Varieties of Just Intonation, cont.

[Continuation from Varieties of Just Intonation]

Moving between rough and informed idioms

As with all the subcategories of ratio-based composition discussed in this text, the rough and informed styles of strict style as a ‘tuning’ represent two idealized poles along a continuous spectrum. Many works move fluidly between them. Nattviol, nattviol is, as we saw above, an example of a piece that at times sounds informed and at other times rough.

Another composition in which I explicitly set out to explore the contrast between these two idioms is the clavichord suite I Sommarluft, whose overall form is built around this very oscillation. Across its movements, the music alternates between rough and informed styles—some movements inhabiting the extremes of the spectrum, others hovering ambiguously in between. When examining the distinctions between these styles, this piece is particularly illuminating, since its transitional movements allow us to pinpoint the precise moments where the rough begins to lean into the informed.

The tuning of I Sommarluft is shown in Figure 15. From this twelve-note chromatic gamut, each of the suite’s six movements centers on a different traditional Western diatonic subset—such as C minor, D major, and F major. Some of these subsets, like A major ["1/1", "9/8", "5/4", "4/3", "3/2", "5/3", "15/8"], are highly tunable. This A major collection is in fact identical to the gamut used in the informed-style piece Vårbris, porslinsvas (Figure 6).

Other scales in I Sommarluft, however, contain fewer tunable intervals and therefore sound less distinctly just. The opening movement’s C minor is one such example: ["32/27", "4/3", "45/32", "128/81", "16/9", "15/8", "135/128"]. This is effectively a Pythagorean Aeolian mode built on "32/27" (C), but with its minor third, minor sixth, and minor seventh all a schisma (32805/32768, ≈ 1.95 cents) too wide. Heard in the suite’s opening, this key conveys little indication that the music is in Just Intonation. No intervals other than the octaves, fifths, and fourths fuse into periodic entities; the tuning thus sounds more like a temperament—even though no pitch has been tempered.

As in Stenskrift, we might gather from listening to the opening movement that the composer took into account the temperament-like characteristics of the tuning—that is, which intervals come to rest more easily and which do not—when writing the music. Nothing strikes the ear as jarring; the phrases and melodies seem to speak in unison with the tuning. Beyond this, however, there is little sense of an intimate engagement with JI itself, as most pitches lack tunable relationships. The first movement of I Sommarluft therefore clearly exemplifies the rough style of strict style as a 'tuning'. 

Figure 15

The second movement continues in a similar manner, employing another largely untunable scale—F major—while the third movement, In Nomine, likewise begins with predominantly untunable chords and intervals, giving particular prominence to the Pythagorean minor third between D ("4/3") and B ("9/8"). A few minutes into the third movement, however, the introduction of 5-limit pitches ("5/4" and "5/3") and their corresponding harmonic and melodic intervals transforms the music. At this point, the sound world begins to approach the informed style of strict style as a ‘tuning’, at times coming very close to true integrated JI 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 passage is entirely tunable: an initial D–F#- just major third is followed by a C#-–A just minor sixth, resolving to a D that forms 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 its sixth degree is a Pythagorean sixth (27/16), whereas A major employed a just sixth (5/3). In other words, the D major collection contains slightly fewer tunable intervals, though it remains closely related. This difference, however subtle, is what positions the movement midway between the informed and rough styles. Many passages—such as the third measure in the excerpted section of In Nomine—could readily be performed by intonating musicians, yet the carefree use of the sixth degree, B, reveals that the music was conceived for a pre-tuned keyboard.

Immediately after that tunable third measure, the fourth measure of Figure 16 introduces a clear Pythagorean third (G–B). While the G is tunable to the preceding D, the B would need to be lowered to B- in order to maintain tunability with the previous context. Another instance of this kind of carefreeness appears in measure 84, where a Pythagorean minor third (B–D) is attacked directly after an F#- that forms a wolf fifth with B. These moments—all involving the same sixth degree—tilt the music toward the rough style, producing an uneven distribution of tunable and untunable intervals (in precisely this sense of the word rough).

Figure 16

The idea behind I Sommarluft was to explore the sounds and affects of scales that do not quite sound like JI (for instance, the first movement’s C minor) and to juxtapose them with scales that clearly do (such as the fifth movement’s E major). Between these two poles lie the intermediate scales (for example, the third movement’s D major), through which the piece enacts a gradual process of entering into and emerging out of JI. In other words, the artistic aim was to traverse the spectrum between the rough and informed styles of strict style as a 'tuning'.

One subtle consequence of this design is that the work may be said to possess two overlapping tonal centers. On one hand, there is the musical tonality—C minor—in which the suite both begins and ends, giving a sense of return and closure. On the other, there is a tuning tonality, gravitating toward E major and A major, where the instrument resonates most strongly and the greatest number of pitches fuse to form clear modal centers. The composer can thus explore two distinct centers of gravity: the contextual root established compositionally, and the inherent root of the tuning system itself, which tends toward resonance and tunability.

This artistic premise makes creative use of the inherent limitations of the keyboard instrument—specifically, the impossibility of tuning all twelve keys to sound equally consonant within a single octave. The resulting music is one that could not meaningfully exist for intonating performers: it depends on the immovable, pre-tuned landscape of the keyboard to bring these two tonal gravities—the musical and the acoustic—into dynamic interplay.

This artistic premise makes creative use of the inherent limitations of the keyboard instrument—specifically, the impossibility of tuning all twelve keys to sound equally consonant within a single octave. The resulting music is one that could not meaningfully exist for intonating performers: it depends on the immovable, pre-tuned landscape of the keyboard to bring this movements between rough and informed styles of JI into play.

Summary of the strict styles

Before moving on to the free style, we can summarize what has been said about the strict style. Music in this style is characterized by the use of small, fixed gamuts. As a result, it often possesses strong affective qualities—distinct moods, rasa-s, or Stimmung-s—that emerge from the tightly defined network of tunable relations. The limited gamut also facilitates intonational precision: because the number of available pitches is small and their relationships are repeatedly reinforced, the performer’s accuracy increases, and the composer can therefore write with greater freedom in melodic and contrapuntal design than in the two styles we will examine next.

The strict style can be subdivided into two sub-categories. The first is the integrated strict style, typically written for intonating instruments, in which the music is constructed as a tunable path. The second is the strict style as a ‘tuning’, generally written for fixed-pitch instruments, where the composer need not account for tunable paths to the same degree. Within this latter sub-category, we can distinguish between two further approaches: the informed style and the rough style.

The informed approach to strict style as a ‘tuning’ is one in which the music is composed so that pitches still audibly retain their harmonic and relational articulation within a tightly knit tunable matrix. This approach differs from integrated JI through the higher levels of harmonic and melodic complexity made possible by the automatic precision of fixed-pitch instruments. Although the resulting music may be far too intricate to be performed by intonating musicians, it nonetheless constructs an audibly tunable context for the listener—one in which complex ratios remain intelligible as JI.

Secondly, there is a rough approach to strict style as a ‘tuning’. When taken to its extreme, the pitches in pieces adopting this approach almost cease to sound as if they were in JI, since the composer neither constructs the pitch sequences as tunable paths nor articulates them as belonging to an interconnected harmonic space. As we have noted, this approach resembles the one that, when applied to intonating instruments, results in microtonal music using JI-type intervals. In the rough strict style as a ‘tuning’, however, the precise, fixed tuning of the instrument enables a faint emergent tunability to unfold throughout the composition—one that still imparts a subtle yet distinct sense of JI. The rough strict style as a ‘tuning’ thus preserves the perfume of JI that microtonal music using JI-type intervals lacks.

In both the informed and rough approaches, the strict style as a 'tuning' remains defined by its reliance on fixed, pre-determined gamuts and stable modal identities. The distinction lies in how clearly these gamuts are articulated as tunable contexts for the listener. In the free style, by contrast, these fixed modal structures begin to loosen. It is to this freer, more modulatory form of ratio-based composition that we now turn.

Pitch classification in free style

The difference between the free style and the strict style can be understood in a few complementary ways. On the one hand, we might say that the free style is music in which pitches are fluid rather than fixed: a "B" might at any moment function as a "16/15", "135/128", "256/243", or "25/24". This perspective assumes that in moving from the strict to the free style, we are altering the nature of the pitches themselves—whereas in the strict style, each pitch has a fixed intonation, in the free style, it becomes flexible or contextually variable.

On the other hand, we can instead describe this shift as an expansion of the gamut rather than a change in the nature of pitches. In strict style pieces, the gamut is typically small, rarely including pitches separated by less than a sixth of a tone. In the free style, however, we admit a potentially infinite continuum of pitches lying as close together as commas—such as the 21.5-cent syntonic comma—thus vastly increasing the density and malleability of the tonal field.

The second way of conceptualization aligns with the compositional approaches of Harry Partch and Ben Johnston, both of whom derived their pitch materials from extensive ratio lattices. They did not, as I understand it, regard their music as being based on scales with flexible degrees; rather, they treated comma-spaced pitches as activations of distinct nodes within a just-intonation matrix. Larry Polansky (2009) has referred to this as a method of multiplicity. In this context, the traditional diatonic space of seven tones, as found in much Western music, is multiplied into an immense continuum of possible pitches.

However, a crucial question arises as to whether Johnston’s compositional method corresponds to the way our minds actualize pitches when listening to his music. Do we perceive the tones as drawn from a vast gamut, or as a smaller set whose intonation is fluid, even if this is not how they are notated? Addressing how the mind–body represents and categorizes pitch (to use provisional cognitivist terms) is essential if we wish to develop a discourse grounded in how the music sounds, rather than how it is composed. As Huron (2006) aptly observes, "how minds represent music has repercussions for what listeners remember, what listeners judge to be similar, and other musically significant functions" (73). Accurately describing this representational process is crucial if we wish to get the phenomenology right. Since our analytical models should align with auditory perception rather than compositional technique, we must ask whether, in listening, the mind differentiates "9/8" and "10/9" as distinct pitches with separate musical meanings, or instead perceives them as intonational inflections of a single focal pitch class.

When attempting to answer such questions, it is important to recall the earlier point that, in most listening situations, we do not attend to pitches as such. Unless engaged in tasks like melodic transcription, we rarely strain to conceptualize or categorize tones. Rather, we are attuned to a musical world that is, at a gross level, non-reflective and non-conceptual. Once we begin to focus deliberately on intonation, we may notice distinctions that would remain inaudible in a more effortless, attunemental, or 'natural' mode of listening. Such distinctions may be essential for the performer, but the listener need not consciously recognize them. The musician’s listening, in this sense, necessarily differs from the listener’s.

To determine whether we, as listeners, actually perceive a shift from "10/9" to "9/8" as a modulatory event, we must begin from this effortless, unforced mode of hearing—and only from there assess whether pitches are grouped into broader focal pitch classes or perceived as distinct. This is by no means an easy phenomenological exercise, but it is a crucial skill if we are to describe our modes of listening with any fidelity.

An important idea from music cognition to bring into this discussion is Dowling’s (1978) proposal—drawing on Miller’s (1956) classic cognitivist insight—that the diatonic scale contains seven discrete pitch classes because this number aligns with our limited capacity to remember and label items along continuous dimensions such as pitch frequency. This notion of cognitive restraint implies that the mind’s perceptual and memory systems, constrained by their own limits, tend to seek simplified organizations of the complex sensory input encountered when listening to, for example, the vast pitch collections used by Johnston or Partch. According to this view, the mind would categorize such sounds into focal pitch classes. Even though the composer may work within a broad gamut where "10/9" and "9/8" are clearly distinct, the listening mind simplifies them as different tonal shades of the same focal pitch class. In relation to D as the root (1/1), "10/9" and "9/8" share a certain E-ness, much as crimson and ruby share a redness. In other words, we do not hear the move from "10/9" to "9/8" as a modulation: to do so would mean perceiving a shift between distinct scale steps, whereas, in cognitivist terms, these pitches collapse into a single step within the same perceptual category.

But what range of pitches would be eligible to share this E-ness? Would an "8/7" also collapse into the same category as a "10/9"? Sabat (2008/2009) has observed—an observation with which many musicians would likely agree—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 "10/9" and "8/7", relative to a root "1/1", is larger than that threshold (48.77 cents), these two pitches can indeed be heard as distinct. Yet, if a "9/8" is also present in the harmonic space, the situation would perhaps change: since "10/9" lies 21.51 cents below, and "8/7" 27.16 cents above, the "9/8". The presence of this intermediary pitch could thus in theory bind the other two perceptually, serving as a center from which "10/9" and "8/7" appear as deviations smaller than 35 cents—thus allowing all three to cohere around a shared perceptual E-ness. In some compositions this might indeed be what happens, but from my own practice I know that this is not necessarily the case. In my composition Sakta vindar, for example, the E↑ ("8/7") is heard as a new perceptual category, while E- ("10/9") and E ("9/8") act as variations on one another.

Generally speaking, the smaller the interval, the stronger the tendency for its constituent pitches to collapse perceptually into one another. In the JI repertoire, the comma 49:48 is a fairly common melodic interval, measuring just above 35 cents (≈ 35.7 ¢). In my own compositions, this interval appears between the scale steps "7/4" and "12/7" in Av dagg och fattigdom and in the third movement of Andra Segel. Moving between these tones produces a clearly perceptible melodic motion involving two distinct pitches.

In other works, such as Mellan bleka stränder (efter Ni Zan), I have instead used the slightly smaller interval 56/55, occurring between "7/4" and "55/32". At about 31.2 cents, this interval falls below Sabat’s proposed 35 ¢ threshold. It is striking how, in this piece, the two tones begin to behave more like shadings of each other—producing a hazy, enharmonic blur rather than a definite melodic step. While the 49:48 in Av dagg och fattigdom remains a clear two-pitch gesture, the 56:55 in Mellan bleka stränder tends toward a perceptual fusion, the two tones sharing a single, almost 'fused' identity.

Yet it is also possible for intervals smaller than 56:55 to behave as discrete pitches. In the music of Catherine Lamb, the syntonic comma 81:80 (21.51 ¢) functions as a melodic interval, and throughout the JI repertoire one also encounters the septimal comma 64/63 (27.26 ¢) used in similar ways. There is, however, a notable difference in how these two commas are perceived melodically, and the general rule that smaller intervals exhibit a stronger tendency toward fusion helps explain their contrasting uses.

Pitches separated by septimal commas more readily resist enharmonic fusion than those separated by syntonic commas, even though the latter is only 5.75 cents narrower. In my experience, it is comparatively easy to establish a context in which the movement between "9/8" and "8/7"—a 64/63 relation—is heard as a genuine melodic motion, or as a modulation to a new scale step, as between the E and E↑ in Sakta vindar. By contrast, syntonic commas tend to manifest as enharmonic variants of the same pitch, and require special circumstances to be perceived as distinct (examples of this will appear below in the music of Sabat and Lamb). The movement between "9/8" and "10/9" thus more often sounds like a single pitch being re-intoned rather than a stepwise motion. Both relations imply new harmonic regions, yet one is perceived as melodic while the other tends to fuse—an outcome that turns, remarkably, on a mere 5.75 cents.

When dealing with intervals smaller than 35 cents, establishing the supporting conditions that allow pitches to separate becomes crucial. The degree to which tones separated by the 56/55 (31.2 ¢) fuse is highly dependent on context. In Om dagen stilla, the same interval is employed as in Mellan bleka stränder (efter Ni Zan), yet here it is perceived more as two distinct pitches. My intuition is that this difference arises from the more fully articulated harmonic space that operates in Om dagen stilla. Whereas Mellan bleka stränder is built from a limited, largely scalar pitch material, Om dagen stilla unfolds within a richer, more spectral harmonic field. This broader harmonic articulation enables the two tones forming the 56/55 to separate more clearly, as each becomes associated with different spectral sub-sections—different harmonic, tunable paths.

In Om dagen stilla, the 56/55 interval occurs between the pitches labeled "27/22" and "135/112" in the matrix shown in Figure 17. Both pitches are richly embedded within networks of tunable paths to other tones. The "135/112" functions clearly as a fifth partial within a spectral sub-section that also includes its related root as well as the fifth, ninth, and seventh partials. Similarly, "27/22" is supported by numerous tunable relationships: it serves as the seventh partial within another spectral sub-section encompassing the eleventh, ninth, and third partials.

Figure 17. Matrix of Om dagen stilla

In Mellan bleka stränder (efter Ni Zan), the 56/55 interval appears between "55/32" and "7/4". As shown in the matrix, the "7/4" is well integrated into the harmonic space, connected to numerous other nodes, whereas "55/32" (and "165/128") seem to hover at its periphery, each with only a single tunable connection. As a result, "7/4" emerges as the principal pitch, while "55/32" functions more as its microtonal variant. By contrast, in Om dagen stilla the more elaborated harmonic space gives the two tones forming the 56/55 distinct structural roles: both are harmonically significant in their own right. My thesis, then, is that this harmonic intelligibility—the degree to which a pitch is meaningfully anchored within a network of relations—facilitates our ability to perceive closely spaced tones as separate pitches rather than as fused shadings.

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

It is worth mentioning here pieces that employ enharmonic equivalents—intervals smaller than about 5 cents, such as 441/440 (3.93 ¢) or the schisma (1.95 ¢). Repertoire that makes use of such minute distinctions reveals yet another perceptual threshold: intervals so narrow that they do not even register as the kind of enharmonic shadings Sabat associated with intervals below roughly 35 cents. These ultra-small intervals simply sound like the same pitch. Only when the distance exceeds about 5 cents do they begin to produce a sense of distinct 'shades' rather than collapsing completely. 

For instance, the 7.71-cent difference between A#-- ("225/128") and B♭↓ ("7/4") is large enough that the two tones no longer function as enharmonic equivalents. Within a modal context, such a pair creates a loose-style situation in which the pitches appear as slightly different variants of the same tone. By contrast, if a piece employs only enharmonic equivalents smaller than 5 cents—and no comma-sized distinctions such as the syntonic 81/80—it will sound like the strict style rather than the free or loose, even if each pitch in the scale admits multiple spellings and the theoretical gamut is vast. Later in this text, we will analyze the piece Rosor och så liljor, which certainly looks like a free style piece due to the vast amount of pitches that are always changing intonation, yet it sounds like a strict style piece because all these changes happen within a 5 cent radius. 

In summary, the narrow band between roughly 5 and 25 cents appears especially crucial for enharmonic shading. Intervals smaller than this tend to collapse into a single pitch, while larger ones begin to acquire distinct identities as discrete scale steps. When we combine these observations with the cognitivist idea that it is difficult to keep more than seven pitch categories simultaneously 'present', we might hypothesize that, in many free-style contexts, the multitude of frequencies separated by commas smaller than about 35 cents overwhelms perception, leading to a simplified organization into focal pitch classes. Following Sabat, sounds less than approximately 35 cents apart often become perceived as different nuances of the same tone.

Yet this threshold cannot be treated as fixed. Our perceptual orientation toward sound is guided not only by cognitive constraints but also by a search for meaning. The 35-cent boundary is fluid, shifting with harmonic context and with the particular modes of listening that different pieces afford. Some contexts encourage fusion; others foster separation. 

Distinct or fused pitches in the free style

Before presenting examples in which comma-distanced pitches resist organization into focal pitch classes, I will begin with a piece in which such organization does seem to occur: my composition Marc (Sabat). The piece employs fifteen pitches per octave, as shown in Figure 19. Although these fifteen tones are clearly distinct in the compositional process, in performance they are perceived as nine fluid pitch categories (A, B, C, D, D#, E, F#, G, G#). This observation does not imply an endorsement of cognitivism, but rather describes how the music is actually heard—at least by me. The fact that even I, as the composer, intuit the piece in this simplified way suggests that listeners are likely to do so as well.

This does not, however, mean that Marc (Sabat) could be performed equally well in equal temperament, where this perceptual simplification of sense data would be translated into a literal simplification of tuning. The specific intonations carry expressive and psychological significance that must be realized precisely in performance, even if the resulting perception groups some of these pitches into shared categories. I will return to an analysis of this piece below (see Figure 25).

Figure 19

The 5-limit free style used in a piece like Marc (Sabat) may in fact be the form of just intonation that, in sound, most closely resembles the tuning practices of Western Common Practice Period music. In that repertoire—performed, for example, by a string quartet—the pitches are likewise fluid, yet each subtle intonational adjustment does not register as a modulation or the introduction of a new pitch category. A musician trained within the Western tradition will therefore likely find nothing unfamiliar in a free-style piece such as Marc (Sabat), since its intonations remain within the stylistic bounds of Western tuning practice: neither markedly flat nor sharp.

Lou Harrison even suggested that free-style JI corresponds to what string quartets playing classical music naturally do. In the chromatic sections of his Suite for Symphonic Strings, he refrained from notating JI explicitly—unlike in the other movements—and instead instructed the players simply to find "consonant" and "harmonious" intonations themselves (that is, to play it in JI). He claimed the result would not have differed much had he written out the ratios (Miller et al. 1998, 121). In my view, this reflects a certain idealism among JI composers of that generation, who believed JI to be the natural intonation of musicians. Contrary to Harrison’s assumption, research has shown that performers of Western art music tend to adhere surprisingly closely to equal temperament, even when playing without a tempered instrument (Burns 1998, 246). The range of pitch fluidity in such performance is therefore far narrower than in free-style JI—even within a simple 5-limit system, where relatively few comma levels are in play.

If one way of perceiving comma-distanced pitches in free-style music is to group them into focal pitch classes, there are also instances where we intuitively distinguish such pitches more clearly. A prime example is Marc Sabat’s Gioseffo Zarlino, an excerpt of which is shown in Figure 20. When analyzing our listening experience of this piece, we might ask whether the pitches A– and G+ are perceived as distinct from A and G. Based on my own listening, I believe they are—but only after the music has unfolded for some time, gradually attuning the ear to hear them that way.

The piece consists of short, repeated phrases in which pitch intonation shifts by syntonic commas. These subtle alterations, though initially imperceptible in categorical terms, draw our attention through repetition. As we become more attuned, we begin to recognize the microtonal differences as meaningful distinctions—though not necessarily as conceptual categories. The music itself guides us toward what is important to hear. In Gioseffo Zarlino, part of the work’s expressive meaning lies precisely in this categorical separation between comma-distanced tones. Over time, I found myself perceiving G and G+ as distinct pitches, after initially hearing them as variants of the same focal pitch class. As these comma differences began to acquire a sense of freshness, they came to feel almost modulatory—much like how a B♭ might sound within C major in equal temperament.

This sense of modulatory freshness—the categorical separation that occurs, for instance, between "5/3" and "27/16"—does not typically arise in the free style, nor does it occur in a piece like Marc (Sabat). From a cognitivist standpoint, one might follow Dowling and Miller in attributing this to our limited cognitive capacities and our tendency toward categorical perception. Yet, as Sabat’s example demonstrates, certain stylistic choices can invite a different kind of listening altogether. This suggests that we cannot accept a naïvely cognitivist model of perception. What Gioseffo Zarlino reveals is that pitch perception involves a participatory search for meaning—an active co-creation between listener and sound—rather than the passive 'processing' of pre-given sense data into discrete percepts.

Figure 20

Another instance of this phenomenon can be found in the reduced melodic duo version of Catherine Lamb’s Prisma Interius VIII. Although this music employs numerous pitches separated by commas, it does not convey the characteristic fluidity of the free style. Instead, the pitches seem to cohere within a single, expansive mode—a sound world perhaps best described (if we retain the present terminology) as a strict style with a large gamut that happens to include comma-distant tones. Yet the piece does not fit neatly into any of the categories outlined here: strict-style works were defined as having relatively small gamuts and no closely spaced commas, while free-style pieces generally lack the strong modal and affective 'glue' that binds Lamb’s pitches together in a unified Stimmung or rasa.

The reason Prisma Interius VIII manages to sound strict-style-esque despite its large number of pitches may lie in the unique way these pitches are spectrally articulated as part of a single overtone series: all are notated as partials above a very low (inaudible) 5 Hz fundamental. While it is always possible, in principle, to construct a low common root for any JI piece—a notional 1/1 from which all tones can be regarded as overtones—Lamb establishes this principle with unusual clarity. The piece unfolds slowly and pedagogically, beginning with an oscillation between G and G– (81:80) and gradually adding tones that fill out the spectrum. This process insists on the separation of G and G– as distinct partials: they represent the 80th and 81st harmonics, not different shades of the same pitch class.

A section such as the one at rehearsal number 10 (see Figure 21), which in isolation might appear to belong to a 5-limit free-style context, instead functions within what could be described as a single-overtone-series strict style. Interpreting it through the lens of the free style would therefore feel unsatisfactory, since its perceptual coherence arises precisely from the unified spectral framework that governs the entire piece.

Figure 21

A parenthetical point worth noting about Prisma Interius VIII is that its opening section largely consists of very small, untunable intervals—some as narrow as the prominent 81:80 (see Figure 22). This offers a clear illustration of an important point raised in the introduction: the boundary between microtonal music that employs JI-type intervals and what we might call integrated JI cannot be sharply drawn, but represents two idealized poles. In practice, most music moves fluidly and in complex ways between these extremes.

Categorically, Prisma Interius VIII begins as microtonal music using JI-type intervals, where the performer must rely on short-term memory to approximate the intervallic relations. Gradually, however, these microtones are reinterpreted as having always belonged to a tunable matrix—in this case, a single overtone series. During the opening minutes, the piece alternates between these two performance modes—tuning and approximating—but this oscillation is not readily audible to the listener. Even for the performer, the distinction is subtle: at times it becomes unclear whether one is tuning to remembered pitch relations or habitually approximating them through embodied familiarity. Between these two modes there exists a feedback system, each reinforcing and refining the other.

Figure 22

Poetic moods and tunable paths in the free style

If Harrison, in his Suite for Symphonic Strings, regarded the free style as Western musicians’ natural mode of intonation, he at other times considered it impossible to realize without instruments specially built for the purpose. His Simfoni in Free Style exemplifies this conviction: the work employs an array of differently tuned, custom-made flutes (Doty 1987), designed so that the tunings and key placements would guide the performers through the work’s intricate intonational pathways. The symphony has never been performed live, and what remains is a MIDI mock-up—but even in that form, what immediately stands out is the extreme flexibility of its pitch space, which indeed seems to justify the need for specialized instruments.

Within just two measures (8–9), five distinct 5-limit comma levels are traversed, from B- (one comma down) to B♭♭+++ (three commas up). This is, by any standard, radical intonation—astonishingly difficult to execute accurately at such speed. The resulting sound world is one of extreme malleability: de-centered, non-hierarchical, fluid, and almost liquid in its motion. The affective and poetic mood produced by this intonation stands in stark contrast to the clearly defined and unifying rasa-s of the strict style discussed 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 certain discord between the liquid-like tuning and the melodic and motivic language of Simfoni in Free Style. In fact, everything except the intonation sounds like what Miller and Lieberman, in their comprehensive study of Harrison, aptly call "vintage Harrison" (1998, 118). The music resembles his familiar idiom, only refracted through psychedelic intonation—vintage Harrison seen through psychedelic glasses. Despite its rapidly shifting comma levels, the underlying gestures and phrasing remain stylistically consistent with his other works. The piece even employs overt wolf intervals: for instance, a direct 27/20 dyad between D and G appears in measure 5, a striking example of harmonic extremity within an otherwise recognizably Harrisonian texture.

One could argue that the only way to render the intonation of this piece convincingly is if it is supported by an artistic idea and poetic mood wholly centered on this liquid-like stretching of pitch space—an expression that speaks in one voice with its tuning. In other words, it would need to be psychedelic Harrison seen through psychedelic glasses. Simfoni in Free Style, however, does not embrace such a poetic orientation, nor does it invite that corresponding mode of listening. Instead, it remains "vintage Harrison", in which many of the intervals simply sound out of tune.

Simfoni in Free Style thus offers a compelling example of an important principle: the intonation system and the composition must speak in a single voice—share the same artistic intention—if the tuning is to sound integrated rather than jarring. For comparison, an almost equally rapid movement between comma levels occurs in the passage from Prisma Interius VIII shown in Figure 21. There, the music moves swiftly between four comma levels, from B♭ to D#---, yet it sounds entirely natural. This is because the passage articulates a clearly defined harmonic territory—one that has been patiently built and internalized through the piece’s gradual, evolving structure.

Tunable paths in the free style

Even in a free-style composition, only a little is required for temporary reference points to emerge—points that can make certain pitches sound out of tune, even when nothing in the notated succession of tones suggests this. In Figure 24, adapted from Figure 2 above so that the pitch sequence now reads E:D;B–:A–:C#–:G#, the A– will likely sound too low even within a free-style context, as the surrounding D and E provide local tuning references with which it forms wolf intervals. Such contextual anchors inevitably arise—however weak or fleeting—since our perception of pitch is continually shaped by the tones that precede and surround it (see Krumhansl 1990, 283).

Figure 24

This does not change simply because a composer calls the music free style or celebrates the supposed 'liberation' of intervals from fixed gamuts through locally determined intonation. Musical perception is not as purely local as Harrison seemed to believe. Even in free-style writing, the composer must remain sensitive to emerging modalities and hierarchies as they form in the listener’s ear. After all, the very logic of JI presupposes a 1/1. The composer must therefore sense which tones exert the strongest tuning influence and adjust the musical context accordingly. Such adjustments need not always involve altering pitches themselves: in Figure 24, for instance, clarifying the phrasing—by grouping the D and E together and beginning a new phrase on the B–—can already reduce the perceptual pull those tones have on A–. Changes in phrasing, articulation, and context guide the process of enacting meaningfulness, continually reshaping how intonation is heard and understood.

In my free-style pieces, I have sought to emphasize how the absence of a unified mode—and of the affective quality that such a mode typically brings forth—places the listener in a different kind of present moment, less bound to a single modal center or hierarchy. The music thus remains perpetually 'new', modally unpredictable, and affectively shifting. Rather than being unified by a stable, overarching mood, each phrase or section can possess its own distinct affective color, brought about by the freedom of intonation itself. In Marc (Sabat), for instance, the use of free style accentuates the music’s fragmentary character: there is no shared modal hierarchy that binds the fragments together, and this very discontinuity becomes part of the piece’s expressive logic.

An excerpt from Marc (Sabat) is shown in Figure 25. By tracing the tunable paths, we can see clearly how this melodic and harmonic writing differs from the strict style shown in Figure 7. To make sense of the shifts between A and A–, and between D and D–, the performer must react swiftly, grasping the notes that support each modulation. For example, in measure 27, the second violinist must play a D–, whereas in measure 23 the same instrument performed a D. Here, the player must quickly reorient their intonational reference points—ignoring the F#– at the beginning of measure 27 (which would otherwise form a Pythagorean third)—and instead tune to the immediately preceding sonority, where the first violin and trumpet provide a clear E– and A– forming a perfect fourth.

Similarly, the modulations to the major chords [B-, D#--, F#-] and [E-, G#--, B-], which emerge from the initial G Ionian moment [G, A, B-, C, D, E-, F#-], depend on the performers' ability to anchor intonation to specific guiding tones. In measure 27, the musicians must 'catch' and build the modulation upon the emphasized B– (m. 24) and E– (m. 26), allowing the tuning to unfold along these carefully established harmonic reference points.

The tuning in this passage is exceptionally fragile: crucial reference points arise from the immediately surrounding tones rather than from any overarching modal framework. Across the three systems shown in Figure 25, four comma levels are employed. Such rapid modulations—where each pitch connects to the next through a tunable link and new modalities must be built upon tones heard only once—leave virtually no margin for error. The musicians must remain constantly vigilant, maintaining acute awareness of each interval and aggregate to ensure the tuning holds together moment by moment.

By carefully considering instrumentation, composers can assist performers in this demanding task. When writing for string instruments, as in Marc (Sabat), open strings and natural harmonics can serve as vital tuning aids, providing momentary anchor points. Consider, for instance, the stark modulation from [E- A- C] to [F#- A D] in measure 29. The challenging comma movement in the second violin part—from A– to A-—is made manageable through the use of open strings and harmonics: A- is reached via a tunable path, while A is accessed through an open string.

Furthermore, orchestration can be used to clarify these relationships. By emphasizing the top line, C to D, as the main gesture (with the C 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 internalize. The two A's are perceived as belonging to a secondary voice that derives its intonational orientation from the clearly articulated primary line.

Figure 25 (Trumpet in Bb)

Another of my free-style pieces is Radii solis, et sternet (sibi aurum) quasi lutum (see Figure 26). In this work, the music employs five distinct comma levels—one more than in Marc (Sabat). Yet the motion between them unfolds slowly and gradually, primarily along tunable paths. The resulting listening experience is similar to that of Marc (Sabat): the music retains a fragmented quality, emphasizing present-moment harmonic constellations rather than establishing a unified modal mood or sustained affective center.

Figure 26

Summary of the free style

Before turning to the loose style—the final category to be discussed—we can now summarize the characteristics of the free style. In essence, it stands as the opposite of the strict style. Rather than working with a fixed gamut, it operates with an unbounded and fluid collection of pitches. Instead of exhibiting the strong, unifying modal affect typical of small gamuts, free-style pieces often lack a global modal cohesion. This does not mean that they are devoid of affect, but that their affective qualities are shaped more locally: pitches relate less to a shared modal framework and more to their immediate surroundings. Consequently, the craft of composing tunable paths becomes more delicate, as performers can rely less on modal context or scalar familiarity.

The sometimes teeming number of pitches in a free-style piece raises a fundamental cognitive question: how are these distinctions represented in listening? Do we hear "27/16" and "5/3" as the same pitch in different intonations, or as distinct tones altogether? In one of my own works, such distinctions were found to blur perceptually, suggesting that this collapse may be typical of the free style—that we do not perceive a move from "27/16" to "5/3" as a modulation in the way we would a change from B to B♭ in equal temperament. Yet this is not always the case. Composers such as Sabat and Lamb demonstrate listening practices in which pitches separated by commas can be heard as categorically distinct.

Within the free style, then, we might speak of two subcategories marking the ends of a continuum: music that tends to invite a simplified organization into focal pitch classes, and music that invites categorical separation of comma-distanced intervals. Comparable subcategories will also be significant in the loose style, to which we now turn.

Loose style

While the distinction between strict and free style—and the importance of differentiating them—was first articulated by Lou Harrison, the loose style introduced here represents an intermediary category. Its defining feature is a hierarchical ordering of pitches that governs the degree of intonational flexibility. Some pitches, structural in nature, remain fixed, while others are granted freedom. Pitches lower in the hierarchy may be freely replaced by their neighboring variants (pitches separated by commas), whereas the structural tones higher in the hierarchy generally cannot—unless intentionally altered for a modulatory effect, in which case such changes would otherwise sound out of tune.

In my own loose-style compositions, the structural tones are often familiar harmonic anchors such as [1/1, 3/2] or [1/1, 4/3, 3/2]. The four-pitch pattern [1/1, 5/4, 3/2, 15/8], in particular, recurs frequently across my works. Three different realizations of this pattern are shown in Table 2. In this table, these four structural pitches are arranged into three hierarchical levels: "1/1" occupies the highest level, "3/2" the second, and "5/4" and "15/8" the third. These levels illustrate that the higher a tone stands within the hierarchy, the greater the modulatory effect of altering it by a syntonic comma will be. The fourth level, beneath the structural tones, comprises all remaining pitches employed in the pieces.

Table 2

These tables originate from observations made in my compositional practice before the concept of the loose style had taken shape. For instance, while composing mot våren bortom havet (the first piece listed in the table), I noticed that lowering D by a syntonic comma to D- had jarring effects, often sounding out of tune unless prepared with extreme care. By contrast, altering A to A– felt smooth and natural, while changing B- to B produced a clear modulatory shift—but not as strong as altering G to G- or G+, which most readily sounded out of tune.

Through many such practical experiments, I began to realize that these behaviors pointed to an idiom distinct from the free style. The later formulation of the loose style in JI thus provided a theoretical explanation for patterns I had already encountered in practice. Interestingly, I also found that the hierarchical structure represented in the [1/1, 5/4, 3/2, 15/8] pattern corresponds closely to the basic pitch-space models developed by Krumhansl, derived from empirical studies of tonal hierarchies (see Krumhansl & Cuddy 2010)—a convergence that invites further study.

To illustrate how these hierarchies operate in practice, let us examine seven measures from att sjunka i doftande klöver for violin, cello, and piano. Figure 27 shows an early version of these bars. At first glance, the score appears entirely tunable—seemingly written by a composer who has taken great care to maintain coherent tunable paths. The only potential concern might arise in the violin's third pitch: this brief three-note melody outlines a wolf fourth (E to B-), which could seem problematic.

However, a closer look reveals a strong bass motion in both the cello and piano, moving from A to D. Within this context, the violin's E is tuned as a perfect fifth to the A, and its B- as a just major sixth to the D. The relationship to the bass line thus becomes the decisive factor, effectively concealing the apparent wolf interval between E and B-. In performance, this contextual tuning renders the wolf interval unproblematic, showing that its theoretical 'wolfness' is perceptually absorbed within the broader harmonic relation.

Figure 27

Having concluded that the tunable path is smooth, we are therefore surprised, upon listening, to hear that the violin's A- in measure 261 sounds too low—indeed, out of tune. This is not at all evident from simply reading the score. On the contrary, since the A- in Figure 27 is preceded by both a B- and an E-, it appears correct in that position; it forms the third interval in a series of 4/3s. What occurs in listening, however, is the emergence of a hierarchy of pitches characteristic of the loose style of JI.

The emphatic D in the low register of measures 258–259, played by both piano and cello, lingers in memory through measure 261 and causes the ear to expect an unlowered A, forming a 3:2 relationship to that D. There is no doubt that D functions here as the basis pitch—as the fundamental or root—and this shapes the desired intonation of the subsequent A. As Krumhansl (1990) observes, "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 this passage, the D's stability ensures that it remains mentally present long after sounding, exerting a stronger influence than the more recently heard, but less structurally significant, E-.

The A can, in principle, be tuned either to the D (as A) or to the E- (as A-), but not to both. Here it must be tuned to the D, since the mind’s hierarchical organization of pitch exerts greater force than short-term melodic tunability. Through this process, the A assumes a structural role one level beneath the D—serving as its tunable 3/2—and thereby exemplifies how hierarchical relations override local tunable paths in the loose style.

To avoid hearing the pitch as false in relation to the immediately preceding B- and E-, we must perceive it as arising within a modal—or tonal, to use Krumhansl's term—context. If this sense of a modal hierarchy is absent when hearing measures 260–261, the A will inevitably sound out of tune. When composing in the loose style, the composer often encounters passages that, in notation, may appear 'wrong'. Yet these hierarchies create compositionally rich situations in which the ear is more inclined to accept ostensibly untunable intervals—such as wolf fourths—than to tolerate structural pitches shifting by commas.

Figure 28

Figure 28 shows the penultimate version of the passage from Figure 27. The A- in measure 261 has been changed to an A, a revision that also required altering the E- in the same measure to E (though not the preceding E- in the measure before). As a result, the cello’s E in measure 261 now forms a wolf interval with the following B- in measure 262. While this interval does not sound particularly out of tune within the established modal context, the entire passage retains a faint perfume of out-of-tune-ness—an expressive shimmer produced by the network of direct wolf intervals.

When composing this passage, I did not regard these wolfs as a major problem. Earlier in the piece, the frequent use of awkward melodic outlines and skips—often exposing Pythagorean thirds and wolf fourths—had already established a deliberately thorny and angular melodic style. The exposed wolves in Figure 28 therefore fit naturally within this idiom, and I chose not to rewrite the passage completely.

At the same time, I felt that the section in Figure 28 contained slightly too much of this quality. To address this, I added a low D as a pure fifth beneath the A in measure 261. This small adjustment acted as a reminder of the modal center and simultaneously masked the 10:9 between B– and A by redirecting attention to the 5:3 between B– and D. As a result, the perfume of the wolf's out-of-tune-ness dissipated, while the counterpoint retained its thorny and angular character. Finally, to help the performers navigate rapid shifts—such as between E- and E—I ensured that these passages were supported wherever possible by open strings and natural harmonics. The final version is shown in Figure 29.

Figure 29

The solo double bass piece Väntar där dimma uppstår employs a similar use of consciously imperfect JI writing to achieve a particular poetic mood. As in att sjunka i doftande klöver, the 'imperfections' in the tunable paths give rise to expressive effects that would be impossible within completely perfect JI melodic writing—that is, a style relying almost exclusively on tunable intervals.

In measure 187 (Figure 30), a major ninth is played—not the tunable 9/4, but the untunable 20/9. The performer finds this by revoicing the tunable minor seventh (9/5) from the previous measure, keeping the G+ stable while transposing the A up two octaves. Because the pitch classes were already established in the 9/5, the resulting 20/9 sonority in measure 187 does not necessarily sound 'wrong' or out of tune. Both pitches belong to the mode, yet the interval produces a fascinating, energetic pulsation that yearns for resolution.

To perform this 20/9, the musician first tunes the 9/5 in measure 185, keeps the G+ steady, and adds the high A as a natural harmonic—resisting the impulse to lower the G+ to G in order to form the more stable, just major ninth 9/4. The effect is one of delicate tension: an intentional poetic imperfection that animates the stillness of the surrounding sonorities.

Figure 30

While the main purpose of this text is to argue for the importance of JI composers taking tunability seriously, it would be a mistake to demand that all music employ only tunable intervals. As the examples from att sjunka i doftande klöver and Väntar där dimma uppstår show, there is also an expressive place for untunable intervals. Their careful use can open unique poetic and affective dimensions—moments where the ear is drawn not to stability, but to the delicate play of tension and release that arises precisely from imperfection.

Implied modal modulations in the loose style

One of the features of the loose style that I find particularly interesting is how its relatively small gamut of pitches—compared to the free style—often leads to a greater perceptual differentiation between comma-distanced ratios. Listeners are, at times, less inclined to categorize such pitches as mere variants of the same focal pitch class. In the loose style, we can often hear and distinguish, for instance, "8/5" and "128/81" as genuinely different pitches—something we are not usually attuned to in the free style.

When we analyze these moments of heightened discrimination, it becomes clear that the reason lies in the implied modulation they convey. Alterations between comma-distanced pitches in the loose style often suggest shifts with deeper harmonic or modal implications than a mere microtonal adjustment of a single scale degree. In such moments, the change in intonation serves a different purpose: not simply to tune locally beatless intervals, but to articulate a broader transformation within the piece’s modal or harmonic orientation.

Yet this is not always the case in the loose style. In att sjunka i doftande klöver, discussed above, the alteration between E and E- passes unnoticed, without any sense of modulation. There, the pitches on the lowest, fourth level of the hierarchy adapt primarily to tune accurately to the structural tones rather than to articulate modal change.

In other loose-style works, such as Sommarberg, i glömska, and Vid stenmuren blir tanken blomma, the situation is quite different. In these pieces, pitches on the lowest hierarchical level are replaced by their neighboring comma variants to express modulations between closely related modal scales—scales that, in equal temperament, would share the same pitch content, such as D Ionian and B Aeolian. In such moments, the altered pitch stands out vividly, recognized as belonging to a new category and carrying a distinct sense of modulatory freshness.

One might even describe these pieces as continuously modulating among different strict-style scales rather than remaining within a single loose-style mode. Interpreting the music in this way offers an alternative reading of Dowling's suggestion that only about seven pitches can be actively held in memory: each modulation effectively resets the listener's perceptual field, sustaining clarity by constantly reestablishing a limited tonal framework.

Figure 31 (transposed score; trumpet in Bb)

Examples of such modulatory comma changes can be found in Sommarberg, i glömska for violin, viola, and trumpet. In this piece, the music frequently modulates between the D Ionian mode [1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] and the B- Aeolian mode [5/3, 15/8, 1/1, 10/9, 5/4, 4/3, 3/2]. The only difference between these two scales is the alteration of the pitch class E—from 9/8 to 10/9.

An excerpt of such a modulatory passage is shown in Figure 31. In measure 127, the music clearly begins in D Ionian but gradually modulates through the phrase, arriving in B- Aeolian by measure 136. The shift is achieved as the E changes to E- (tunable as a 5/3 to G and a 4/3 to B-) in measure 135. In measure 139, the phrase begins its return to D Ionian, accomplished by changing E- back to E (tunable as a 9/4 to D) in measure 140.

Throughout this passage, the alternation between E- and E makes the modulation between B- Aeolian and D Ionian distinctly perceptible—not only as a change in modal center but also as a shift in affective quality. In equal temperament, such a passage would scarcely register as a modulation; diatonic Western music can easily end a phrase on the sixth degree without implying a move to the relative minor. In JI, however, such comma-sized shifts are strikingly impactful. Conversely, in a free-style context—where comma alterations are constant—the same change would likely go unnoticed. In the loose style, by contrast, these subtle transformations are heard as meaningful: the pitch changes that articulate them, such as E to E-, stand out with genuine modulatory significance.

Figure 32

Another loose-style piece that employs comma movements to indicate modulation is Vid stenmuren blir tanken blomma for violin and viola. In the section shown in Figure 32, the music alternates between D Ionian and E- Aeolian. The difference between these scales is greater than that between D Ionian and B Aeolian discussed above, as it also involves a change of pitch class (C to C#-). Yet the modulation is made perceptible not only through this C-to-C#- shift but equally through the alteration of A- to A.

In measures 180–183, the music is clearly in E- Aeolian. As soon as the natural A (rather than the modal A-) appears in measure 185, it already foreshadows the coming modulation: we anticipate that the C will soon rise to C#- in the next bar. The change in A thus already carries the modulation. When the C#- arrives, it sounds natural and expected, since part of the transition has already taken place with the A. Functionally, this A acts as a Pythagorean sixth—an untunable interval, here performed as an open string—against the low C in measure 184, and it thus clearly marks a modulatory gesture.

In equal temperament, the modulation would only be felt with the introduction of the C#. In JI, however, the weight of the modulation is distributed between the A and the C#-, rather than resting solely on the latter. Later in the section, the modality settles more firmly into E- Aeolian, but between measures 184 and 208 (not excerpted here), the A- continues to alternate with A, suggesting subtle, oscillating modulations between E- Aeolian and D Mixolydian.

In an earlier section of the same piece, beginning at measure 84 (see Figure 33), a modulation that might otherwise sound abrupt—from A Ionian to C Ionian—is rendered smooth through anticipatory intonation. Before the 'modulation proper' occurs, the pitches E and B are already lowered to E- and B- while the music is ostensibly still in A Ionian. Much of the modulatory work thus happens preemptively: by the time the C enters in measure 90, the ear has already adjusted to the new harmonic environment. As a result, what might have sounded jarring in equal temperament instead feels natural and continuous in JI.

Figure 33

A final example can be found in Livets eget bleka flöde on page 21 (see Figure 34). In a melodic sequence moving between D and G, we encounter an E. Why an E and not an E–? In the octave positions used here, an E– would be tunable both to the D as a 9/5 and to the G as a 12/5, yet the passage is written with an E instead. This decision was deliberate: an E- would have implied an undesirable modulation—drawing the harmony toward G or C rather than maintaining its orientation around D or A. In other words, E- would have suggested tonal hierarchies and hinted at harmonic regions foreign to the piece’s intended modal space.

Figure 34

We have now reach a point where we can briefly summarize our findings about the loose style. As a style of integrated JI, the loose style is defined by the presence of modal hierarchies—not because such hierarchies are absent in the strict or free styles, but because here they function in a distinct way that directly shapes the fluidity of pitches and the construction of tunable paths. This distinct organization warrants its classification as a separate style.

A loose-style gamut comprises both structural and non-structural pitches. The structural tones occupy the upper levels of the modal hierarchy, while the non-structural ones reside at the lower levels. Structural pitches exhibit two primary characteristics. First, they are more vividly retained in short-term memory and thus serve as reliable tuning references even after other, potentially conflicting tones have been played. Second, replacing them with their neighboring comma variants produces a pronounced modulatory effect. For these reasons, structural pitches tend to remain fixed in their intonation and are seldom substituted by nearby variants.

Non-structural pitches, by contrast, possess the inverse qualities: they are less easily retained in memory and more readily replaced by neighboring comma variants—that is, they are fluid. Depending on the piece, these neighboring variants may be treated as freely interchangeable, used simply to accommodate local tuning conditions, or they may be given more audible distinctness by articulating subtle modulations between closely related modes (for instance, between C Ionian and D Dorian). Excerpts from my own compositions have illustrated both tendencies.

The loose style, then, stands as a space of balance: between fixity and fluidity, between structural stability and local adaptation. Its poetic power lies in this interplay, where even the smallest comma shift can breathe life into the modal fabric, allowing comma shifts to become a vehicle of expressive and structural meaning.

Modulation as new harmonic relationships or as new microtonal inflections

Before moving on to the conclusion, one final issue arises naturally from the preceding discussion: whether the modulatory 'effect' in integrated JI is primarily achieved through the establishment of new harmonic relationships or through the introduction of new microtonal inflections. In other words, when we hear a passage shift from C Ionian to D Dorian, what makes it sound like a modulation? Is it that the fifth between D and A becomes a pure 3:2—making new melodic and harmonic connections possible—or is it rather the subtle microtonal raising of A (from A– in C Ionian to A in D Dorian) that creates the perceptual freshness?

Since these microtonal adjustments are, in absolute terms, quite small, one might argue that the modulatory effect arises less from the difference in cents and more from the emergence of new harmonic possibilities among tones that previously could not relate tunably. An example of this can be found in my brass nonet Tusen tysta skogar. The first part of the piece is largely situated in the G Hypoaeolian mode ["3/2", "8/5", "9/5", "1/1", "9/8", "6/5", "4/3"], but at crucial cadential points the "4/3" is replaced by "27/20", enabling new harmonic alignments: a 9/5 with the "3/2", or a 4/3 with the "9/5". We may thus say that the freshness of this modulation stems not so much from the slightly higher intonation of the "F" ("4/3") but from the fact that entirely new harmonic relationships become available to the ear. The change is relational before it is microtonal.

But if new harmonic relationships were the only factor responsible for producing the sensation of modulation, then such modulations should stand out just as clearly in equal temperament, since in an equally tempered version the statistical relationships between pitches would remain identical. The harmonic event—the introduction of a previously avoided relation between G and F, for example—would still occur. Indeed, some trace of the modulatory quality remains even in ET, yet it is markedly weaker. This suggests that the microtonal inflections themselves also play a crucial role.

The effect of modulation does not merely come about because the just intervals dictate certain 'rules' of melodic and harmonic writing, but also because of the microtonal profiles of the pitches that clarify these rules. It is only in the justly tuned context where ratios are related as simple ratios as this effect truly becomes clear. 

I am reminded here of a statement by Terry Riley: 

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

Riley’s remark illuminates precisely this phenomenon: tuning does not merely “adjust” relationships; it reveals them, sharpening the harmonic image until previously hidden nuances become perceptible.

Music with enharmonic equivalents

The pieces in which I have employed 'enharmonic equivalents' offer valuable insight into the question of whether modulation arises primarily from a reconfiguration of harmonic space or from new microtonal placements of the same 'scale steps'. As discussed earlier, the term enharmonic equivalent here refers to pitches that lie less than five cents apart. When tones produced by very different tunable paths—such as "315/176" and "25/14"—land within this narrow frequency range, the music can modulate through them to distant harmonic regions almost imperceptibly, without audibly altering the microtonal profile of its pitch classes.

In Ljusomflutna, sakta vindar, for example, the viola’s pitch in measure 56 (a "12/11") is enharmonically equivalent to the violin’s pitch in measure 60 (a "35/32"), as shown in Figure 35. If modulation were primarily the result of reconfigured harmonic relationships, we would expect the "35/32"—functioning as a 5-limit interval relative to the viola’s 7-limit tone in m. 60, rather than, as "12/11", an 11-limit interval (11/4) to a Pythagorean tone—to sound distinctly modulatory. Yet it does not. The piece retains the impression of being in the strict style, though one more complex and tonally ambiguous than the strict style of Vårbris, porslinsvas.

Figure 35

Another example can be found in the excerpt from I luftens svala dunkel shown in Figure 36. Here, two distinct types of E-flats, C’s, and F’s occur. Although these enharmonic equivalents imply vastly different harmonic relationships, I would argue that their replacements do not produce any noticeable modulatory effect. Rather, the passage retains the character of a strict-style piece.

Figure 36

Experiences from these pieces—where enharmonic equivalents make possible new harmonic implications yet still sound as though they belong within a strict-style context—suggest to me that the implied modal modulations characteristic of the loose style are, to a significant extent, a product of changes in microtonal scale positioning.

Among the styles discussed so far, the enharmonically flexible strict style occupies a particularly revealing position. It appears, in my own practice, most frequently in non–5-limit JI works (for instance, in 7-, 11-, and 13-limit pieces), but its importance extends beyond my compositional output: it exposes how just intonation can sustain scalar clarity and flexibility simultaneously. This style sounds 'strict' yet its harmonic space remains quietly unstable and polysemic.

The enharmonically flexible strict style music achieves a polysemic and ambiguous sense of mode—one with an inherently center-shifting character—despite sounding, on the surface, like a strict-style piece that exploits only a single scale. In this sense, the enharmonically flexible strict style closely resembles the loose style, as both occupy a position between the strict and free styles.

In the loose style, the fluidity of certain pitch classes—able to drift by syntonic commas—creates music that continually shifts its implied harmonic center, oscillating between modes such as D Dorian, C Ionian, and A Aeolian. In the enharmonically flexible strict style, similar subtle shifts of tonal centers occur, but here without the pitches drifting by more than five cents. The result, for me, is a stimulating balance between the stability of a fixed scale and the openness of an ambiguous, polysemic pitch space.

Figure 37

The piece in which I have perhaps taken this idea the furthest is the viola or violin solo Rosor och så liljor. This work employs a simple seven-tone strict-style scale, but by allowing each pitch to move within a radius of no more than five cents, a remarkably rich field of harmonic possibilities opens up. Owing to the complexity of this particular scale, I decided not to notate the piece exclusively with JI accidentals, but instead to combine cent deviations, JI accidentals, and ratios.

A short excerpt of how the notation would appear in standard JI notation is shown in Figure 37; there, the music looks like an unrestrained free-style piece. The chosen notation in Figure 38, however, reveals this apparent freedom to be contained within a highly disciplined framework—a strict seven-tone grid in which each scale degree may flex only within a narrow band of five cents.

Figure 38

A similar effect occurs in my large keyboard piece Som regn. Although the pitches in this work cannot physically move because of the fixed tuning, I chose to notate the implied harmonies in a manner similar to Rosor och så liljor. Here, the cent deviations indicate the pitches' out-of-tuneness rather than the amount by which they must be adjusted. Som regn thus operates, in many ways, like an enharmonically flexible strict-style piece—but with the crucial difference that the necessary adjustments take place in the listener’s or performer’s mind rather than on the instrument. 

Figure 39 shows a brief excerpt from Som regn. Note how the A and E are notated differently depending on their harmonic context: the A functions both as an 11/9 to C and as a slightly out-of-tune 8/7 to B-, and it is often difficult to decide which interpretation should take precedence. The A is, in other words, polysemic and multi-stable—both readings are equally valid. For this reason, it is appropriate to describe such tunings as a kind of temperament—one in which pitches can possess multiple spellings and simultaneous identities—even though no pitch has, in a technical sense, been tempered.

Because Som regn is written for a fixed-tuning instrument, certain idiomatic features—such as the use of direct, untunable melodic intervals—suggest that the piece is not merely an instance of the enharmonically flexible strict style, but rather belongs to a subset of the informed style of strict style as a 'tuning'. The informal term I have given to this subcategory—applied to works for pre-tuned instruments that employ enharmonic equivalents—is JI-temperaments. Other pieces in this category include Nästan lyrik for keyboard and piano, and Mellan vita stammar for keyboard, percussion, and double bass.

Figure 39

Conclusion

My main purpose in this text has been to give the reader an insight into the craft of writing tunable music within an idiomatic and integrated justly tuned idiom. The discussion began with a basic description of what JI is, emphasizing that just intonation concerns the tuning of sounds into resting, fused, and periodic entities. As a microtonal system, JI certainly introduces new interval sizes and microtones and asks performers to approximate these as closely as possible—but it is much more than that.

Because rational notation is often used primarily for microtonal purposes, this practice needed to be clearly distinguished from the integrated JI explored here, which employs JI as a way to create tunable contexts for melodies and harmonies. The former was bracketed as microtonal music that uses JI-type intervals, while the latter—integrated JI—was treated as a tuning-based compositional practice in which the very grammar of the music arises from tunability itself.

Research has shown that most music around the world is performed by reproducing pitches in accordance with cultural and stylistic norms. The musician performing integrated Just Intonation, however, is not merely reproducing pitches (though they too must rely on mental representations), but continually tuning them to one another within an interconnected, tunable harmonic space. The very act of producing pitch is therefore fundamentally different in integrated JI than in most other musical traditions.

The compositional craft required to support this kind of performance is likewise unique. It is ultimately the composer’s task to ensure tunability—to connect tunable intervals into coherent tunable paths through which the music can be realized.

If every interval in a JI piece had to link directly to another through an immediately tunable connection—like a domino chain or a relay race—it would be impossible to play even a simple ascending 5-limit diatonic scale, since both major seconds (9:8 and 10:9) as well as the minor second (16:15) are untunable. It was therefore established that a piece in JI need not be restricted to the exclusive use of immediately tunable intervals.

The justification for this rests, on the one hand, in our capacity to tune by short-term retention of previously heard pitches, and, on the other, in the possibility of contextual, modal tuning—a kind of emergent tunability that arises within a given mode. These two forms of non-simultaneous tuning—short-term retention and contextual, modal tuning—are weaker and less stable than simultaneous tuning, yet their implications for both the performance and composition of JI remain profound.

Given these premises, a large part of the compositional craft in JI was shown to lie in achieving what might be called a proper balance between tunable and untunable intervals. What constitutes this proper balance cannot yet be defined by any fixed rules; the composer must instead rely on a cultivated sensitivity to tunability.

After an initial demonstration of how short-term retention can make identical pitch collections sound either well or poorly balanced depending on interval order, the discussion turned to the main topic of modality. It was observed that the balance between tunable and untunable intervals depends on the strength of a piece’s modality. The simple claim advanced was that strong modalities tolerate a higher proportion of untunable intervals, whereas weak modalities require a greater presence of immediately tunable ones. This continuum of modal strength was then articulated into three basic styles of integrated JI composition—strict, loose, and free—which formed the main focus of the text.

In the integrated strict style, a strong modality affords a balance between tunable and untunable intervals that leans generously toward the untunable side. Intervals that are technically untunable within this system—such as Pythagorean thirds and sixths—can sound more correct than their unavailable, tunable 5-limit counterparts, provided they are properly prepared within the mode. Pieces in the integrated strict style achieve this strong sense of modality through the exclusive use of a small, fixed gamut of pitches.

Pieces in the free style are generally characterized by a vast—indeed, theoretically unlimited—gamut of pitches. In this style, the composer cannot rely on any unifying mode or tonal hierarchy beyond those that arise momentarily in the course of the music; intonation depends primarily on the immediately surrounding tones and on temporary reference points. The free style is therefore the most fragile of the three and demands the most careful and deliberate writing. Here, the balance between tunable and untunable intervals must generally be such that the majority of pitches are directly tunable.

In the loose style, a hierarchical ordering of pitches determines the intonational fluidity of pitch classes. This hierarchy often creates a preference for preserving certain untunable intervals, rather than altering them to their tunable comma variants, when the pitches involved are high in the modal hierarchy and thus structural in nature. Pitches at these higher levels are vividly retained in short-term memory and can serve as tuning references even after other, potentially conflicting tones have sounded.

Pitches lower in the hierarchy exhibit the inverse characteristics. For these, tunable Ptolemaic intervals are generally preferred over untunable Pythagorean ones, and their short-term retention is weaker. In practice, the structural pitches at the top behave much like those in the strict style—immovable, since altering them would imply a strong modulation—while the lower-level tones behave more like those in the free style, where such changes may pass unnoticed or produce only subtle, local modulations. The loose style thus occupies the middle ground between the strict and free styles, incorporating elements of both.

An important theme throughout this text has been the search for a mode of theorizing that corresponds to how we actually perceive pitches when listening to JI music, rather than one that merely mirrors the compositional techniques used to produce them. The impetus for this theme arose directly from the challenges encountered in the three styles and the phenomenon of enharmonic equivalents.

In the free and loose styles, the question arose whether we hear pitches separated by commas as distinct categories, or—as in Western classical music—as different intonations of the same focal pitch classes. It became clear that no universal answer could be given: the outcome depends on the specific musical context. In many cases—perhaps in most—the listener’s perception tends to group nearby ratios into shared focal pitch classes. Yet certain stylistic conditions, often associated with minimalistic or highly reduced aesthetics, can resist such grouping and instead make the comma-separated pitches perceptually discrete. Within both the free and loose styles, we may therefore gesture toward two sub-categories: one that invites categorical separation of comma-distanced intervals, and another that favors their simplified organization into broader focal pitch classes.

When discussing the strict style as a tuning—that is, the strict style written for fixed-pitch instruments—I noted the possibility of constructing complex rational tunings in which the tunable origins of the intervals become obscured. This raised the analytical question of whether such music can still be said to be in Just Intonation. When the ratios between pitches—each separated by several tunable steps—grow too complex, we no longer perceive the result as belonging to JI.

In I Sommarluft, for instance, the seemingly ordinary minor third between "128/81" and "15/8" yields the compound ratio 1215/1024—one schisma higher than the already untunable Pythagorean minor third of 32/27. The result is an out-of-tune version of an already untunable interval. In the first movement of I Sommarluft, where such relations abound, little perceptual information signals to the listener that the piece is in JI at all. Within the strict style as a tuning, I therefore proposed the subcategory rough strict style as a 'tuning' to account for this phenomenon. It contrasts with the informed strict style as a 'tuning', in which the ratios retain their tunable relationships thanks to closer lattice proximities and more compact, clearly articulated harmonic spaces.

When discussing the phenomenon of enharmonic equivalents, we saw that music which, on paper, appears to belong to the free style may in practice sound like the strict style—or at times like the loose style, since enharmonic equivalents can occur there as well, though this has not been examined in detail here. This happens when the total gamut of pitches remains tightly centered: if all tones fall within roughly five cents of seven focal pitches, the ear tends to perceive the result as a single, coherent mode rather than as a fluid or modulating field. Such music reveals that what defines a style is not the number of ratios employed but how the ear organizes and enacts them. The categories and subcategories summarized in Table 3 are therefore not merely compositional techniques but perceptual forms of attunement—distinct modes of listening through which intonation becomes meaning.

Table 3

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 categories in Table 3 should not be regarded as rigid genres of JI music. Many works defy these classifications, moving fluidly between them. Rather than fixed types, they are best understood as points along a flexible continuum—behavioral tendencies that describe how a piece acts within its intonational environment. A single work may at times display the characteristics of one style and at other times those of another.

For example, I Sommarluft oscillates between the rough and informed styles of the strict style as a 'tuning', containing numerous passages that fall somewhere in between. Similarly, Vid stenmuren blir tanken blomma begins in a loose style and ends in something close to the strict style, though the final section does not map perfectly onto it: its pitch space contains two notes a syntonic comma apart, which technically situates it within the loose style. Other pieces, such as Nattviol, nattviol, are not so much transitional as intrinsically ambiguous, embodying features of both the rough and informed variants of the strict style as a 'tuning'.

When used in this way, the categories in Table 3 allow us to describe passages as exhibiting strict-style behavior or loose-style behavior, even when they do not fit neatly into either category. Loose-style behavior implies a hierarchical ordering of pitches that leads the ear to favor certain untunable intervals rather than adjusting them by commas to make them tunable. Strict-style behavior, by contrast, implies a clearly defined mode in which each pitch has a precise and stable location. When this mode functions effectively, comma adjustments are not even an imagined possibility for the listener.

When we examined Prisma Interius VIII by Catherine Lamb, describing the music as exhibiting both free-style characteristics (its large gamut of comma-distanced pitches) and strict-style characteristics (the unified feel of this gamut), as well as moving between integrated JI and microtonal music that uses JI-type intervals, offered a more nuanced way of understanding its use of rational intervals and intonation. The categories, though rarely fitting perfectly, thus serve their purpose: they illuminate how a piece behaves—how its intonational logic is enacted in perception.

To think of these styles as compositional techniques rather than perceptual tendencies would risk justifying poor writing through theoretical reference. One could, for instance, include an untunable interval and claim its validity by appealing to the loose style and a table of structural intervals—without truly relying on the ear. Or one might let a passage ascend rapidly through several comma levels, producing a jarring effect, and defend it by pointing to the score, where everything appears coherent because each interval is directly connected and tunable on paper.

The concept of distinct styles of JI composition can, however, offer valuable insight into our subjective interpretations and serve as practical compositional aids. They help us understand why a passage may appear tunable yet sound otherwise (as in Figure 27, the early version of the piano trio att sjunka i doftande klöver), or why a chord may seem jarring in notation but not in sound (as in Figure 8, the 'major triad' from the violin and viola duo Vid stenmuren blir tanken blomma). Beyond analysis, these stylistic distinctions also provide a shared vocabulary for collaboration between composers and performers—conceptual tools that can clarify how tuning, perception, and intention interrelate in practice.

Looking at Table 3, it becomes clear that when a composer describes herself as 'working with Just Intonation', the practice can mean many different things. One composer might focus solely on retuning or building instruments, treating their sonorities as unique collections of microtonal pitches—without concern for tunable paths or for balancing tunable and untunable intervals. Another, writing for intonating instruments, may approach JI simply as a microtonal system, unconcerned with whether intervals interlock to form the fused identities characteristic of justly tuned harmony. Yet another composer may work from the opposite premise, centering their practice on the creation of tunable paths and tunable harmonies through a specialized counterpoint and melodic technique.

These are three fundamentally different musical crafts. By extending Lou Harrison’s original distinction between the strict and free styles, I have sought to provide a framework and a shared language through which such diverse practices can be meaningfully related.

Besides tunability and pitch-category perception, a third theme running through this text has been the poetic and affective character of the different styles. We saw how strict-style pieces often embody clear affects or rasa-s, in which each pitch carries a distinct expressive flavor. In the loose style, subtle yet powerfully charged modulations arise when the intonation of certain pitch classes shifts by commas, allowing affect itself to move and transform. The free style, by contrast, can give rise to a fluid, almost liquid sensation of pitch—ever in the present moment, unpredictable, and lacking the affective cohesion that binds the strict style into a single, unifying mood.

Throughout this text, I have emphasized how essential it is that the composition and the intonation system speak in a single voice—striving toward the same artistic goal. When they do not, even finely tuned ratios can sound arbitrary; when they do, even rough tunings can feel inevitable. A work such as Julia Wolfe’s STEAM, for instance, employs Harry Partch instruments as 'found objects', without subtle concern for their tuning—and this poses no problem, for its artistic force lies in its raw, percussive physicality, its deliberate roughness. By contrast, Harrison’s Simfoni in Free Style takes melodic and motivic materials from his familiar idiom and overlays them with a radically fluid, freely drifting intonation. The result is a disjunction: the elements of the piece no longer speak in a single voice, and the tuning ceases to feel integral to its expressive world.

Just Intonation composition remains in its early stages, even though Harry Partch’s seminal Genesis of a Music appeared as early as 1949. Few musicians today possess the full range of skills required to perform it, and few works receive performances that reveal its true potential. Much more practical research—through composition and performance alike—is needed before we can adequately answer the two fundamental questions that have underlain this text: What constitutes a reasonable balance between tunable and untunable intervals in order to provide the musician with a coherent tunable path? And to what extent can we, as musicians, truly tune to non-simultaneous sounds?

It seems to me that any future formulation of the rules of JI composition will depend on how we come to answer these questions—through continued listening, performing, and composing within the living field of Just Intonation itself.

In this text, I have sought—through excerpts from my own compositions—to advance an understanding of how the balance between tunable and untunable intervals shapes our perception of pitch. Yet I must admit that my own works are full of flawed tunable paths. There are several reasons for this. First, intonation is, after all, only one parameter of music—and, some would say, a subtle one. It is not uncommon for other artistic priorities to take precedence, leading to decisions that compromise optimal tunability. Second, a piece in which every interval is always harmonically tunable produces a very specific mode of listening—one perpetually focused on fused, resonant sounds. In works such as I den gyllne luften, Offerblommor, and Minnets svala flod, almost every sonority is tunable, creating an extremely narrow margin for error: the smallest deviation becomes perceptible. In these pieces, the listener cannot help but hear the tuning itself, and the music risks settling into a rasa of crystalline purity. When taken too far, this can become exhausting for both performer and listener alike.

A more spacious, easeful mode of listening emerges in pieces such as mot våren bortom havet or Vid stenmuren blir tanken blomma, where the presence of untunable intervals interrupts the formation of that pristine rasa. In I den gyllne luften, one can at worst stop hearing the music and begin to hear only the act of tuning. By contrast, the careful interpolation of untunable intervals can soften this focus, dispersing attention and inviting a more relaxed, open mode of listening.

I therefore believe that a piece of music need not consist solely of directly tunable intervals, but that untunable intervals are meaningfully related to a tunable context, where they are introduced and perceived as having a tunable origin. In my own compositions, I strive for the piece itself to teach its microtonal characteristics, allowing performers and listeners alike to learn its intonational logic through the act of playing and hearing the piece itself, rather than learning it from somewhere else.

To maintain a balance between melodic flexibility and harmonic clarity, I work modally. A completely free style is not practical, yet neither is a purely microtonal approach. The most compelling path, for me, lies in composing with clearly articulated modes—systems in which pitches are bound by tunable relationships while also possessing their own microtonal identities and affective qualia.

In such pieces, Just Intonation gradually recedes from conscious awareness and becomes indistinguishable from the world—the fundamental attunement—of the music itself. In both German and Swedish, the word Stimmung or stämning refers not only to a specific musical tuning but also to the non-conceptual mood or disposition of a world. This is no coincidence: the two ideally arise together. A tuning system is not merely a method for selecting or intonating pitches as perceptual objects; it is not what the piece is about. Rather, it serves as a quiet framework through which a mode of listening can emerge—one in which sound, world, and self converge into a single relational field, a space that might also be called emptiness.

Notes

This text was first presented as a lecture in Tel Aviv on July 12, 2017, at the Tzlil Meudcan Festival, and was expanded and reworked between 2017 and 2019 into several informal versions circulated under the title Modality and Tunability in Strict, Free, and Loose Style Just Intonation. The current version, retitled Varieties of Just Intonation, was published in 2020 and has continued to evolve through subsequent revisions. The most recent update was completed on April 10, 2023.

[1] In this text, harmonic ratios are notated with a "/" and melodic ratios with a ":". When ratios are used to signify scale degrees within a scale or gamut, double quotation marks (" ") are used. Sets of scale degrees forming a mode or scale are enclosed 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 primarily discusses music in 5-limit Just Intonation, the minus and plus signs (– / +) are used throughout to indicate syntonic commas (≈ 21.5 cents).