In this week’s episode, Thomas B. Yee discusses non-octave repeating scales, exploring precedents in the ideas of theorists from outside mainstream music theory and the application of non-octave repeating scales in the works of living composers.

This episode was produced by Katrina Roush along with Team Lead Jennifer Weavers. Special thanks to Craig Weston, Luis Javier Obregon, David Forrest, Liam Hynes-Tawa, and Jenny Beavers for literature recommendations, draft feedback, and insights helpful in producing this episode.

SMT-Pod Theme music by Zhangcheng Lu; Closing music "hnna" by David Voss. For supplementary materials on this episode and more information on our authors and composers, check out our website: https://smt-pod.org/episodes/season03/

[Intro Theme: Zhangcheng Lu, “BGM Scales”]

Welcome to SMT-Pod, the premier audio publication of the Society for Music Theory. In this week’s episode, Thomas B. Yee discusses non-octave repeating scales, exploring precedents in the ideas of theorists from outside mainstream music theory and the application of non-octave repeating scales in the works of living composers.

From the beginning of our music education, we are taught octave equivalence as a foundational given. For example – pitch names run from A to G, and the next pitch restarts at A to indicate that it is, in some sense, the ‘same’ as the previous A. A C major scale repeats its seven pitches starting on C in each octave. Octave equivalence is assumed in pitch naming, the layout of keyboard instruments, equal temperament tuning, scales, and chord analysis. Even twelve-tone rows and serial techniques assume octave equivalence, as a C in any octave ‘counts’ as a C for the purposes of the tone row. Pitch organization and harmonic theory as we know them largely depend on the assumption of octave equivalence.

But what if octaves aren’t equivalent? Or, to state my meaning more precisely – what if octaves aren’t treated as equivalent in terms of a piece’s pitch organization, as a compositional choice? As a composer and theorist, this question has increasingly occupied my research and creative activities, and I am fascinated by the musical possibilities posed by this question. Let me clarify my meaning. I am not primarily interested in ontological claims – for example, whether each pitch we call C is, in some metaphysical sense, ‘the same’ as the others. Nor is it a pedagogical claim – I do not doubt the usefulness of octave equivalence in teaching music, as I regularly do. Nor is it an acoustic claim, concerning frequency ratios and tuning systems. Rather, my interest is a practical and creative one – namely, if composers eschew octave equivalence in scalar and pitch organization, what innovative compositional possibilities may be discovered?

In this episode, I present my research on non-octave-repeating scales, focusing on their historically, sonically, and compositionally unique qualities. True to their name, non-octave-repeating scales are those that repeat, not at the octave as conventional scales do, but at some other interval. For example, a conventional C major scale repeats its intervallic content at each C pitch [C major scale across three octaves]. A non-octave-repeating scale using similar intervallic content might repeat at an interval of a major ninth – repeating at C, then D, then E, and so on – sounding like this [major scale across three octaves starting on C, then repeating at D and E]. A non-octave repeating scale like this one enables unique harmonic possibilities when arranged vertically as a chord voicing. Let’s listen to one of my favorites.

[Omnisphere “Redemption Piano”: C E F♯ B | D F♯ G♯ C♯ | E G♯ A♯ D♯ | F♯ A♯ B ♯ E♯]

We just heard the chord voicing C, E, F♯ , B repeating at the major ninth – at D, E, F♯, and so on. If we were speaking in person, I would ask what you heard in that sonority. Perhaps you heard it as dissonant, with cross-relations occurring across different octaves, such as the C3 and C♯5 or D4 and D♯6. Perhaps you heard it as a ‘tall’ chord voicing utilizing jazz extensions, containing a ♯11, ♭13, ♭7 and ♮7, and ♭9, ♮9, and ♯9. Or perhaps you heard polytonality, with C, D, E, and F♯ Lydian tonal centers articulated across different registers. However, I heard a single repeating chord voicing, as if one played a C major triad in multiple octaves – but without repeating at the octave. Rather, the initial chord voicing (C, E, F♯, B) does repeat verbatim in higher registers – only repeating at the major ninth.

Non-octave-repeating scales are both undertheorized and underused. Speaking anecdotally, every composer and theorist I have spoken to about non-octave-repeating scales either had not considered them before or agreed that there has been little formally theorized about them. Indeed, I am broadcasting this episode across SMT in part to hear from the community about other relevant work that exists. Among living composers, Craig Weston, Luis Obregón, and I compose using non-octave-repeating scales – and examples from these composers’ pieces will be heard later in the episode. If other living and 20th/21st-century composers make use of non-octave-repeating scales, I would love to hear about them. One well-known moment from Maurice Ravel’s Boléro could serve as a precedent for non-octave-repeating scales. At rehearsal 8, after several statements of the primary theme in C major, a solo French Horn restates the theme in C major with two Piccolos doubling the melody not in octaves, but transposed to G major and E major. Let’s listen now.

[Maurice Ravel, Boléro, rehearsal 8]

This moment is much commented on by Ravel scholars – for example, Jenny Beavers hears the Horn and Piccolos fusing into a single timbral stream akin to an organ stop, and Stephen McAdams interprets the G major and E major Piccolo doublings as sounding partials of the overtone series as if the Horn melody were their fundamental (Beavers 2019, [6.2]; McAdams 1999, 186). It is beyond the scope of this episode to analyze Boléro in-depth, but when I listen to this striking moment, I hear a sonority produced by a non-octave-repeating scale. [Major scale repeating at C5, D6, E7] To be clear, I am not claiming that Ravel conceived of this moment as using non-octave-repeating scales. The presence of key signatures for E major and G major in the Piccolo staves indicates that Ravel most likely thought of this moment as using polytonality. Still, because the E major Piccolo doubling occurs two octaves and a third above the French Horns’ C major melody, and because the F♯ in the G major Piccolo doubling can be derived from a major scale repeating at D, this iconic musical moment could be analyzed as a non-octave-repeating major scale repeating at the major ninth (at C5, D6, and E7).

The next segment outlines theoretical precedents of non-octave-repeating scales that in some way conceptualize pitch organization according to intervals other than the octave, but stop short of taking the plunge into full non-octave-repetition.

[Bumper: Thomas B. Yee, Earthrise for Saxophone Quartet]

It is important to distinguish the non-octave-repeating scales in view from theoretical and compositional approaches which, while similarly challenging conventional scalar organization, do so in distinct ways. First, spectralism organizes pitch according to sound spectra, often based on acoustic analysis of physical sounds and objects, as well as ring modulation (Anderson 2009). Spectralism is associated with European composers including Gérard Grisey, Tristan Murail, and the late Kaija Saariaho. Like non-octave-repeating scales, spectralism produces harmonic frameworks that are register-dependent and do not feature octave equivalence – a given ‘C’ pitch may be included in a spectrum while another ‘C’ is not. However, sound spectra are based in physical acoustic properties, not intervallic relationships as non-octave-repeating scales are.

Second, microtonality and alternative tuning systems, such as just intonation, subvert octave equivalence, and/or conventional scalar construction, such as in the works of La Monte Young, Harry Partch, Ben Johnston, and many others. I do not wish to detract at all from the importance of researching and composing using microtonality and alternate tuning systems. However, I am convinced that there is vast potential in exploring non-octave-repeating pitch organization within equal temperament tuning. Because equal temperament broadly remains the standard in music education, performance, and composition, expanding equal temperament pitch organization beyond the assumption of octave equivalence is critical.

Third, tonal polymodality is related to non-octave-repeating scales in an inverse relationship. In the second season of SMT-Pod, Frank Nawrot and Matthew Ferrandino theorized tonal polymodality as “music that has a clear tonic and stacks multiple modes with the same tonic and moves between two or more modes over the course of a piece” (Nawrot and Ferrandino 2023, 00:40). For example, the pre-chorus of Tool’s song ‘Stinkfist’ is centered on an E tonal center but uses Aeolian mode in the electric guitar riff while the vocalist sings a Mixolydian melody. [Play E Aeolian from E3 and E Mixolydian from E4] Like non-octave-repeating scales, the modes used often depend on registration, as in ‘Stinkfist.’ Tonal polymodality is a fascinating analytical tool with much theoretical and compositional potential – I myself have inadvertently composed using this technique in several pieces. However, it is the exact inverse of non-octave-repeating scales: while tonal polymodality features the same note of repetition with different modes in different registers, non-octave-repeating scales use different notes of repetition in different registers with the same mode in each.

With those distinctions made, we can now explore theoretical precedents of non-octave repeating scales, all of which originate from outside the European-American canon known colloquially as ‘classical music.’ To put a finer point on it, it took voices outside the mainstream edifice of music theory to plant the seeds of a non-octave-repeating harmonic system. I’m happy to be proven wrong in this claim, but I am unaware of any so-called ‘in-house’ precedents. This list is meant to survey the landscape of non-octave-repetition precedents and is by no means exhaustive. For example, I am curious whether Hindustani raag and Indonesian gamelan scales might subvert octave equivalence and repetition, but am not familiar enough with these traditions to comment on them myself. I will explore the work of three music theorists: Jing Fang (京房), Koizumi Fumio (小泉文夫), and George Russell.

Jing Fang was a renowned Chinese acoustic theorist and mathematician operating in the 1st-century BCE Han Dynasty (Modirzadeh 2001, 85). Around 45 BCE, Jing Fang theorized a 60-tone tuning structure using a system that can be translated as “Three-scale Rise/Fall Tuning” (Chen 2021, 337). This references a Chinese tuning method used since the second millennium BCE, using “a sequence of falling fourths and rising fifths” to generate twelve pitches, or lü (Modirzadeh 2001, 81). Each rise uses the tuning ratio 3:2, generating an ascending perfect fifth; each fall uses 4:3, a descending perfect fourth, which together sounds like this (Chen, 9): [Insert lü spiral generated from Rise/Fall Tuning starting and ending on C].

Rise/Fall Tuning generates pitch collections that are more spiral in nature than linear or cyclical (Modirzadeh, 85). Jing Fang’s contribution was deriving a pentatonic scale collection from each lü in the spiral, totaling to 60 distinctly-tuned pitches – 5 x 12 = 60 (Obregón 2012, 15). Jing Fang’s 60-tone system can be linearized in equal temperament by combining pentatonic scales generated from a sequence of consecutive rising fifths, instead of rising and falling. As Obregón shows, a pentatonic scale from C, C-D-E-G-A, overlapping slightly with one from G, G-A-B-D-E, could continue upwards in a sequence sounding like this:

[Alchemy “Soft Synth Bells”: non-octave-equivalent pentatonic collections starting from B♭1: B♭, C, D | F, G, A | C, D, E | G, A, B… Example 2.3 from Obregón 2012]

However, as Jing Fang’s primary concern was with mathematical tuning calculations, his 60-tone spiral of pentatonic collections, which could have produced a non-octave-repeating scale at the interval of the perfect fifth, remained a theoretical framework only. As Hafez Modirzadeh writes, Jing Fang and other Chinese acousticians balanced theory and practice, “between the pursuit of absolute perfection on the one hand, and the acceptance of common-sense simplification on the other” (2001, 85). In other words, Jing Fang’s sequence of 53-fifths spanning an astounding range of 31 octaves was an acoustic, mathematical heuristic only – not meant to be actualized in performed music.

Koizumi Fumio was a Japanese music theorist known for shaping how traditional Japanese melodic modes are widely understood today. In 1958, Koizumi published the treatise Nihon dentō ongaku no kenkyū, “Research on Japanese traditional music.” As Liam Hynes-Tawa writes, though previous writers conceived of Japanese modes as pentatonic or heptatonic scalar collections, Koizumi argued that “Japanese folk music is not bounded by an octave at all, but rather by a perfect fourth,” facilitating what Hynes-Tawa terms the “rise of the tetrachord” in Japanese music theory (Hynes Tawa 2021, 27). Koizumi’s key insight was realizing that the Japanese modes he studied when generated from the same root, shared in common a perfect fourth, perfect fifth, and octave above the root. Koizumi viewed these as two intervals of a perfect fourth: from the root to the fourth, and from the fifth to the octave. What varied between modes, giving each its distinct identity, was which interval filled in each of the two tetrachords. The miyakobushi mode fills in a semitone above the root and fifth, the ritsu mode a tone above the root and fifth, the min’yō mode a minor third above the root and fifth, and the Ryūkyū mode a major third above the root and fifth. Let’s listen to the miyakobushi and min’yō modes, noting how the two-tetrachord structure stays constant between them.

[E miyakobushi and min’yō modes: E-F-A | B-C-E; E-G-A | B-D-E]

Koizumi’s four modes exhaust the possibilities for filling in the two tetrachords. Already, the relevance of Koizumi’s tetrachordal theory to non-octave-repeating scales is clear – Koizumi conceived of the perfect fourth, rather than the octave, as the basic unit of pitch organization for Japanese modes. This flowed naturally from the melodies Koizumi studied, as they predominantly exhibited a very narrow range – often much narrower than an octave. Also, many Japanese melodies combine tetrachords from different modes – such as a min’yō lower tetrachord and a miyakobushi upper tetrachord – showing the usefulness of seeing tetrachords as the basic unit of pitch organization.

However, Koizumi’s modes remained octave-repeating, alternating between disjunct tetrachords (between the fourth and fifth) and conjunct tetrachords (the octave overlapping as the end of the second tetrachord and the start of its first repetition). This is because Koizumi’s theories described traditional Japanese music that uses octave repetition. But what if Koizumi had not alternated between disjunct and conjunct tetrachords, eschewing the octave altogether? Hynes-Tawa includes a tantalizing footnote to this effect: “By contrast, if all tetrachords were conjunct… or, if they were all disjunct… a spiral of new pitch classes would be produced as the music went higher or lower” (Hynes-Tawa, 27). Hynes-Tawa does not pursue this thought further, but the result of this conjecture would be a non-octave-repeating scale, based either on perfect fourths (if conjunct) or perfect fifths (if disjunct). The verbal resonance of Hynes-Tawa’s “spiral” of pitches to Ching Fang’s 60-tone pentatonic spiral is striking.

[miyakobushi mode composed entirelyof disjunct tetrachords:

C D♭ F | G A♭ C | D E♭ G | A B♭ D…]

George Russell was a Black American jazz musician and theorist known for his 1953 treatise, Lydian Chromatic Concept of Tonal Organization. Though Russell’s Lydian Chromatic Concept, or LCC, was foundational to jazz theory and seminal records like Miles Davis’ Kind of Blue, Russell was marginalized by the music academy in his time and has only recently begun to receive the recognition as a music theorist he deserves. Russell’s LCC proposed two key claims: first, a chord is defined not by its structure of notes, but by the mode it implies (Forrest 2021). This observation is routinely put into practice by jazz musicians, who read chord symbols in fake books and solo over them using the appropriate mode scales.

Second, comes Russell’s revolutionary claim: “The Lydian scale is the sound of its tonic major chord” (Forrest 2021). In other words, the mode implied by a major triad or major seventh chord is not Ionian – commonly known as the major scale – but rather the Lydian mode. This is because the Ionian mode has an “avoid note” – its fourth – while the raised Lydian fourth still sounds consonant over a major triad or seventh chord. However, perhaps the most fascinating part is how Russell arrived at the Lydian mode for his theory. Russell posited a sequence of perfect fifths resonating upwards from the root note of the scale over three and a half octaves. If based on a root of C, these stacked fifths eventually yield an F♯ – the raised Lydian fourth, or ♯11 in chord symbol terms – rather than the F♮ used in the major, or Ionian, scale. Collapsing these fifths into the span of an octave then yields the Lydian mode, the basis for the LCC. Let’s listen to Russell’s sequence of fifths, followed by the Lydian scale derived from it.

[Piano: George Russell’s Lydian Chromatic Concept, sequence of perfect fifths and Lydian scale]

As I have studied the LCC and applied Russell’s insights in my own compositions, a recurring question arises: why did Russell not continue his sequence of perfect fifths past the Lydian ♯11? Doing so from a root of C would have yielded after the F♯ a C♯, then G♯, D♯, A♯ (or B♭), F♮, before returning to the root C like this: [sequence of perfect fifths extending to repeat of the root]. As to why Russell did not do this, the ready-at-hand answer – that Russell already had a C♮ and thus could not also have a C♯– assumes octave equivalence and the context of an octave-repeating scale.

Through private correspondence, George Russell scholar David Forrest agreed that Russell’s findings assume octave equivalence, as they were designed for use in an octave-equivalent style. Russell’s priority, according to Forrest, was arguing for the “Lydian mode as the primary diatonic order” – his sequence of fifths was a means to achieving that end, and he saw no need to carry it further (Forrest 2023, private correspondence). Russell wasn’t trying to construct the complete tonal space possible but to draw attention to a striking aspect of a part of that space. Like Jing Fang’s 60-tone spiral of pentatonic scales, Russell’s stacked fifths was a theoretical heuristic for grounding and explaining practical musical outcomes. Significantly for non-octave-repeating scales, the underlying foundation of Russell’s LCC is the perfect fifth, making “no mention of the octave whatsoever” (Forrest 2023). In other words, if Russell had not collapsed the sequence of stacked fifths into the span of an octave for pragmatic reasons, he might have arrived at non-octave-repeating scales based on the perfect fifth.

[Piano: Lydian tetrachords repeating at the perfect fifth from C0:

C D E F♯ | G A B C♯ | D E F♯ G♯ | A B C♯ D♯…]

The infinite extensibility of a non-octave-repeating tonal space aligns well with the philosophy of the LCC. These three music theorists proposed harmonic structures based on intervals other than the octave. Each stopped short of full non-octave repetition for pragmatic reasons, theorizing or performing in a repertoire that assumed octave repetition. But what if Koizumi had gone further, arranging his modal tetrachords as all disjunct or all conjunct? What if Russell had extended his the sequence of perfect fifths past the Lydian ♯11, creating a fifths-based harmonic spectrum instead of collapsing the resulting pitches within the span of an octave? We can only speculate about these counterfactual histories – but exploring non-octave-repeating scales will gesture towards what the results might have sounded like.

[Bumper: Thomas B. Yee, Concerto Ludus for Piano and Gameboy]

The final section of this episode explores properties of non-octave-repeating scales, examples of non-octave-repeating scales I have found effective, and audio recordings of non-octave-repeating scales used in the works of three living composers. I draw on Craig Weston’s paper “Some Properties of Non-Octave-Repeating Scales, and Why Composers Might Care,” given at the 2012 Region VI Society of Composers, Inc. conference. Weston’s pitch-class analysis of possible non-octave-repeating scales goes into further detail than I can here, and I recommend reading the text version of Weston’s paper, linked in the episode notes. Following Weston, I will classify the repeating interval of a scale according to the level of transposition in semitones. For example, repetition at the perfect fifth would be T7, repetition at the octave T12, and repetition at the major ninth T14. This avoids centralizing the octave, as conventional interval labels do, instead treating octave repetition like any other level of transposition.

Not all non-octave-repeating scales have equal compositional potential. T1 and T2 are too small to produce unique results, producing the chromatic scale and whole-tone scale, respectively. Intervals that divide evenly into T12, such as T3, T4, and T6, lead to repetition too quickly and produce octave-repeating scales. For example, the whole-half and half-whole octatonic scales are T3-repeating scales that repeat at the octave after four cycles. T8 and T9 – minor and major sixths – also repeat after 3 and 4 respective cycles, with the notes of repetition forming an augmented triad and fully diminished seventh chord, respectively.

[Piano: T8 and T9 repetition across two and three octaves, respectively]

However, because the interval of repetition is wider, spanning two octaves for T8-repeating scales and three octaves for T9-repeating scales, the non-repeating space may be wide enough to be compositionally interesting – especially for instruments with a relatively narrow range. Weston does favor T9-repeating scales, such as this one, which spans three octaves before repeating:

[Alchemy “Soft Synth Bells”: Weston’s 2-2-2-2-1 T9-repeating scale from C3: C D E F♯ G♯ | A B C♯ D♯ E♯ | G♭ A♭ B♭ C D | E♭ F G A B]

Thomas

There is also a T8-repeating chord voicing I find useful: a minor triad repeating at T8:

[Omnisphere “Adagio Transparent Strings Warm”: T8-repeating minor triad from E♭2: E♭ G♭ B♭ | B D F♯ | G B♭ D | E♭ G♭ B♭…]

The lugubrious result sounds like a minor-major seventh chord with split thirds and an emphasized dissonance on the Aeolian sixth. However, it does not seem to expand much when moving through registers. Functionally, it sounds like a “tall” jazz chord voicing repeating every two octaves. The most compositionally interesting non-octave-repeating scales are those based on fourths (T5, and T10 by extension), fifths (T7, and T14 by extension), and the major seventh or minor ninth (T11 and T13). T5-repeating scales span five octaves before reaching its root note again, and T7-repeating scales span seven octaves – almost the entire range of the piano. T11- and T13-repeating scales extend beyond the piano, spanning eleven and thirteen octaves respectively – more than is pragmatically usable on acoustic instruments. I have found T5-/T10-, and T7-/T14-repeating collections most useful in my own compositions.

Now, let’s listen to some examples. The first set consists of modal scales, along with their typical chord voicings, repeating at the T14, or major ninth. Though each mode is a seven-note collection repeating at the octave, for simplicity’s sake I will include the octave repetition of the root as the eighth note of the collection before it starts over at the ninth. First is a T14-repeating Lydian mode with a major seventh chord voicing:

[Piano: T14 Lydian scale and major 7th chord voicing repeating at C3, D4, E5, F♯6]

I like this non-octave-repeating scale’s progression towards the sharp side keys, seeming to expand the higher one traverses across registers. The multiple major sevenths built into the chord voicing give it an ethereal, dreamy quality. Next is a T14-repeating Dorian mode with a minor seventh chord voicing:

[Piano: T14 Dorian scale and minor 7th chord voicing repeating at C3, D4, E5, F♯6]

In the Dorian mode, any scale note sounds good over its tonic minor seventh chord. Sounding two iterations of this minor seventh chord voicing generates all the notes of a Dorian scale. Also, to my ear at least, the second scale iteration is quite usable over the seventh chord of the first scale iteration – D Dorian above a C minor seventh, for example:

[soloing D Dorian from D4 over a C minor 7th chord at C3].

Next is a T14-repeating Mixolydian mode, with a dominant seventh chord voicing:

[Piano: T14 Mixolydian scale and dominant 7th chord voicing repeating at C3, D4, E5, F♯6]

Used as a standalone vertical chord sonority, this is one of my favorite non-octave-repeating scales. Even at the outer range of the spectrum, the included pitches support each other well, making this voicing effective for wide textures and sonorities spanning the full range of the ensemble. Next is a T14-repeating Lydian-Mixolydian mode – sometimes called the acoustic scale because it is derived from the overtone series – with a chord voicing that in isolation sounds like the whole-tone collection:

[Omnisphere “Redemption Piano” and “Adagio Transparent Strings Warm”: T14 Lydian-Mixolydian scale from C3: C D E F♯ G A B♭ C | D E F♯ G♯ A B C D…]

This chord voicing sounds derived from the whole-tone collection, including both the ♯4 and ♭7 of Lydian and Mixolydian, respectively. But melodically, the fifth and sixth scale degrees provide the scale an interesting asymmetry, so that melodies are not simply made of whole steps and their derivative intervals. And as the scale expands across registers, the placement of this asymmetry shifts, creating melodies that evolve in fascinating ways. Now, to make good on the non-octave-repeating potential of Koizumi’s tetrachords theory, here are two non-octave-repeating adaptations of traditional Japanese modes – first, a T7-repeating miyakobushi mode using only disjunct tetrachords, then a T5-repeating ritsu mode using conjunct tetrachords.

[Alchemy “Soft Synth Bells”: miyakobushi mode in disjunct tetrachords from C3: C D♭ F | G A♭ C | D E♭ G | A B♭ D…, ritsu mode in conjunct tetrachords from D3: D E G | G A C | C D F | F G B♭…]

I have previously used each of these non-octave-repeating scales to generate complex melodies that evolve as they expand across registers. Finally, we will hear four excerpts of music from living composers who have researched and composed with non-octave-repeating scales. First is Luis Obregón’s Bells of Silk and Wood for Piano and Erhu. As the presence of the Erhu suggests, this piece is based on Chinese harmonic frameworks – specifically, the Three-scale Rise/Fall Tuning sequence and Jing Fang’s 60-tone spiral of pentatonic scales. Listen for the right-hand Piano ostinato outlining the first six notes of the Rise/Fall Tuning

sequence and the T7-repeating pentatonic collections that expand to novel harmonic areas in higher and lower registers.

[Luis Obregón, Bells of Silk and Wood for Piano and Erhu, II. “ligero” mm. 14-31]

Second is my composition Concerto Ludus for Piano and Gameboy. This piece recreates the soundworld of retro video game soundtracks with electronic sound sources derived from plugins emulating the sound chip of the original Nintendo Gameboy. This passage uses a T14-repeating Mixolydian mode, with a suspended 7-chord voicing arpeggiated across the piano showcasing most of the expanding tonal space.

[Thomas B. Yee, Concerto Ludus for Piano and Gameboy mm. 230-241]

Third is Craig Weston’s Glancing Spirals, for Violin, Clarinet, and Piano. This piece uses the T9- repeating whole-tone collection showcased earlier. The harmonic material of this lyrical passage may seem to teeter between tonality and non-tonality – because, of course, it is non-octave-repeating tonality. This reflects Weston’s fundamental boredom with the “false dichotomy between tonal and atonal music” (Weston 2012, 13). Listen for the pitch spectrum of this passage becoming gradually revealed, metamorphizing as the instruments stretch into higher or lower registers.

[Craig Weston, Glancing Spirals for Violin, Clarinet, and Piano, III. “Sweetly Singing” mm. 4-17]

Fourth is a passage from my Holocaust Remembrance Opera, Eva and the Angel of Death, telling the story of Auschwitz survivor Eva Mozes Kor. In this passage, mezzo-soprano Eva Mozes Kor, age 10 in the narrative, resolves to survive and walk out of Auschwitz alive with her sister Miriam. At the moment of her climactic declaration, swelling T7-repeating stacked fifths voiced throughout the orchestra punctuate Eva’s courageous statements, suggesting with their expansive tonal space the freedom that Eva envisions.

[Thomas B. Yee, Eva and the Angel of Death, a Holocaust Remembrance Opera, mm. 64-77]

This episode has posed the question “What if octaves aren’t equivalent?” as a provocative challenge to the assumed octave-repetition underlying much melodic and harmonic analysis. The voices of three music theorists speaking from outside the traditional European-American classical music canon – Jing Fang, Koizumi Fumio, and George Russell – have offered theoretical alternatives to octave-based pitch organization, though each for their own reasons did not embrace total non

octave-repetition. Then, we heard several non-octave-repeating scales and four excerpts from the music of living composers highlighting the compositional potential of non-octave-repeating scales. Let’s conclude with five, out of many, reasons to use non-octave-repeating scales:

1. Melodic: Using expanded resources for melodic writing based on register, beyond a consonance-dissonance binary based on the current chord. 2. Harmonic: Non-octave-repeating scales offer virtually unlimited and largely untapped resources for devising new chordal timbres. 3. Formal: It enables music to modulate across an unfolding spectrum based on a continuum of nearness and distantness of register (Weston, 13). 4. Semiotic: Because non-octave-repeating scales do not currently have much presence in the musical landscape, using them offers fresh potential for signification – the strategic rather than the stylistic or conventional, in Robert Hatten’s terms (Hatten 2004, 186). and 5. Historical: Weston writes that there are at least two versions of music history: the official version and the chutzpah version, where all prior musics lead logically to one’s own music (13). In one sense, I have offered a small-scale chutzpah version of music history, where non-octave-repeating scales offer some of the familiar structure of common-practice tonality while freeing composers from the tyranny of the octave and a single tonal center. I believe there is value in non-octave-repeating scales falling somewhere between the passé false dichotomy between tonal and atonal music.

This episode is an invitation: to research, to dialogue about, and to create using non-octave-repeating scales. Perhaps one day, listeners can, like myself, come to hear not only this sound: [Piano: C3, C4, C5, C6] but also this one: [Omnisphere “Redemption Piano”: C3, D4, E5, F♯6] as equivalent – just not octave-equivalent.

[Outro Theme: David Voss, “hnna”] I would like to thank my collaborative peer reviewer, Craig Weston, as well as Luis Javier Obregon, David Forrest, Liam Hynes-Tawa, and Jenny Beavers for literature recommendations and draft feedback. Thanks also go to SMT's co-chairs Jenny Beavers and Megan Lyons, my episode producer Jen Weaver, and all the SMT-Pod board members. The compositions heard between segments of this episode include Earthrise for Saxophone Quartet and Concerto Ludus for Piano and Gameboy, both composed by me. The compositions featured in the third segment include Bells of Silk and Wood for Piano and Erhu, Mvmt. II: Ligero by Luis Javier Obregon, Glancing Spirals for Violin, Clarinet, and Piano, Mvmt. III: Sweetly Swinging by Craig Weston, Concerto Ludus for Piano and Gameboy, and Eva and the Angel of Death – A Holocaust Remembrance Opera composed by me. Check out the episode notes for scores, recordings, and articles referenced, the episode transcript, and additional materials for further research.

[Outro Theme: David Voss, “hnna”]

Visit our website smt-pod.org for supplemental materials related to this episode and to learn how to submit an episode proposal. And join in on the conversation by tweeting your questions and comments about this episode @SMT_Pod. SMT Pod's theme music was written by Zhangcheng Lu with closing music by David Voss. Thanks for listening!