nuance—Renaissance polyphony and Hindustani classical music—affects and rasas emerge through the particular movement of tones. In Renaissance music, the affect was not inherent in the modes themselves (Dorian, Mixolydian, etc.) but arose through compositional constraint: by limiting melodic contours and leaps to specific scale degrees. Likewise, in North Indian Hindustani music—based on rational intervals and, like JI, characterized by unequal scales—the rasa does not reside in the scale per se. It arises from how the degrees are combined and from the particular paths taken through them (Mathur et al. 2015).
In strict style JI, the rasa functions in a similar way: it is not inherent in the gamut, but in the compositional constraints the gamut imposes, and in the ways these restrictions compel the composer to engage with the predetermined tunable paths. Through this disciplined relationality, the music acquires its distinctive rasa—its clear and tangible Stimmung.
Tuning to the affect in strict style
At this point, it is valuable to acknowledge that any piece—whether written in JI or in an irrational system of intonation—will have its intonation facilitated by the two aspects identified here as central features of the strict style.
Firstly, because the number of pitches is small and they recur frequently, the musician quickly learns and internalizes their positions. Repetition fosters the formation of stable mental representations of the gamut, allowing the performer to 'tune' contextually—to the gamut as a whole—rather than solely in a local sense, relative to immediately preceding tones. Numerous studies have shown that both the establishment of stable interrelationships (Krumhansl 1979) and the frequent recurrence of pitches (Deutsch 1972, 1982; Dewar, Cuddy, & Mewhort 1977) strengthen this effect.
Secondly, the distinct qualia of each scale degree further facilitate intonation. One can even speak of the musician as 'tuning to the affect' associated with each pitch. Importantly, these benefits do not depend on the intervals being tuned to low-integer ratios. When I studied Sundanese vocal music, both of these aspects—the limited pitch set and the affective familiarity of each tone—were crucial for accurate intonation, even though the intervals themselves were far from justly tuned.
What JI adds to this picture is twofold. On the one hand, just intervals, by virtue of being tunable into composite and resting wholes, possess more clearly defined acoustic identities. They are therefore easier to learn than irrational intervals: asking a musician to master a 7/6 (267 cents) is less demanding than asking her to reproduce a narrow third of 275 cents. On the other hand, JI makes possible the emergence of the modal field effect described in the previous section—where pitches support one another’s intonation by reinforcing their mutual tendencies. Tunability seeps outward from harmonic simultaneities and guides the placement of subsequent notes within the modal framework. We saw an example of this in Figure 4.
Having the gamut tuned in JI therefore greatly facilitates accurate performance. This echoes Ben Johnston's argument that just intervals are simply more intelligible than irrational ones. Employing ratio-scale thinking enhances both clarity and coherence 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 why the strict style is the form of JI in which the composer can afford to be most careless when constructing tunable paths. Figure 2, previously described as having an 'imperfect' balance between tunable and untunable intervals, could still function moderately well within a strict style composition for the simple reason that a strong modality renders the act of playing this music less a matter of tuning each pitch individually and more a matter of relating them to a unified modal field and its emergent tunability.
This does not, however, mean that an interval such as 81/64—the so-called Pythagorean major third—becomes tunable merely by virtue of appearing in the strict style. The balance between tunable and untunable intervals must still be maintained: if the proportion tilts too far, the music risks becoming entirely untunable. For instance, performers should not be asked to sustain untunable intervals for too long, since the retention of the necessary reference points required to tune these complex intervals fades quickly when sustaining them.
The relative carelessness afforded by the strict style simply means that untunable intervals can be introduced more directly and with less preparation than in the other styles. We need not always be as cautious as in measure 160 of Figure 7, where the performance of a Pythagorean major sixth (27/16) was carefully prepared through a series of directly tunable steps.
Balancing between the strict style and microtonal music
The risk we run by not exercising the degree of caution exemplified in Figure 7 is that complex intervals may become detached from their harmonic origins and their positions within a tunable matrix. Since such intervals lack stable tunable identities of their own, they depend on their modal context for intelligibility within JI. When the context fails to provide this support, the untunable intervals cease to participate in any emergent tunability; they instead function merely as scale degrees with particular microtonal inflections. In Johnston’s terms, we have then shifted from ratio-scale listening to interval-scale listening.
In other words, the intervals no longer sound as being in JI but rather resemble microtonal music that uses JI-type intervals. Within a strict style piece, the frequent repetition of a small number of non-tempered scale degrees still makes such microtonal JI-derived intervals relatively easy to play accurately—but we can no longer claim to hear them as JI.
Consider again the Pythagorean third (81/64). Because it is only a syntonic comma wider than the simple 5/4, it can easily be perceived as a 'too wide', slightly out-of-tune variant of it. In integrated JI—across all styles—the composer who wishes to employ an 81/64 without having it heard as a 'bad 5/4' must use careful melodic and contrapuntal design, taking into account both tunable paths and modal strength.
The task is essentially pedagogical: the composer must enable the performer and listener to grasp, through passive perceptual learning, that an aggregate built from a Pythagorean third is not meant to form a major third (5/4) or a septimal third (9/7). Instead, it should be enacted as two tones whose individual melodic trajectories and positions within temporary modal frameworks yield a distinct ditone—an interval with its own character and function. Achieving this requires the clear establishment of tuning references to which each pitch relates. In short, the 81/64 must be carefully prepared.
Writing an 81/64 to be heard as a ditone is especially challenging in the free style, where a constant micro-adjustment of pitches by commas creates a stylistic environment in which the 5/4 variant is always a possible alternative. In the strict style, however—where such alterations are not stylistically permitted—an ill-prepared 81/64 is not necessarily heard as an out-of-tune 5/4, even if it is not fully articulated as a ditone either. The rapid beating—the 'out-of-tune-ness'—that arises between the component pitches of such an 81/64 can, in fact, still be perceived as a defining quality that contributes to the piece's particular mood or rasa, much like the 32/27 encountered in Figure 7.
In the strict style, the 81/64 may thus be enacted by the listener as two tones which, by virtue of their fixed positions within a strong gamut, form a kind of 'third' characterized precisely by its fast-beating quality. In other words, even as the 81/64 ceases to function as a ditone (in ratio-scale listening) and begins to behave as a 'wide third' (in interval-scale listening), it can still sound correct within the strict style due to the constrained and characteristic modal field in which it occurs.
At such moments, however, it is important to recognize that we have stepped outside integrated JI and entered the realm of microtonal music that uses JI-type intervals—even if the resulting 81/64 is perceived as a 'correct' and contextually appropriate sound.
Additional caution should be exercised when composing music reminiscent of Western tonal idioms, for in such contexts the strict style proves severely limited. Tonal composition requires a far greater degree of intonational flexibility to accommodate its many possible chord formations. It is no coincidence that the rise of tonal harmony coincided with the development of keyboard temperaments: a modal system based on pure ratios simply cannot absorb the numerous commas required for tonal music.
Because of our ingrained stylistic expectations of tonal music, transposing a 'tonal' passage into a strict style gamut—such as the one shown in Figure 7—often makes it sound out of tune. When the musical texture becomes too chordal or too reminiscent of functional harmony, the "81" in a 64/81/96 formation easily begins to sound like an out-of-tune third, unredeemed by either strong modality or affect.
Figure 8
An example of 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 final section creates a musical context in which the Pythagorean major third between C and E in measure 543 (see Figure 8) does not sound excessively wide but instead precisely appropriate to the mode and its affect. Adjusting this interval to 5/4 sounds contextually incorrect. When performers accidentally do so in rehearsal, the resulting sonority is immediately recognizable as wrong—despite their having 'corrected' a non-tunable interval to a tunable one.
Although the notated pitches in measure 543 appear to form a conventional C-major triad in second inversion (G–C–E), the music—if accurately intoned—does not sound chordal enough for the E to benefit from being lowered by a syntonic comma. The modal context is one in which this 'C-major chord with a high E' sounds precisely right.
It can furthermore be noted that careful orchestration can provide vital intonational support in passages such as these. In this section of Vid stenmuren blir tanken blomma, open strings and their natural harmonics are used extensively to facilitate a predominantly Pythagorean intonation: only these pitches and their octave transpositions appear throughout.
In the violin dyad of measure 543, the E functions as the second harmonic of the violin’s first string. In measures 541–542, it sounds together with the second harmonic of the violin’s second string (A), forming a 3:2. The pitch C ultimately derives from the viola’s fourth string but appears here as a stopped note forming a 4:3 with the viola’s G—performed as the first harmonic on the third string. The simultaneous presence of the tone a 3:2 above A and the tone a 4:3 above G produces the perceptual effect of a ditone, not an out-of-tune 5/4.
That the E enters last and is played as a harmonic further supports its correct intonation by clarifying its overtone relationships. The violinist first tunes the C to the preceding and simultaneous G as a 3/2, and only then adds the E. Of course, playing the E as a harmonic does not guarantee perfect intonation, as the pitch of harmonics can be subtly adjusted. Nevertheless, their use is invaluable in rehearsal, where they help the musicians internalize and stabilize the intended harmonic relationships by ear.
Two types of strict styles
As noted above, the strict style most commonly appears in pieces for fixed-pitch instruments. The craft of writing for fixed-pitch instruments differs from writing for intonating instruments in several important ways.
First, because of the clarity and mechanical precision of fixed intonation, the composer can write passages of high intonational complexity—passages that would be impractical or even impossible for intonating performers to execute with comparable clarity—and still hear them rendered with exact tuning.
Second, the difference lies not only in the composer’s assurance of accuracy but also in how we listen to such instruments. This is due in part to musical convention: listeners are accustomed to hearing tempered and irrational intervals on keyboards, metallophones, and other pre-tuned instruments. Yet this is not the whole reason. When listening to music performed on a fixed-pitch instrument, we also hear that the performer has no agency over the tuning—the pitches sound automatic, devoid of the micro-adjustments and expressive fluctuations that signal intonational intent. As a result, the listener does not evaluate the intonation in the same way as when hearing an intonating musician.
Unprepared uses of untunable intervals—such as direct Pythagorean sixths or thirds—that would sound conspicuously out of tune on intonating instruments rarely appear as audible 'problems' here. A 128/81, for example, is far less likely to be perceived as a mistuned 8/5 when played on a fixed-pitch instrument than when performed by intonating musicians.
Because of these two factors, strict style pieces written for fixed-pitch instruments often exhibit a distinctly different character from those written for intonating musicians. To account for this difference, the strict style is here subdivided into two categories: the integrated strict style—paradigmatically written for intonating musicians and designed with their limitations in mind when constructing tunable paths (as exemplified by Vårbris, porslinsvas)—and the strict style as tuning, paradigmatically written for fixed-pitch instruments.
Strict style as a 'tuning' and its two sub-categories
Simply transcribing an integrated strict style piece such as Vårbris, porslinsvas for a fixed-pitch instrument—as in the fourth movement of the kacapi piece Andra Segel—does not in itself yield a piece that employs strict style as a 'tuning'. The transcription still retains all the tunable paths that make it performable on an intonating instrument.
What is meant by strict style as a 'tuning' are those pieces in which a composer tunes an instrument to specific just intervals and writes music for that tuning, but without needing to consider tunable intervals and paths in the same way she would if composing for intonating musicians. Since the accurate tuning is already built into the instrument, there is no need to ensure tunability through compositional design.
It would be a mistake, however, to regard this type of strict style as inferior to the integrated strict style. On the contrary, it opens up artistic possibilities that are impossible to achieve when the composer must constantly account for tunable paths. Indeed, some of the pieces we now consider among the great ‘masterworks’ of Just Intonation belong to this very idiom.
Within strict style as a ‘tuning’, an enormous range of compositional practices and musical types is encompassed. Many—but not all—works written for precisely tuned instruments can be situated somewhere along this spectrum.
At one end of the spectrum lie pieces such as La Monte Young’s The Well-Tuned Piano and Michael Harrison’s Revelation. I refer to this end as the informed style of strict style as a ‘tuning’, since the pitches in these works arise within tunable contexts. Not every interval may be directly tunable, yet the tones are perceived as emerging from a latent, justly tuned modal field. These compositions thus embody and reveal a profound understanding of their tuning systems.
At the other end lie works such as my Stenskrift for justly tuned piano. In such pieces, pitches do not clearly arise within tunable contexts. The piano is justly tuned, but rather than treating the pitches as points within intricate webs of tunable relations—overlapping overtone structures or latent modal hierarchies—they are used more for their affective qualia than for their relationships within harmonic structures. The justly tuned instrument is thus approached less as a vehicle for exploring tunable relationships and more as a found object. I refer to this end of the spectrum as the rough style of strict style as a ‘tuning’.
The rough style of strict style as a 'tuning'
When writing for intonating instruments, a compositional approach that disregards tunability and treats ratios merely as a collection of pitches often ends up sounding like microtonal music that uses JI-type intervals. Applying the same compositional strategy to instruments that are already tuned, such as keyboards and zithers, however, tends to yield a somewhat different sonic outcome. The resulting music frequently retains the quality of being in JI even when it contains only incidental tunable paths or long stretches without directly tunable intervals. The precision of the fixed tuning exerts a subtle yet powerful influence: it perfumes the music with the sound of JI in a way that distinguishes it from microtonal music that uses JI-type intervals.
In other words, the exactness of the tuning allows the music—such as that in Figures 9 and 11—to continue sounding like JI, something that would not occur, or at least not as naturally or effortlessly, if the same music were performed by intonating musicians. It is for this reason that a clear difference must be maintained 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 merely because it appears less informed about articulating a justly tuned harmonic space—the tunable origin of pitches. Both ends of the spectrum outlined above possess distinct merits and artistic potential. In my own compositions, I have explored both. In Stenskrift for piano, I employed a 5-limit JI tuning in this rougher or ‘uninformed’ way to achieve a specific effect. Listening to the piece, one hears not only a disregard for tunable paths and for the tunability of simultaneous sounds, but also a disregard for the modal functions and origins of the pitches within the tuning (what this means will become clearer below).
Yet this disregard is not audible as a problem, because the music is not constructed from modal harmony or contrapuntal melody but from aggregates and complex, often chromatic, sound-blocks (see Figure 9). Owing to their mixture of tunable and untunable intervals, the aggregates in Stenskrift exhibit differing degrees of stillness and activity. The oscillation between active aggregates—whose predominantly Pythagorean thirds and sixths produce fast beating—and resting aggregates—whose Ptolemaic thirds and sixths embody stillness—creates the music’s breathing pattern and forms its syntax.
It almost sounds as though the composer had found the piano already tuned in this way and allowed the points of rest and tension discovered within the tuning to dictate the music’s flow. The phrasing and the tuning thus speak with a single voice, pursuing the same aim despite the apparent lack of overt awareness of the tuning’s underlying structure.
This rough treatment of JI is not without its own qualities. Certain types of chordal relationships, sharp modulations, and chromatic movements that would be impossible to conceive when thinking in terms of tunable paths become available through this approach. Its unpredictable, thorny, and unpolished nature possesses a poetic force of its own—one that can be not only effective but essential to many compositions.
Figure 10 presents another example from a strict style piece—Ribuan Matahari for kacapi and violin—that leans toward the rough style. In this work, most intervals lose their harmonic, justly tuned functions and operate instead as microtonal intervals. The scale itself derives from creating JI approximations of my piece Laras Sunyi for gamelan degung, tuned to a traditional Sundanese tuning. Like Harrison’s JI gamelan works mentioned above, the piece treats its collection of pitches more as a Sundanese laras (scale) than as a system of JI ratios.
Even though a tunable path can in theory be traced between all pitches in the gamut, these tunable intervals are never articulated clearly enough for the listener to perceive them as such. The minor third between the violin’s opening two notes is a simple, tunable 6/5, and in measure 7 the kacapi’s bass line outlines a tunable minor ninth (11/5). In practice, however, it is the interval sizes—not their composite periodicities—that carry musical meaning. The pitches lose their function as tunable intervals and instead act almost entirely as scalar tones. Yet the sound of JI persists, perfuming the music with its unmistakable resonance.
The informed style of strict style as a 'tuning'
In Stenskrift, many intervals are so complex and widely spaced in harmonic space that they cease to sound like JI at all. The notated ratios, though rational, lack the contextual support that would allow them to be heard as such. Some intervals could theoretically be reinterpreted as simpler—yet still untunable—ratios (the interval between Bb and F#- is 405.87 cents–1.95 cents out of tune from 81:64), but since the piece provides no modal framework through which to make sense of them, they appear merely as microtonal inflections. The result is a sound world where many pitches only barely retain the quality of JI.
In contrast, a defining feature of the informed style is that even highly complex intervals continue to sound in accordance with their rational notation: the original modal functions of pitches remain perceptible. La Monte Young’s The Well-Tuned Piano exemplifies this distinction. As Robert Hasegawa (2008) observes, complex intervals such as 256/147, 147/128, 49/32, and 63/32 would, outside of Young’s harmonic context, likely be heard as out-of-tune versions of simpler ratios like 7/4, 8/7, 3/2, or 2/1. He writes that when we hear 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 exists "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 8/7—a relatively large comma that can itself be obtained by stacking two smaller commas often used to create 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 8/7 but precisely as 147/128. What is 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 precisely by this quality that Hasegawa identifies in Young’s work: the music provides the ear with perceptual footholds, enabling it to follow the harmonic steps and grasp intervals that, in another context, might be reinterpreted as simpler ratios.
It is not, of course, that the ear literally traces these steps one by one; rather, the harmonic field of the piece supports the listener in enacting complex intervals that would otherwise collapse into out-of-tune versions of simpler ones. This reasoning should sound familiar from our earlier discussion of the 81/64 in Vid stenmuren blir tanken blomma: there, too, the composer must guide the listener toward perceiving the interval as a ditone rather than a 'third' in order for it to remain intelligible as JI. In both cases, 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 my Andra Segel for kacapi siter. This passage exemplifies the informed style of strict style as a ‘tuning’. All intervals lie only a few steps apart in the ratio lattice; even the most distant tones—"8/7" and "35/32"—are connected by just four tunable steps (7/4 + 7/4 + 5/4 + 3/2). Such a compact harmonic space, in which every pitch can be reached through a short series of tunable relationships, allows the music to articulate its harmonic connections with clarity and ease.
For example, "35/32" relates to "8/7" as the major third of the seventh's seventh, a relationship that, while never conceptually calculated by the listener, becomes passively and intuitively graspable through the flow of the music itself.
What distinguishes this movement from Andra Segel—and The Well-Tuned Piano as well—from integrated strict style pieces is the high degree of harmonic and idiomatic complexity, which makes such music impractical to transcribe for an intonating ensemble. Although the tuning paths appear relatively straightforward on paper, strict style as a ‘tuning’ pieces are often playable only on fixed-pitch instruments because of the complexities that arise from instrumental idioms and composite intervallic structures.
For instance, the first three notes in Figure 11 are formed by stacking two 7/4 intervals. While each 7/4 is easily tunable, the frame interval between the first and third pitch is 49/16—a sixth tone higher than a perfect fifth plus an octave. When sustained, this interval produces significant beating and a perceptible gravitational pull toward an implicit "3:1", making it challenging for an intonating performer despite its seemingly simple construction and the clarity which with it resounds on the kacapi.
Later in the excerpt, the interval between the fourth and fifth pitches is a familiar, tunable minor sixth (8/5), but the span between the third and fifth pitches forms an untunable tritone of 626 cents—8 cents sharp of 10/7 and 10 cents flat of 13/9—again producing prominent beating that could interfere with the tuning of the otherwise stable 8/5. On a fixed-pitch instrument, however, such passages pose no practical or perceptual difficulty: both performer and listener experience them as clear and coherent.
This clarity is precisely what gives strict style as a ‘tuning’ its distinctive musical appeal. Freed from the practical constraints of tunability, the composer can exploit highly complex harmonic constellations, rapid successions of beating intervals, and dense contrapuntal textures that would be untenable in music for intonating performers. The fixed-pitch instrument becomes a field in which the pure ratios themselves—rather than their performative tunability—can unfold as intricate, luminous sonorities.
Yet the fast and complex music in Figure 11 appears only toward the end of the movement and is preceded by a slow unfolding in which pitches are gradually added to the harmonic space. Formal features such as these profoundly influence how the pitches are perceived: the slow, deliberate establishment of the modality enables the later, faster passages to remain intelligible. Because the modality has been so clearly established by this point, the third pitch retains its identity within the harmonic space even though it forms an untunable 294/128 (a 147/128 with an added octave) to the preceding note. As discussed earlier, this interval is just 8.4 cents narrower than the equally untunable 16/7. The same pitch also forms an untunable 21/16 to the phrase’s second note.
When played in perfectly accurate tuning, however, and after the modality has been fully established, the ear apprehends the fourth pitch as a 4/3 to the phrase’s first note, and thus as forming a 294/128 with the third. Because we do not hear this fourth pitch as an out-of-tune 16/7, an out-of-tune 9/4 to the third pitch, or an out-of-tune 4/3 to the second, we can say that the piece exemplifies the informed style of strict style as a ‘tuning’.
Composing the context 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 can tolerate greater mistuning while still retaining their identity, whereas more complex intervals require far greater precision to be recognized as themselves. In other words, an interval such as 147/128 has a much smaller margin of perceptual error than 8/7 and therefore demands more exact tuning.
When writing for fixed-pitch instruments, the composer can rely on this precision and thus take greater liberties in melodic writing. Strict style as a 'tuning' allows the composer to work with those intervals that are highly sensitive to mistuning yet remain audible as themselves—rather than as mistuned simpler intervals—when tuned with great exactness. Precision alone, however, is not sufficient: the interval must also arise within a musical context that guides the listener’s ear "securely" toward grasping the complex ratio. The surrounding music must support the perception of a 147/128 as 147/128, and not as a slightly mistuned 8/7.
What the process of composing such contextual support looks like can be observed by comparing Figures 12 and 14, which present the unfinished and finished versions of the same passage from the keyboard piece Nattviol, nattviol. This work might be described as one that aspires to the informed style of strict style as a ‘tuning’ but frequently lapses into the rough style. One of the main factors behind this slippage is the chromatic idiom that pervades the piece. Precisely because of the larger, twelve-tone gamut involved, it becomes difficult for the ear and mind to make sense of the untunable intervals and to actualize them as JI intervals—unlike the 294/128 in Andra Segel or the 81/64 in Vid stenmuren blir tanken blomma.
In this chromatic idiom, no strong modality arises to guide the perception of untunable intervals as tunable. Twelve chromatic pitches may simply be too many to cohere into a unified modal field. Figure 12 shows an early version of the final section of Nattviol, nattviol.
Figure 12
At the point in the piece from which this excerpt is taken, the listener has already spent nearly fifteen minutes within this tuning, hearing it articulated both through tunable paths and through untunable leaps. All the pitches of the gamut shown in Figure 13 have been clearly established. Yet when the "9/7" (the C#↑) appears in measure 259 as a 24/7 (8/7 + two octaves octave) to the low B, it strikes the ear as out of tune—as though a "9/2" were expected in its place. Even though the listener has repeatedly encountered the pitch class "9/7" as the seventh below "9/8", its untunable inversion still resists acceptance: it sounds awkward, unsettled.
This moment reveals that the music is far more sensitive to the use of untunable intervals than Andra Segel, where such inversions were more readily absorbed without sounding 'wrong'. It also shows that this passage indeed belongs to the informed style: although other parts of the piece may lean toward the rough, the ear’s refusal to hear this interval merely as a microtonally inflected scale step—and its insistence on making harmonic sense of it—demonstrates that the conditions for informed-style listening have been established.
Figure 13
The solution to the problem of the out-of-tune-sounding "9/7" was to add another B ("9/8") an octave above the "9/7", articulating it slightly after the lower B ("9/8") by means of arpeggiation rather than a simultaneous attack, and sustaining this upper B over the "9/7". The interval between the high B and the C#↑ is a tunable 7/4. In this revised version, shown in Figure 14, the ear receives sufficient contextual support to perceive the C#↑ as a “9/7”—that is, as forming a 24/7 to the low B—rather than as an out-of-tune ninth (9/2).
The contrast between Figures 12 and 14 demonstrates how a composer must work to produce music that genuinely sounds in the informed style of strict style as a ‘tuning’. Without this intervention, the passage in Figure 12 leans toward the rough style: it sounds more microtonal than just, even though the tuning of the instrument is identical in both versions.
Figure 14
This example from Nattviol, nattviol demonstrates that writing in the informed style of strict style as a ‘tuning’ requires not only precise tuning but also careful compositional support. The listener’s ability to perceive complex ratios as themselves—rather than as mistuned simpler intervals—depends on the surrounding harmonic context, on how the piece guides the ear through its lattice of relations. The composer’s task is thus less to display the tuning system than to compose the conditions under which it can be heard.
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