[Continuation from Varieties of Just Intonation]
Moving between rough and informed idioms
The second movement continues in the same way with another untunable scale, F major, and the third movement, titled In Nomine, also begins with mainly untunable chords and intervals; it gives special prominence to the Pythagorean minor third between D ("4/3") and B ("9/8"). A few minutes into the third movement, however, the inclusion of tunable 5-limit pitches ("5/4" and "5/3") to form 5-limit harmonic and melodic intervals transforms the music. At this point, the music starts to sound more like the informed style of strict style as a 'tuning', or even very close to true integrated JI music in D major ["4/3", "3/2", "5/3", "16/9", "1/1", "9/8", "5/4"]. In the third measure of Figure 16, the music is entirely tunable: an initial D and F#- just major third is followed by a C#- and A just minor sixth and then followed by a D as a perfect fifth (3/2) below the A.
The D major scale of In Nomine differs from the A major scale of Vårbris, porslinsvas (Figure 6) in that the sixth degree is a Pythagorean sixth (27/16), whereas A major had a just sixth (5/3). In other words, it does not have as many tunable intervals as the A major but is very close. Yet, this difference is what situates this movement between the informed style and the rough style. Many passages, such as the third measure from this excerpted section of In Nomine, could be performed by intonating musicians. Yet, the carefree usage of the sixth scale degree, B, still reveals that this music was written for a pre-tuned keyboard. For example, immediately following that tunable third measure, the fourth measure of Figure 16 presents a blatant Pythagorean third (G and B). The G is tunable to the preceding D, but the B would have to have been a B- to be tunable to the preceding notes. Another example of this kind of carefreeness can be seen in the simultaneous attack of the Pythagorean minor third B/D directly following an F#- (wolf-fifth to B) in measure 84 in Figure 16. These moments, all involving B, move the music closer to the rough style because they create an uneven (remember that this is the sense in which the word rough is used) distribution of tunable and untunable intervals.
Figure 16
The idea behind I Sommarluft was to explore the sounds and affects of the scales that do not quite sound like JI (e.g. the first movement's C minor) and juxtapose these with the scales that clearly sound like JI (e.g., the fifth's movement's E major), and then shape the piece as a process of going in and out of JI by working with the scales in between (e.g., the third movement's D major). In other words, the artistic idea was to transition between the rough and informed styles of strict style as a 'tuning'. One of the subtle results of such an idea is that the music can be said to have two primary tonalities. On the one hand, the tonality is C minor in which the piece begins and ends; when the last movement returns to C minor, it feels like a return to the home tonality. On the other hand, the tuning itself can be said to gravitate towards E major and A major because this is where the instrument is the most resonant and where the most pitches fuse and enforce strong modal centers. We might describe this as the root or tonality inherent in the tuning system itself. The composer can explore these two opposing centers of gravity; on the one hand, there is the gravity set by the musical context (the root of the key established by the composer), and, on the other hand, there is the gravity of the intonation system itself, which leans towards tunability and resonance. This kind of artic idea utilizes the limitations of the keyboard instruments and the fact that it anyway is impossible to tune all different common Western key signatures to 'sound good' with only twelve keys to an octave. The result is a kind of music that would be impossible to realize with intonating instruments.
Summary of the strict styles
Before moving on to the free style, we can summarize what has been said about the strict style so far. We have seen how music in this style is characterized by the usage of small, fixed gamuts. Because of this, the music usually embodies strong affective qualities (like moods, rasa-s, or Stimmung-s). The usage of small gamuts facilitates intonational accuracy, and the composer can therefore be more carefree in terms of melodic and contrapuntal writing compared to the two other styles we will examine below.
The strict style can be subdivided into two sub-categories: strict style as integrated JI, usually written for intonating instruments and where the music is constructed as a tunable path, and strict style as a 'tuning', usually written for fixed-pitch instruments and where the composer does not consider tunable paths to the same degree. Within the latter sub-category, we can find two different approaches.
First, there is an informed approach to strict style as a 'tuning'. When this approach is taken, the music is composed in a way so that pitches still audibly retain their harmonic, relational articulation within a tight, tunable matrix. This approach differs from integrated JI in the higher levels of complexity and freedom in harmonic and melodic writing that the automatic intonation of fixed-pitch instruments affords. The informed approach manages to still construct the music as a tunable path for the listener even though it might be too difficult for musicians to actually intonate.
Secondly, there is a rough approach to strict style as a 'tuning'. When taken to the extreme, pitches in the pieces that takes this approach almost no longer sound as if in JI because the composer neither constructs the series of pitches as tunable paths nor articulates the pitches as belonging to an interconnected, tunable harmonic space. We noted that the rough approach is a similar approach that when applied to intonating instruments leads to microtonal music that uses JI-type intervals. In the rough strict style as a 'tuning', however, the precise, exact tuning facilitates (as the composition unfolds) an emergent tunability that still provides the music with a (sometimes subtle) quality of being in JI. The rough strict style as a 'tuning' still provides the music with the perfume of JI that microtonal music that uses JI-type intervals lacks.
Pitch classification in free style
An important idea from the field of music cognition to bring into this discussion is Dowling's (1978) suggestion, made by drawing upon Miller's (1956) classic cognitivist idea, that the diatonic scale contains only seven discrete pitch classes because this matches our limited ability to remember and label items reliably along continuous dimensions such as pitch frequency. This idea of cognitive restraint comes with the implication that the mind’s perceptual and memory systems seek, because of its limited ability, a simplified organization of the sensory information received when listening to, say, the huge collection of pitches in the work of Johnston or Partch. In these pieces, the mind would according to this idea categorize the sounds into focal pitch classes. Even though the composer works from a huge gamut of pitches where "10/9" and "9/8" clearly are distinct, they are nonetheless simplified as different tonal shades of the same focal pitch classes in the listening experience. In relation to C as the root ("1/1"), "10/9" and "9/8" share a 'D-ness' in the same way as the colors crimson and ruby share a redness. In other words, we do not hear the shift from "10/9" to a "9/8" as a modulation. To hear it as a modulation would mean that we hear it as a shift from one 'scale' step to another, but according to a cognitivist thesis, these pitches are collapsed into a single scale step.
But what range of pitches would be eligible to share this D-ness? Is also an "8/7" collapsed into the same category as a "10/9"? Sabat (2008/2009) has observed in relation to this, and I believe that many musicians will agree with him on this observation, that "intervals smaller than 1/6 of a tone (approximately 35 ¢) begin to take on the character of enharmonic shadings of pitch rather than functioning as distinct tones" (1). Since the difference between a "10/9" and "8/7" in relation to a root "1/1" is larger than that, 48.77 cents, this means that these two pitches can be heard as distinct. However, if there is also a "9/8" present, the situation might change since "10/9" is 21.51 cents lower and "8/7" is 27.16 cents higher than the "9/8". The presence of "9/8" could then 'bring together' these two pitches by functioning as the center from which "10/9" and "8/7" are both deviations smaller than 35 cents.
Generally speaking, the smaller the interval, the greater the tendency for the pitches that make up the interval to collapse into each other. In the JI repertoire, the comma "49/48" is a quite common melodic interval, and this interval is indeed just above 35 cents at circa 35.7 cents. In my own compositions, this interval is found between the scale steps "7/4" and "12/7" in Av dagg och fattigdom and the third movement of Andra Segel. Moving between these scale steps in these pieces is a very clear melodic movement involving two separate pitches. In other compositions, such as Mellan bleka stränder (efter Ni Zan), I have used the slightly smaller "56/55" found between a "7/4" and "55/32". As this "56/55" is 31.2 cents wide, it falls below Sabat's postulated 35¢ threshold. Indeed, it is interesting to note how in this piece, when compared to Av dagg och fattigdom, the two pitches making up the 56/55 behave slightly more like 'shadings' of each other, creating more of a hazy and fuzzy enharmonic effect rather than clearly functioning like two separate pitches. While the pitches making up 49/48 in Av dagg och fattigdom are clearly separated tones, the pitches making up "56/55" in Mellan bleka stränder (efter Ni Zan) have more of a shared 'fused' identity.
Yet, it is also possible for intervals smaller than "56/55" to behave like discrete pitches. In the music by Catherine Lamb, the syntonic comma 81/80 of 21.51 cents has been used as a melodic interval, and it is also not uncommon in the JI repertoire at large to find the septimal comma 64/63 of 27.26 cents used in this way too. There is, however, a big difference between these two commas in terms of the ease in which they are perceived melodically, and the rule that the smaller the interval the greater the tendency for fusion helps explain the different general usages of the septimal and syntonic commas that we find in the JI repertoire. Pitches separated by septimal commas have a greater proclivity to not sound like enharmonic equivalents compared to the syntonic comma, even though the syntonic comma is only 5.75 cents more narrow. In my own experience, I would claim that it is quite 'easy', in terms of establishing the adequate circumstances, to make the pitches making up a 64/63 actualized as separate pitches; it is possible to establish a context in which the movement between "9/8" and "8/7" is heard as a melody–albeit very 'microtonal' sounding. Syntonic commas, however, are instead 'easy' to actualize as enharmonic variations and require, in my experience, special circumstances to separate (we will see examples of such special circumstances below in the music of Sabat and Lamb in the examples below). It is not as easy to articulate the melodic movement between "9/8" and "10/9" without it sounding like 'the same pitch being re-intonated'. Both the differences between 9/8 and 10/9, and 9/8 and 8/7 imply new harmonic regions, yet one is more distinguished as a melody than the other. 5.75 cents is big enough of a difference for the tendency to 'distinguish' to flip over into 'fusing'.
In dealing with intervals smaller than 35 cents, establishing the supporting condition for pitches to separate is crucial. To what extent pitches that are separated by the "56/55" of 31.2 cents fuse is very much a matter of context. In the harmonic context of the piece Om dagen stilla, the same 56/55 of 31.2 cents is used but sounds in this piece more like separate pitches than was the case in Mellan bleka stränder (efter Ni Zan). My intuition tells me that this has to do with the more fleshed-out harmonic space that operates in Om dagen stilla. While Mellan bleka stränder uses a quite small harmonic material that is very scalar, Om dagen stilla has a more fully articulated harmonic space that is more spectral in orientation. This allows for the two different pitches that make up the 56/55 to more easily be separated since they are clearly associated with different spectral sub-sections–with different harmonic tunable paths.
In Om dagen stilla, the 56/55 interval is found between the pitches called "27/22" and "135/112" in the matrix in Figure 17. Both of these pitches have many tunable paths to other pitches. "135/112" clearly functions as a 5th partial in a spectral sub-section. Both the root as well as the fifth, ninth, and seventh related to this 5th partial are present in the scale. "27/22" is in a similar way scaffolded by many tunable intervals–it is the 7th partial in a spectral sub-section where also the 11th, 9th, and 3rd partials are represented.
Furthermore, some pieces use 'enharmonic equivalents'—intervals below 5 cents, such as 441/440 of 3.93 or the schisma of 1.61 cents, and this kind of repertoire illuminates another threshold: where intervals are so narrow that they do not even sound like the kind of 'enharmonic shadings' that Sabat referred to with intervals below approximately 35 cents. Narrow enharmonic equivalents merely sound like the same pitch. It is only when the sizes are larger than 5 cents that they start to sound like different 'shades' rather than collapsing completely. The distance of 7.71 cents between A# - - ("225/128") and Bb ↓ ("7/4") is, for example, too big to sound like an enharmonic equivalent. Using these two pitches in a modal context creates a loose style context where these two pitches appear as different variants of the same pitch. However, if only using narrow enharmonic equivalents that are smaller than 5 cents and no pitches separated by syntonic-sized commas, the music will sound like the strict style rather than free or loose, even though each pitch in the scale might have multiple spellings and the theoretical gamut be quite large.
In summary, it seems to me that the narrow band between 5 and 25 cents is especially crucial for enharmonic shadings. Smaller intervals than this are too small to sound like an intonational 'variation' whereas larger intervals can sound like discrete scale steps as they acquire more of an identity of their own. Combining these observations with the above cognitivist idea about the difficulty in keeping more than seven pitches 'present' at the same time might lead us to want to at least as a hypothesis say that in many free style styles, the large number of frequencies separated by commas smaller than 35 cents overwhelms the perception and gives rise to a simplified organization of pitches into focal pitch classes; sounds that are less than approximately 35 cents apart (to follow Sabat) become different 'nuances' of the same tone. Unfortunately, it will not be that easy to postulate that kind of rule, and this is because our perceptual orientation toward sound is not only guided by cognitive restraints but also by a search for meaning. The 35-cent threshold is fluid and impacted by the harmonic contexts and particular modes of listening that different pieces of music afford. Some harmonic contexts and modes of listening will allow for a proclivity of 'fusion' while other harmonic contexts will easier allow for separation. We can even draw to mind how in the realm of Equal Tempered music, composers such as Takemitsu have even achieved to make intervals as large as quarter tones sound like enharmonic variations of each other.
Before giving some examples where the organization of comma-distanced pitches in focal pitch classes does not happen, I will first give an example of a piece where I think that it happens. This is a piece of mine coincidentally named Marc (Sabat). This piece uses fifteen pitches per octave as shown in Figure 19. Despite clearly being different pitches in the composition process, these fifteen are in performance heard as 9 'fluid' pitches (A, B, C, D, D#, E, F#, G, G#). This does not mean an endorsement of cognitivism, but simply reflects how I perceive the music. The fact that even I, who composed the music and know about all the differences, intuit the music in this way suggests that other people likely hear it this simplified way as well. This certainly does not mean that a piece like Marc (Sabat) could equally well be played in ET, where the proposed simplified organization of sense data is literally translated into a simplified, tempered, system of pitch classes. The different intonations carry important meanings and psychological cues that are very important to play precisely, but this does not translate into perceiving the pitches distanced by commas categorically as different. Below, at Figure 25, I will return to an analysis of this piece.
Figure 19
The 5-limit free style used in a piece like Marc (Sabat) might actually be the style of JI that in sound comes closest to Western Common Practice Period music. In that musical style, performed for example by a string quartet, the pitches are also fluid without each new intonational iteration sounding like a modulation or a completely new category. The musician accustomed to Western music will probably not hear anything odd about a free style piece like Marc (Sabat) as the pitches are neither significantly flat nor sharp from their stylistic boundaries within Western music. Lou Harrison even implied at times that he thought that free style JI is what string quartets who play classical music naturally do. In the chromatic sections of his Suite for Symphonic Strings, Harrison refrained from notating the music in JI despite the other movements being in JI. Instead, he left it up to the musicians to find 'consonant’ and 'harmonious' intonation themselves (i.e., "play it in JI"). The result, he said, was not very different were he to write out the ratios himself (Miller et al. 1998, 121). In my opinion, this is expressive of the unfortunate idealism associated with some JI composers of that generation; they believed that JI was the 'natural’ intonation of musicians. Contrary to Harrison's claim, research has shown that performers of Western art music adhere surprisingly close to ET with no seeming preference for JI, even when playing without a tempered instrument (Burns 1998, 246); the 'span' of the pitch classes’ fluidity is not nearly as vast as they typically are in free style JI, even when only in 5-limit with relatively few comma-levels operating.
If one way of perceiving comma-distanced pitches in free style music is to categorize them into focal pitch classes, there are also musical instances where we intuitively distinguish between comma-distanced pitches more clearly. A prime example through which one can explore this phenomenon is Marc Sabat's Gioseffo Zarlino, an excerpt of which is shownin Figure 20. While analyzing our listening experience to this music, we should question whether the pitches A- and G+ are perceived as distinct from A and G. Based on my own listening experience, I found this to be true, but only after the music had progressed for a while, as the music gradually attuned me to that way of hearing. The piece features repetitions of brief, simple phrases, in which the intonation of pitches changes by syntonic commas. These subtle changes between repetitions enter our awareness, even when we do not know what they are, and they encourage detailed listening. The resultant attentiveness allows us to recognize, in a non-conceptual way, the microtonal differences as important distinctions (but not necessarily categorized as such). We become attuned to these subtle details as the music guides us toward what is important to hear; it is in the categorical separation between comma-distanced pitches where part of the piece's meaning resides. Eventually, I started to perceive G and G+ as different pitches after initially fusing them (hearing them as part of the same focal pitch class) as these comma differences started to take on a greater sense of freshness, almost akin to how a Bb sounds in C major in Equal Temperament—a sensation of modulation.
This modulatory freshness and categorical separation when going from a "5/3" to "27/16" does not generally happen in the free style, and it does not happen in a piece like Marc (Sabat). If we are inclined to cognitivistic explanatory models, we might want to follow Dowling and Miller and say that this is due to our limited cognitive capacities and our propensities for categorical perception. As seen in the example by Sabat, however, certain stylistic choices might invite us to a different kind of listening. This means that we cannot accept a naive cognitivist view of perception. As this example by Sabat shows, pitch perception has to do with searching out and as a listener co-create what is meaningful in music, not about 'processing' some kind of postulated 'sense-data' into perceived pitches.
Figure 20
An additional instance of this phenomenon is found in the reduced 'melodic duo' version of Catherine Lamb’s Prisma Interius VIII. Even though this music uses many small pitches separated by commas, it does not have the feel of being in the free style. The pitches feel like they all belong together as if constituting a single, large mode, which is perhaps best described (if sticking with the categories used in this text) as a strict style with a large gamut of pitches, some of which are comma-distances apart. The music does not neatly fit the categories outlined in this text since strict style pieces were defined as having generally small gamuts and no pitches separated by small commas, and free style pieces do not have the strong modal and affective 'glue' that bind the pitches together in a single Stimmung or rasa.
The reason for Lamb's piece managing to sound strict style-esque despite its large number of pitches is perhaps found in the unique way the pitches are spectrally articulated as part of a single overtone series; all pitches are notated as partials over the low (inaudible) 5 Hz. While it is true that one can always construct a low, common root to pieces in JI (a low 1/1 from which all pitches can be considered overtones), this principle is here established with unusual clarity for the listener. The slow and pedagogical unfolding of the piece, beginning with a slow oscillation between G and G- (81:80) and slowly adding pitches that fill out the spectrum, insists on the separation of G and G- as different partials: they constitute the difference between the 80th and 81st partial rather than different shades of the same pitch class. A section like the one at rehearsal number 10, seen in Figure 21, which in isolation looks like a 5-limit free style passage appears in something like a 'single-overtone series-strict style' context that makes the analysis of it in the free style dissatisfying.
Figure 21
A parenthetical point to note in connection with this piece is how its mentioned opening section largely consists of small untunable intervals–some of them as small as the prominent 81:80 (Figure 22). This piece provides a clear example of an important point mentioned in the introduction; a definite line between microtonal music that uses JI-type intervals and what we call integrated JI is impossible to establish but rather represents two idealized extremes. In reality, much music moves back and forth in complex ways between these two styles. Categorically, we must say that Prisma Interius VIII begins as microtonal music that uses JI-type-intervals where the musician must rely on her ability to memorize the sound of the intervals, but that these microtones are later re-interpreted as 'always having been' part of a tunable matrix (in this case a single overtone series). But even as the music in the first minutes of the piece goes back and forth between the two performance modes of tuning and approximating pitches, this is not an oscillation that is readily audible to the listener. For the musician as well, this distinction is blurry; at times we can not tell if we are tuning to short-term memory, or if we are habitually approximating pitches in reliance on mental/embodied representations. Between these two modes of performance, there is a feedback system where one supports the other.
Figure 22
Poetic moods and tunable paths in the free style
If Harrison in his Suite for Symphonic Strings considered free style to be Western musicians’ natural mode of intonation, Harrison at other times deemed free style to be impossible to achieve without instruments specially built for the purpose. As an example of this, his Simfoni in Free Style employed an array of differently-tuned custom-made flutes (Doty 1987) with the idea that the tunings and key placements would guide the musicians through the complex intonation. Simfoni in Free Style has never been performed live, so the only thing we can listen to is a MIDI mockup, and the first thing that strikes me when listening to this mockup is indeed the extreme flexibility of intonation that seems to warrant the need for intonational guides in the form of specifically built instruments. The pitch level rises rapidly through the different comma levels. In just two measures (8-9), five 5-limit comma levels are present from B- (one comma down) to Bbb+++ (three commas up). This is, by all means, radical intonation and very difficult to achieve accurately at such a fast pace. The result of this kind of intonation is the sound of a highly malleable pitch space. It sounds de-centered, non-hierarchical, fluid, and liquid-like. The affective and poetic 'mood' produced by this intonation is the very opposite of the clearly defined and unifying rasa-s of the strict style that we studied above. The 'rasa' of Simfoni in Free Style is one of almost psychedelic liquidity.
Figure 23 (Measure 8-9 of Simfoni in Free Style)
For me, there is a discord between the liquid-like tuning and the actual melodies and motifs used in Simfoni in Free Style. In fact, everything but the intonation sounds like what Miller and Lieberman in their comprehensive study of Harrison have rightfully dubbed "vintage Harrison" (1998, 118). The music sounds just like Harrison’s other music but now with very spectacular intonation—it is vintage Harrison seen through psychedelic glasses. The music is not only swiftly moving between comma-levels but there is even a blatant use of wolf-intervals. A direct 27/20 dyad between D and G is, for example, called for in measure 5.
The only way to really play the kind of intonation used in this piece convincingly, one could argue, is if it is backed up by an artistic idea and poetic mood that is 'all about' this kind of liquid-like stretching of the pitch space and that speaks in a single voice together with the tuning—it should be psychedelic Harrison seen through psychedelic glasses. Harrison’s Simfoni in Free Style does not embrace that kind of poetic expression and does not invite such a mode of listening. Instead, it is "vintage Harrison" in which many intervals plainly sound out-of-tune.
Simfoni in Free Style thus provides us with another important example of the rule that the intonation system and the composition must speak in a single voice–speak with the same artistic intention–for the intonation not to sound jarring. As a comparison, an almost equally fast movement between comma levels happens in the passage from Prisma Interius VIII shown in Figure 21. Here, the music swiftly moves between four comma levels from Bb to D#- - - but sounds completely natural because of the way it articulates a well-defined harmonic territory that has been slowly and gradually built up and established throughout this slowly evolving piece.
Figure 24
Even in a free style composition, it has to be noted that it does not take much for temporary reference points to arise—reference points that can make certain pitches sound out-of-tune even when they do not conspicuously appear as such from just reading the succession of notes in the score. In Figure 24, modified from Figure 2 above, the A- will likely sound too low even in a free style context as the D and E provides local tuning references. Such local reference points will always arise—weak and temporary as they may be—as the perception of musical pitches always is colored by other pitches previously heard (see further discussion in Krumhansl 1990, 283). This does not change just because one calls the music 'free style' and glorifies it through a rhetoric of having 'freed' intervals from musical gamuts by working out the intonation of each interval locally. The perception of music is not as local as Harrison seem to have thought. Even when writing free style music, the composer must be attentive to local modalities and modal hierarchies as they arise. It is, after all, the very nature of JI to have a 1/1. The composer must be able to feel which notes influence tuning most and then adjust the music accordingly. Such adjustments do not necessarily always entail replacing pitches; by simply clarifying the phrasing in Figure 24, we can diminish the importance that the D and E have on the A- by, for example, grouping the D and E together and clearly starting a new phrase on the B-. Again, pitch perception is about enacting meaningfulness, not about processing data, and shifts in phrasing and articulation will guide us in this endeavor and will therefore change our perception of intonation.
Successful pieces in free style do not by any means have to go as far as to embrace the kind of psychedelic liquidity of Simfoni in Free Style, but they must speak in a single voice with the intonation system. From the perspective of my own practice, the main quality of the free style is the freedom from strong modal contexts. In my free style pieces, I have been inspired to emphasize how the lack of a unified mode and lack of such a mode's co-emergent affective quality puts the listener in a 'present moment' that is less in the grip of a single modal center or modal hierarchy. The music stays 'new' and becomes modally unpredictable. Such pieces are not unified by strong, affective modes, but each new phrase and section can have contrasting affective qualities brought about by the free intonation. In Marc (Sabat), the fact that the music is in free style helps to emphasize the very fragmented nature of this music as there is no unified mood or affect in terms of intonation that connects the fragments.
An excerpt from Marc (Sabat) is shown in Figure 25. By tracing the tunable paths, we can clearly see how this melodic and harmonic writing differs from the strict style seen in Figure 7. In order to make sense of the shifts between A and A- and D and D-, the musician must be quick to grab onto the notes that support these modulations. Such is the case in measure 27, where the second violinist must perform a D- whereas in measure 23 there had been a D. Here, the violinist must quickly re-orient the intonational reference points, ignore the F#- at the beginning of measure 27 (to which it will form a Pythagorean third), and tune only to the immediately preceding sonority where the first violin and the trumpet provides a clear E- and A- perfect fourth. In the same way, to accurately perform the modulations to the major chords [B-, D#--, F#-] and [E-, G#--, B-] from what at the beginning of the excerpt is a G Ionian [G, A, B-, C, D, E-, F#-] moment. The musicians must in measure 27 catch and build this modulation upon the emphasized pitches B- (m. 24) and E- (m. 26).
The tuning in this passage is fragile; crucial reference points are found in the immediately surrounding notes rather than from a modal framework. In the three systems of Figure 25, four comma levels are used. Swift modulations such as these where each pitch connects to another in a tunable link and where new modalities must be built on pitches only heard once leaves little room for mistakes. The musicians must stay vigilant with regard to performing each aggregate accurately.
By carefully considering the instrumentation, composers can help the musicians in this endeavor. When writing for stringed instruments as in Marc (Sabat), they can, for example, make use of the open strings and natural harmonics to aid the tuning and at certain moments provide anchor points. Consider, for example, the stark modulation from a [E- A- C] to a [F#- A D]-harmony in measure 29. The difficult comma movement in the second violin part from A- to A is facilitated by the use of open strings and harmonics; A- is arrived at through a tunable path, but A is arrived at through using an open string. Additionally, by using orchestration to emphasize the top line C to D as the important part of this gesture (the C is doubled by violin I and trumpet, and the D by violin I and II in octaves), the intonation of A- to A becomes easier to grasp because its pitches are clearly in a second voice attributed to and dependent upon the first voice.
Figure 25 (Trumpet in Bb)
Another piece of mine in the free style is Radii solis, et sternet (sibi aurum) quasi lutum (Figure 26). In this piece, the music uses five different comma levels, which is one more than Marc (Sabat). Yet, the music moves slowly and gradually through the comma levels through mainly tunable paths. The effect of listening to this music is one in which, just like Marc (Sabat), the music retains a fragmented sound and emphasis on the present-moment harmonic constellations rather than establishing a strong modal mood.
Figure 26
Before moving on to the loose style, the last style to be discussed in this text, we can now summarize the free style. Basically, it is a style that is the total opposite of the strict style. Instead of working with a fixed gamut, it works with an unlimited and unrestrained number of pitches. Instead of having the kind of strong unifying modal affect that is associated with small gamuts, free style pieces can potentially have no global, unifying modal affect of this kind. This does not mean that the music lacks modal affect, but rather that the modal affects are sculpted more locally: the pitches themselves are less related to a unifying mode and more related locally to their immediately surrounding pitches. This makes the craft of composing a tunable path more feeble as the musician can rely less upon contextual 'modal' tuning and scalar familiarity.
The sometimes teeming number of pitches in a free style piece begs the question of how these are cognitively represented by the listener. Do we hear "27/16" and "5/3" as the same pitch in different intonations or as different pitches altogether? In a piece of mine, such distinctions were shown to be categorically blurred in the listening experience. It was argued that while this blurring may be the most normal for pieces in free style—that is, we do not perceive a move from "27/16" to "5/3" as a modulation in the same way as changing a B to Bb in Equal Temperament would—this can change depending on how the composer constructs her piece. Pieces by Sabat and Lamb suggest listening practices in which pitches commas apart can be distinguished categorically. Within the free style, perhaps we can talk of two sub-categories that represent the two ends of a possible continuum: music that readily invites a simplified organization into focal pitch classes and music that readily invites separating comma-distanced intervals categorically. Similar sub-categories will also play an important part in the loose style when now moving onto the loose style, which we now move on to consider.
Loose style
While the strict and free style, and the importance of properly distinguishing between them, was already introduced by Lou Harrison, the intermittent loose style is introduced here for the first time. Its most salient characteristic is a hierarchical ordering of pitches that dictates intonational fluidity. Some pitches, structural in nature, are fixed, while the remaining pitches are free. Pitches low in the hierarchy can be freely replaced by their neighboring variants (distanced by commas), while the structural pitches higher up in the hierarchy can under regular circumstances not (if not intentionally used for a modulatory effect, they will sound out-of-tune). In my loose style pieces, the structural pitches are often 'typical' structural pitches such as ["1/1","3/2"] or ["1/1","4/3", "3/2"]. The ["1/1","5/4","3/2","15/8"]-pattern, in particular, has found its use in many compositions. Three different iterations of this pattern are shown in Table 2. In this table, these four structural pitches are themselves ordered into three hierarchical levels; “1/1” is on the top level, “3/2” is on the second level, and “5/4” and “15/8” are on the third. These three levels reflect how the higher one moves in the hierarchy, the greater the modulatory effect of changing pitches by syntonic commas will be. The fourth level under the structural pitches in the table includes all the other pitches used in the pieces.
Table 2
These tables have their origin in observations done in my compositional practice prior to formulating the idea of the loose style. For example, when composing the first piece shown in the table, mot våren bortom havet, I noticed that lowering the D by one syntonic comma to D- had jarring effects, often sounding out-of-tune if not prepared with extreme caution, while changing the A to an A- was very smooth and easy; changing the B- to a B had a big modulatory effect, but not as big as changing the G to G- or G+ had—the alteration which most easily sounded jarring. I wrote many pieces making practical observations like these before I started to realize that maybe this was an idiom distinct from the free style. With the eventual formulation of the loose style of JI, these practical observations found a corresponding theoretical explanation. Later on, I noticed that these tables for the ["1/1","5/4","3/2","15/8"]-structure largely corresponds with the basic pitch space tables used by Krumhansl based on empirical measures of tonal hierarchies (see for example Krumhansl and Cuddy 2010)—a convergence that has to be further studied in the future.
To illustrate the workings of these hierarchies in practice, let us look at seven measures from att sjunka i doftande klöver for violin, cello, and piano. Figure 27 shows an early version of these bars. When just reading this score for the first time, we might conclude that everything looks very tunable; we might indeed think it is written by a composer who has taken great care of the tunable paths. Our only objection is possibly the violin line’s third pitch—we can complain that this three-note melody frames a wolf fourth (E to B-), and we might ask ourselves if this is really tunable. Then, however, we see that there is a strong bass line in both the cello and piano that moves from A to D and that the violin tunes the E as a perfect fifth to the A, and then a B- as a just major sixth to the D. The relationship to the bass line becomes the important factor here and the fact that the violin melody is outlining a wolf-fifth is obscured and hidden in performance and thus not important.
Having concluded that the tunable path is smooth, we will surely be greatly surprised when listening to this music and hearing how the violin’s A- in measure 261 sounds too low—indeed out-of-tune. This is not at all evident from just reading the score. On the contrary, the fact that the A- in Figure 27 is preceded by both a B- and an E- makes it look correct with an A- in that position; it is the third interval in a series of 4/3s. What has happened in listening to account for this phenomenon is the arising of a hierarchy of pitches characteristic of the loose style of JI. The emphatic D in a low register in 258 and 259 in both the piano and cello linger in the memory through 261 and makes the ear want to hear the A as unlowered because it forms a 3:2 to it. There is no doubt that it is the D that serves as the basis-pitch of this passage as a 'fundamental' or 'root', and this fact influences the desired intonation of the A. As Krumhansl (1990) notes, the "recognition memory for a tone depends on its position in the tonal hierarchy, with more stable tones in the tonal hierarchy more stable in the memory trace" (148). In other words, because of its position as the basis pitch of this passage, the D remains mentally present for a long time and has, therefore, a strong influence even as E-, not a basis tone, has sounded more recently. The A can be tuned either to the D (as an A) or the E- (as an A-), but not both. It 'had' to be tuned to the D because the mind’s hierarchical categorization of pitches was more important than melodic linearity and short-term tunability. As a result of this inflexibility of the A, the A comes, throughout the piece, to take on a role as a structural pitch one step down in the hierarchy from D as a tunable 3/2 to it.
In order to not hear the pitch as false to the immediately preceding B- and E-, we must hear it as arising within a modal–or tonal, to follow Krumhansl’s usage of this word–context. If the experience of a modal hierarchy is not present when we hear measures 260-261, the A will sound out-of-tune. When composing in the loose style, the composer will often end up writing passages such as these that in notation looks 'bad'. These hierarchies result in compositionally interesting moments where the ear is more willing to hear untunable intervals like wolf-fourths than structural pitches adapting by commas.
Figure 28
Figure 28 shows the penultimate version of the passage in Figure 27. The A- in 261 is changed to an A, and this change further necessitated a change from E- to E in the same measure (but not the preceding E- in the measure before). The cello’s E in measure 261 now forms a wolf-interval to the following B- in measure 262. Although this does not sound too out-of-tune in the modal context, the whole passage still has a slight perfume of out-of-tune-ness caused by all the direct wolf intervals.
When I composed this passage, I did, on the one hand, not consider this to be a big problem because prior to this moment, the employment of many 'awkward' outlines and skips that exposed Pythagorean thirds and wolf-fourths had already established a rather thorny and angular melodic style. The exposed wolves in Figure 28 fit within this style very well. I thus decided to keep this passage without re-writing it completely. On the other hand, the passage in Figure 28 had a little bit too much of this quality. To solve this problem, I found that it was enough to just add an additional D as a pure fifth under the A in measure 261 as this functioned as a reminder to the ear of the modal center. It also served the purpose of masking the 10:9 between B- and A by instead drawing attention to the 5:3 between B- and D. By so doing, the perfume of the wolf’s out-of-tune-ness evaporated but still left us with a rather 'thorny' and 'angular' sounding counterpoint. Lastly, in order to help the musician navigate the fast shifts between, for example, E- and E, I made sure that these kinds of passages were always supported by the use of natural harmonics and open strings. The final version is shown in Figure 29.
Figure 29
The solo double bass piece Väntar där dimma uppstår employs a similar usage of consciously 'imperfect' JI-writing to achieve a particular poetic mood. In both this piece and in att sjunka i doftande klöver studied above, the 'imperfections' in the tunable paths give rise to specific poetic effects that would not be possible with completely 'perfect' JI melodic writing (i.e., a JI that almost exclusively uses tunable intervals). In measure 187 (Figure 30) a major ninth is played, but it is not the tunable major ninth 9/4 but rather the untunable 20/9. The performer is asked to find this by re-voicing the tunable minor seventh 9/5 in the previous measure by keeping the G+ stable while transposing the A up two octaves. Because the pitch classes are already established in the 9/5, the sonority 20/9 in measure 187 does not necessarily sound 'wrong' or out-of-tune; both the pitches involved are in-mode pitch classes, but it has a fascinating, energetic pulsation to it that wants to be resolved. To perform this 20/9, the musician first tunes the 9/5 in 185, then keeps the G+ stable, and adds to high A as a natural harmonic and while doing so, resist the temptation to lower the G+ to a G in order to form the still, just major ninth 9/4.
Implied modal modulations in the loose style
Another example can be found in Livets eget bleka flöde on page 21 (Figure 34). In a melodic sequence between a D and a G, we find a E. Why an E and not an E-? In the octave positions given in this music, a E- would be tunable to both the D as a 9/5 and to the G as a 12/5, but instead, the music is written using an E. I made this choice because the E- would have implied a undesirable modulation. E- would have suggested harmonic hierarchies and hinted at harmonic regions to which the piece does not strive. Just as with the example above, the only way to answer why an E would make more sense than the E- is found by looking at the modal context and tonal hierarchies.
As we have seen with all of these examples, there is a wide range of approaches to modulation in the loose style, and all modulations do not have to be as smooth as possible. While pieces like Sommarberg, i glömska and Vid stenmuren blir tanken blomma attempts for a very smooth effect, other pieces such as att sjunka i doftande klöver and Väntar där dimma uppstår that we looked at above have a rougher treatment of modulations. Just as in non-JI music, modulations can be prepared and smooth, or unprepared and jarring, sometimes noticeable and sometimes unnoticeable. Making poetic use of the kinds of subtle modulations discussed here is, however, according to me, one of the great strengths of the loose style.
Figure 34
The piece in which I took this idea perhaps the furthest was the viola or violin solo piece Rosor och så liljor. This piece uses a simple 7-tone strict style scale, but by allowing each pitch to move within no more than a 5 cent radius, a very rich gamut of other harmonic possibilities is opened up. Because of the complexity of this particular scale, I even decided to not notate this piece exactly with JI-accidentals, but rather to introduce a combination of cent-deviations, JI-accidentals, and ratios. A brief excerpt of how the notation would look in normal JI notation is given in Figure 37; it looks like a very untrammeled free style piece. The chosen notation in Figure 38 instead reveals this freedom to be contained within a very strict 'grid' that is the 7-tone basic scale with only a 5 cent flexibility to each scale step.
Figure 38
A similar effect happens in my big keyboard piece Som regn. In that piece, the pitches obviously cannot move because of the fixed tuning, but I decided to anyway notate the implied harmonies in a similar way as in Rosor och så liljor. Here, the cent deviations show pitches' out-of-tuneness rather than how much they have to be adjusted. Som regn operates in many ways like an enharmonically flexible strict style piece but the pitches have to be adjusted 'in our minds' rather than on the instruments. In that sense, this music functions a lot like a kind of temperament in the sense that pitches can have multiple spellings.
Figure 39 shows a brief excerpt from Som regn; note how the A and E are notated in different ways depending on the harmonies. The A is both an 11/9 to C as well as a slightly out-of-tune 8/7 to the B-, and it is often difficult to choose which interpretation should be favored. The A is in other words polysemic and multi-stable; both interpretations are correct. This is why it makes sense to call these kinds of tunings temperaments even though no pitch technically has been 'tempered'.
Because the piece is written for a tuned instrument, there are also instrumental idioms such as the use of untunable melodic intervals that suggests that the music should be classifiable as being in the informed style of strict style as a 'tuning'. In Table 3 where many of the different styles are summarized, this kind of 'JI-temperament' is listed as a sub-category to both the informed style of strict style as a 'tuning' and the enharmonically flexible strict style.
Many of the categories and subcategories mentioned in this text are summarized in Table 3. These categories should not be viewed as rigid 'genres' of JI (just intonation) music; there are many instances where JI music defies these classifications. Instead, they are best understood as points on a flexible continuum. Many pieces move back and forth along this continuum: at certain moments, they may strongly exhibit characteristics associated with one style, and at other times, they may embody the typical features of a different style. For example, the piece Vid stenmuren blir tanken blomma begins in a loose style and ends in something that most closely resembles the strict style. At this point, however, the pitch space features two pitches that are only a syntonic comma apart, which technically classifies it as being in the loose style. Another piece discussed that transitions between different styles is I Sommarluft. The music of this piece oscillates between the rough and informed styles of strict style as a 'tuning'.
[1] In this text, harmonic ratios are notated with a “/” and melodic ratios with a “:”. When ratios are used to signify scale degrees in a scale or gamut, double quotes “ ” are used. Sets of scale degrees to form a mode or scale are notated in brackets “ [ ] ”. For example: In the mode [“9/8”, “5/4”, “3/2”, “5/3”], the melodic leap between “9/8” and “3/2” is a 4:3 and the harmonic sound of playing them together is 4/3.
[2] Because this text will mostly discuss music in 5-limit Just Intonation, the minus and plus signs “- / +” are used in the body of this text to indicate syntonic commas (21.5 cents).
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