Structural properties of multi-octave scales

Whilst not widely extended, non-octave-repeating scales are present in a variety of musical settings, yet have received scarce attention in the existing literature. This paper provides a brief general historical contextualization before focusing on a specific group of two-octave scales based on properties in common with the most widely used scales in Western music. After characterizing them in mathematical terms, an exhaustive list of such scales is provided, being the first exhaustive list of non-octave-repeating scales of any given characteristics. A scale endowed with structural properties attributed to the diatonic collection in the field of diatonic theory – such as well-formed, Myhill property, maximally even or diatonic – is singled out for the first time in this paper.


Organizing pitch content in music
Music cannot occur without the existence of a set of organizational principles. In particular, one vital aspect of a musical system is how it organizes pitch content. A music's tonal content is generally defined by exploiting a limited selection amongst a range of possible pitches. By ordering the subset of selected pitches by pitch height we arrive at a musical scale. Yet, whilst the world's scales differ greatly regarding both their construction and usage, they usually have much in common. In Music, Language and the Brain, Patel (2008, p. 19-20) finds the following commonalities amongst musical scale systems: • Although scales vary in terms of pitch number, most are comprised of 5-7 pitches per octave (Gill and Purves 2009, p. 8). • Although scales differ with regard to their intervallic construction, most scales consist of adjacent tones comprised of 1-3 semitones, with two semitones being the most common scale step used in the construction of melodies (Nettl 2000, p. 468). • Most scales are generally not symmetrical in their intervallic construction (Savage et al. 2015, p. 20). Pelofi and Farbood (2021) present empirical evidence that the learning of novel musical systems is related to the degree of symmetry in a scale. This fact may explain the prevalence of non-uniform scales across musical cultures.
One common trait amongst the world's scales is so universal that it tends to be part of the very definition of scale in most texts, rather than an extended commonality, namely, that scales are an intervallic structure that repeat at the octave; see, for example, Patel (2008, p. 14) and Graue (2017) and the references therein. In effect, this tends to be the case in all advanced musical cultures (Burns 1999, p. 252). Whilst it is clear that a seemingly endless myriad of musical scales exist within the boundaries of a single octave, one could ask whether other options may also have musical potential? It is at least theoretically possible to devise scales in which pitch content differs throughout successive octaves. Yet, could such scales be a viable form of tonal organization? Have human beings ever innately strived towards such constructs or is the concept not intuitively appealing?
The fact is that non-octave-repeating scales, although not common, exist in a variety of musical cultures and several twentieth century and contemporary composers as well as improvisors have explored the possibilities inherent in these structures in terms of organizing pitch content; see Nikolsky (2016), Shillinger (1946), Slonimsky (1947), Persichetti (1961), Squibbs (2002), Dixon (2007), Baker (1990), Liebman (1991), McGill (2013) and Yamaguchi (2013). 1 Yet, the literature dealing with these scales is scarce and there are many questions regarding this topic which remain unanswered or even unposed. Which general characteristics could be appealing in non-octave-repeating scales? Can non-octave-repeating scales be devised and defined based on these general characteristics? By which criteria could they be categorized, organized, and compared? Is it possible to construct non-octave-repeating scales endowed with structural properties attributed to certain octave-repeating sales in the field of diatonic theory, specifically to the diatonic collection? If so, given the particularities of non-octave-repeating scales, must one adapt the methods of analysis in order to reveal said properties?

Non-octave-repeating scales in the musical practice
Given our apparent natural disposition towards octave-repeating scales, it would be fair to ask whether there could be any advantages in a non-octave-repeating scale as a form of organizing pitch content. Many readers are possibly familiar with Nicolas Slonimsky's Thesaurus of scales and melodic patterns, listing hundreds of non-octave-repeating symmetric scales based on the equal division of a given octave-range, but have these ever proven to be useful for organizing pitch content in a musical work or is the notion a mere, somewhat extravagant theoretical concept? The fact is that non-octave-repeating scales, whilst generally associated with twentieth century Western Art Music, are not as uncommon as it may appear and have existed for considerably longer than what most might expect. Scales comprised of a series of successive similar tetrachords -often fourth-, sometimes fifth-repeating, as opposed to octave-repeating -have existed for centuries in the liturgical and secular music of different cultures.

Non-octave-repeating scales in non-Western traditions
What is probably the earliest known example can be found in the anonymous ninth century treatise Musica Enchiriadis; an 18-pitch scale, spanning two and a half octaves, known as the Enchiriadis Scale or Daseian Scale, due to the fact that it is presented in Daseian notation (Erickson 2001), consisting of a series of four disjunct T-S-T tetrachords repeating at the fifth with an incomplete tetrachord of only two pitches at the top; see Figure 1. Nikolsky (2016, p. 9-10) explains that scales consisting of a string of consecutive trichords/tetrachords are found in Medieval Western Europe, Byzantium, Russia, Armenia, Georgia, Azerbaijan, and Bulgaria, and were used to organize pitch content in Christian plainchant, which subsequently shaped the folk music of many Eastern Orthodox nations and ethnicities.  The scale employed in the Znamenny rospev -the principal chant of the Russian church, most likely of Bizantine origin (Swan 1940, pp. 232) -, which in turn found its way into much Russian folk music (Belianski 2015, p. 21), is a prime example of a fourth-repeating scale comprised of a succession of major trichords (see Figure 2). Similar constructs can also be found in traditional Armenian music (Pahlevanian, Kerovpyan and Sarkisyan 2001), in the Adonai Malakh -one of the three principal modes or shteygers employed in Jewish liturgical singing (Seroussi et al. 2001) -or in Georgian traditional vocal polyphony (Gogotishvili 2010).
Some maqamat, 2 such as the Maqam Bastanikar are also non-octave-repeating (Farraj and Shumays 2019, p. 390). With regard to our previous question of whether non-octave-repeating scales were but a mere theoretical concept without any tangible musical value, the highly spiritual nature and emotional power of much of the music discussed, as well its longevity within a variety of musical cultures over several centuries, would surely speak towards the contrary.

Non-octave-repeating scales in Western traditions
In Western Music, non-octave-repeating scales have been of interest to various composers and theorists since the first half of the twentieth-century, such as Nikolas Slonimsky, Joseph Shillinger, Elliot Carter, Vincent Persichetti or Alfred Schnittke; see Slonimsky (1947), Shillinger (1946), Derby (1981), Persichetti (1961), and Dixon (2007). Carter's Duo for Violin and Piano (Derby 1981) and Schnittke's Symphony No. 4, which draws its pitch-content exclusively from three distinct octave-repeating scales (Dixon 2007, p. 161), are examples of works that make extensive use of such scales.
Returning to whether or not the scales in Slonimsky's Thesaurus have ever proven to be useful for organizing pitch content in a musical work, two of the three scales employed by Schnittke in Symphony No. 4 are contained in the text, namely scales no. 833 (Figure 3(a)) and 831 (Figure 3(b)), as is the two-octave scale in Carter's in Duo for Violin and Piano, listed as scale no. 698 (Figure 3(d)).This is not to say that these composers where necessarily aware of this fact -in the case of Schnittke's Symphony No. 4, for instance, the scales developed out of motifs holding a religious symbolic meaning in line with the spirit of the work (Dixon 2007, p. 168-174) -, but it does speak towards the validity of these formations as a way of organizing pitch content in music. Figure 3 shows some of the non-octave-repeating scales found in the work of Schnittke and Carter. Iannis Xenakis also frequently employed non-octave repeating constructs he referred to as pitch sieves in his compositions, devised through a quite different mathematical process described in Formalized Music (Xenakis 1992). Several jazz musicians -e.g. Dennis Sandole, David Baker, David Liebman as documented in McGill (2013), Baker (1990), and Liebman (1991) -have also shown interest in non-octave-repeating scales in the context of improvisation. In the 21st century, there has been a growing interest amongst several contemporary composers, improvisers and researchers such as Joel Hoffman, Gao Weijie, Craig Weston, and Anders Lønne Grønseth, who have developed different ways to construct and employ such scales in their work; see Brown (2019), Weston (2020), and Grønseth (2015).
Whilst Schnittke employs three different non-octave-repeating scales to organize pitch content in Symphony No. 4, there are also several examples of pieces based on a single non-octaverepeating scale, such as Xennakis' Komboï (1981), written for harpsichord and percussion. Ian Pace's analysis shows that most of the piece derives its pitch material from a single non-octaverepeating scale spanning six octaves, with few exceptions of pitches alien to the scale being used throughout the work (Pace 2001, p. 129). Other pieces based on a single scale of this nature include Gao Weijie's The Road (1996) or Joel Hoffman's 6-8-2-4-5-8 (2006), String Quartet 3 (2005), or Piano Concerto (2008; see Brown (2019), Hoffman (2005), Hoffman (2008 and Zheng (2017). Similarly to the scales in Slonimsky's Thesaurus, Hoffman's and Weijie's symmetric scales are devised by the repetition of similar scale fragments. Yet, unlike in Slonimsky, these composers use scale fragments themselves larger than an octave, leading to complex structures with a huge intervallic span, such as the seven-octave scale used in  consisting of six intervalically identical scale fragments spanning a major ninth (Brown 2019, p. 43-44). As is to be expected in scales of such proportion, these generally include the full pitch-class (pc) set.
According to Weston (2020, p. 13), a large part of the appeal of using non-octave-repeating scales is the possibility of composing in a highly chromatic context while still maintaining the advantages regarding structure and modulation found in composing with traditional scales. That a scale of any intervalic range may be a useful tool in providing structure, enabling musicians to draw from a limited subset of pitches and establish hierarchal relationships between them, is fairly obvious. Yet whether non-octave-repeating scales, which often include the full pc aggregate, allow for meaningful modulation, is maybe less so. One could rightly argue that if a scale contains all 12 pcs, the ear is likely to perceive the scale to consist of all of these arranged in the order they occur when mapped to a single octave, rendering the concept of modulation somewhat irrelevant. Modulation implies introducing notes that were not previously present, a transition from one subset of pitches to another, which may seem rather contradictory in a scale which includes all pcs. Yet, as Weston (2020, p. 8) explains, "these scales are structures in pitch space (as opposed to pitch class space), so we need to examine common pitches (rather than pitch classes), in order to describe the nearness/distantness continuum in modulation space." So, while different modulations of a non-octave-repeating scale containing the full aggregate will be equal in terms of pc content, the placement of the pitches in pitch space will differ. This can be observed in Figure 4 which shows one of Weston's scales along with its first three transpositions. The original scale consists of four consecutive identical scale fragments of a major sixth, spanning a total intervalic range of three octaves and includes the full pc-set. The transpositions have been extended at the bottom and truncated at the top, thereby showing each transposition in the same pitch range for easier comparison.
It is likely that modulation in pitch space as opposed to in pc space in such a highly chromatic setting may be less perceptible than in a diatonic context. However, the fact is that composers have found modulation in non-octave-repeating scales to be a useful compositional tool, even when these do include the complete aggregate. Weston (2020) offers an example of the movement Sweetly Singing from his own composition Glass Spirals to illustrate what he describes as "fairly obvious modulations" (p. 14) within the three-octave scale shown in Figure 4. For the solo piano piece Mists (1980), Iannis Xenakis created a 30-pitch sieve (see Figure 5) with a total interval span of 90 semitones -two more than actually exist on a standard piano keyboard, for which reason the scale is not used in its entirety throughout the piece (Squibbs 2002, p. 92) -, which also includes the full pc aggregate. Throughout the work, the composer makes use of 11 of the 90 possible transpositions of the scale as the basis of the piece's tonal content (Squibbs 2002, p. 95).
In any case and as is to be expected, the strategies composers adopt when working with such constructs are manifold and depend on musical context. The three scales in Schnittke's Symphony No. 4 appear in different modulations and are used both separately and simultaneously, in which  case each scale tends to be assigned to a specific instrument or instrumental section. 3 A further example of how a non-octave-repeating scale may be used to provide structure in an atonal context can be found in the first 83 measures of Carter's Duo for Violin and Piano where the violin part is solely based on the two-octave scale in Figure 3, whereas the piano part plays notes outside the violin's fixed octave scheme (Derby 1981, p. 153).
In the context of jazz improvisation, Liebman (1991) proposes the use of non-octave-repeating scales as a way of improvising over add-on voicings -a term used to describe a chord that include notes not usually found in its upper structure -and polychords. An example of a piece of music in which the improvisation is developed in this manner is Liebman's Carissima, based on polytonal harmonies and analysed in Liebman 1991 (p. 131). The author provides various possible scale options for improvising over each of the five polychords that constitute the piece's chord progression, most of which are non-octave-repeating. Further situations where non-octaverepeating scales may prove useful for improvisors include playing outside 4 (McGill 2013, p. 18) or in a "melodic-modal" context (Grønseth 2015, p. 13). Yamaguchi (2013, p. 16) also suggests that multi-octave scales may work over diatonic chord changes provided the melodic lines fit the harmony and provides examples of scales that could function over dominant chords, such as the scale in Figure 6. The fact that the upper structure of dominant chords can be altered in several ways in a jazz setting makes the possibility of devising a fitting non-octave-repeating scale more feasible due to the increased amount of possible note choices.

Construction of non-octave-repeating scales
With this in mind, it is safe to say that non-octave-repeating scales have proven to be a timetested, effective and useful way of tonal organization in a variety of musical contexts throughout several centuries. Yet, the existing literature dealing with the subject is scarce and focuses primarily on methods of construction and with the application of these scales in specific musical situations. The three most common methods of construction are: (1) by forming scales based on interval cycles that equally divide a given octave range, as in Slonimsky (1947), Shillinger (1946), Yamaguchi (2013), or Weston (2020); (2) by forming scales based on a succession of similar or dissimilar tetrachords, as in Persichetti (1961), Liebman (1991), or Baker (1990); (3) by combining two one-octave scales with a common tonic, as in Persichetti (1961) or Yamaguchi (2013).
The most common applications tend to be either: (a) organizing pitch-content in an atonal context, as in Persichetti (1961) or Weston (2020), or (b) providing vertical structures for improvising over polychords or for playing outside, as in Liebman (1991), Baker (1990), or Yamaguchi (2013.

The present work on multi-octave scales
Whilst all that research provides valuable information concerning the construction and application of non-octave-repeating scales, no thorough comparative study has as of yet, to our knowledge, been undertaken that deals with the musical and structural properties of these scales, or attempts to provide a taxonomy beyond the basic methods of construction. In light of the growing interest in non-octave-repeating scales and their tested musical potential, evidenced by the wide stylistic range of music which employs such constructs, we believe that such a study is indeed relevant.
In the present paper we isolate a group of non-octave-repeating scales based on a series of definite intervallic characteristics shared by the most widely employed scales in Western music, allowing composers and improvisers to expand their musical palette with scales that, whilst intrinsically different, share many of the properties of those they are accustomed to. In the Appendix 1, an exhaustive list of such scales is provided. To our knowledge, this is the first exhaustive list of non-octave-repeating scales of any kind. 5 In the second part of the paper we consider pitch-class (pc) equivalence within the scale's modular interval-span as well as standard pc equivalence as a means of analysing the structural properties of these scales. Whilst the latter provides us with information regarding the pc content and chromatic density of a scale, the former reveals information regarding the interval-class (ic) content useful to assess the overall sonority and thereby a means of comparison between different scales. It also brings to light structural properties which would otherwise pass unnoticed, allowing us to establish similarities between certain non-octave-repeating scales and the diatonic collection, based on properties discovered by researchers in diatonic theory, such as Clough and Douthett (1991), Clough, Engebretsen, and Kochavi (1999), Carey and Clampitt (1989), and Gamer (1967), included wellformedness and Myhill's property. We believe that the existence of these different properties in non-octave-repeating scales is discussed in a peer-reviewed journal for the first time in the present paper.

Two-octave Pressing scales
Firstly, we must determine criteria by which to filter amongst the immense number of possible non-octave-repeating scales. An obvious way is to discriminate by range, so we will commence by limiting our study to two-octave scales. But even within a two-octave span there are 8,388,606 possible 2-to 23-pitch scales (8,388,606 comes from the computation 2 23 − ( 23 0 + 23 23 ). Furthermore, whilst theoretically feasible, not all options are necessarily musically interesting; consider, for example, the set consisting of 12 consecutive semitones and one octave. What is considered musically interesting may logically vary depending on the musical context or culture. In order to limit the scope of this research, we will begin by focusing our attention on twooctave scales that share the same attributes in terms of intervallic structure as those most widely employed in Western music. The assumption is that if a majority of musicians have expressed a predominant preference for certain properties in musical scales, it is useful to search for these same properties when considering two-octave scales. Naturally, future research may include a different or additional set of properties to satisfy other musical priorities. After defining the intervallic commonalities of most scales in the context of Western music, we will provide an exhaustive list of possible two-octave scales that share these attributes. Our hope is that this list may serve as a point of reference for musicians interested in such scales.
We will adopt the terminology used by Clough and Douthett (1991) to refer to generic and specific interval size, by using the terms dlen to refer to diatonic length, or generic interval size, and clen to refer to chromatic length, or specific interval size. Thus, dlen 1 spans one scale-step, as in a diatonic second, dlen 2 spans two scale-steps, as in a diatonic third, etc., while clen 1 spans one semitone, clen 2 spans two semitones, and so on. The spectrum of a dlen specifies the clens corresponding to it, that is, the possible specific interval sizes for a given generic interval, and is expressed as < dlen >= {clen1, clen2, . . .}. 6 We will convey scales both as a series of successive intervals expressed by integer numbers indicating the inter-onset distances as well as standard integer notation indicating the inter-onset's positions, substituting numbers 10 and 11 with letters t and e in order to avoid confusion. For example, the major scale is represented as (2212221) or [024579e], respectively. When employing the latter form to convey two-octave scales, we will consider integers 0-23 to cover a two-octave range and will separate numbers by hyphens.
For the sake of clarity we will introduce the following definitions.
Definition 2.1 A mode is a specific arrangement of pitches in scalar order.
Definition 2.2 Two modes are said to be rotationally equivalent if one is a rotation of the other.
For example, (1313131312122) and (1212213131313) are rotationally equivalent because the latter is a rotation of the former by 8 positions. The relation of being rotationally equivalent is in fact an equivalence relation (that is, a reflexive, symmetric, and transitive binary relation).

Definition 2.3
A scale is a specific collection encompassing all possible rotations of a given arrangement of pitches in scalar order.
Thus, the diatonic collection is a scale and its seven specific rotations (Ionian, Dorian, etc.) are referred to as modes. A scale is an equivalence class of the relation given by the rotations of a mode.

Commonalities between widely used scales -Pressing scales
Among the 11 6 = 462 theoretically feasible heptatonic modes, but a few are widely employed. Tymoczko (2004) identifies three scalar constraints to which the most commonly used scales in Western musical practice adhere: • Diatonic seconds (DS): The specific size of one scale-step is to be either one or two semitones: < 1 >= {1, 2}. • Non-consecutive semitones (NCS): A scale cannot contain successive one-semitone steps.
Hence, it cannot contain a sequence of the type [012]. • Diatonic thirds (DT): The specific size of two scale-steps is to be either three or four semitones: There are but four octave-repeating collections that comply with these constraints, namely, the diatonic [024579e], octatonic [0134679t], whole-tone [02468t], and acoustic scale [024679t]. Lifting the DS constraint allows scale-steps to span three-but not four semitones. If a scale step were to span four semitones, the interval of a third could span no less than five semitones, thus violating the DT constraint. Hence, lifting the DS constraint expands the scale-step spectrum to < 1 >= {1, 2, 3}, allowing for the inclusion of three further widely used collections -the harmonic minor [023578e], harmonic major [024578e] and hexatonic [014589].
These seven scales, referred to by Tymoczko (2004) as Pressing scales, since they were first singled out by Jeff Pressing (1978), constitute the most widely used collections in Western music, from the classical tradition to impressionism and contemporary jazz (Tymoczko 2004, p. 136 ). The former includes an additional pitch in the standard pentatonic, often referred to as a "blue note"; the latter "presents a systematic way of adding chromatic passing tones to standard diatonic scales" (Tymoczko 2004, p. 148). 7 While these scales are no doubt widely employed, it is fair to state that the overall harmony in jazz and jazz related musics is generally derived from the seven Pressing scales detailed above. An important part of jazz pedagogy is often based on a set of rules associating appropriate scales with chords in order to melodically define the underlying harmony. This approach is defined as "chord-scale relationship " or "chord-scale compatibility ." Tymoczko (1997, p. 148) observes that virtually all these rules involve one of the seven Pressing scales.
In conclusion, although the vocabulary of Western music may be enriched by the inclusion of scales of varying intervallic characteristics, the scales that adhere to the Pressing scale restrictions are by far the most commonly employed and constitute the scalar basis of most Western music. The NCS and DT constraints ensure that the scales progress in a step-wise fashion and that tertian chords will posses a similar structure and sonority, being "stacks of three-or four-chromaticsemitone intervals" (Tymoczko 2004, p. 225), undoubtedly one of the reasons these scales have received so much acceptance within a wide range of musical settings. We will now proceed to search for two-octave scales which share the same intervallic characteristics as these seven Pressing scales.

Two-octave Pressing scales and pentatonic equivalents
We start by presenting a few definitions in order to formalize Pressing scales with a view to elaborating an exhaustive list of two-octave Pressing scales (TOPS). For this purpose, it is useful to define the Pressing scale restrictions (PSR) in terms of which interval sequences are excluded.
Interval sequence 11 must be avoided due to the NCS constraint. In order to comply with the DT constraint, the sum of two successive intervals must not exceed four semitones, thereby 7 The methodical study of bebop scales has been widely developed by jazz educators such as Barry Harris or Baker (1985). excluding interval sequences 33, 32, and 23. The spectrum of scale steps must be confined to < 1 >= {1, 2, 3}, since scale-steps of four semitones and above are not compatible with the DT constraint, as diatonic thirds would have to be larger than four semitones. As mentioned previously, these restrictions ensure that scales are intervallically similar to those most employed in Western music and that tertian harmony maintains its characteristic sonority and still sounds "tertian" (Tymoczko 2004, p. 225). Naturally, TOPS are those scales that comply with the PSR and have a two-octave modular interval-span, thereby excluding the seven octave-repeating Pressing scales discussed previously.
Definition 2.5 More generally, Pressing Scales (PS) are scales of arbitrary length complying with the PSR.
Given a scale S, the sequence of notes of S is denoted by [n 1 , . . . , n d ], where d is the number of notes in S; d is said to be the cardinality of S. The scale S can also be described by the sequence of its consecutive distances (p 1 , . . . , p d ). The sum d i=1 p i equals the span (range) of the scale. In particular, if S is a two-octave scale, then d i=1 p i = 24. Certain consecutive distances summing 3 and 4, the spectrum of a diatonic third due to the DT constraint, will be important in the following results. Therefore, let us define the following sets of consecutive distances: The following theorem characterizes TOPS in terms of the distances in D by showing the possible sequences of distances that comply with the PSR and can therefore form a PS. Proof =⇒) If S is a PS, then the following restrictions are present in S: distance 1 can only be followed by either 2 or 3; 2 can only be followed by either 1 or 2; and 3 can only be followed by 1. Based on these restrictions, the only possible distance sequences of length 2 are (12), (13), (21), (22), (31), which are precisely the distances in D. By applying the restrictions in the definition of PS to these sequences again, the restrictions (1)-(5) appear. Indeed, the trees in Figure 7 show all possible distance sequences of length 4 for a PS.
By traveling all the paths down from the root to the leaves for the three trees, the restrictions (1)-(5) are established.

Cardinality of TOPS
We now proceed to categorize Pressing scales in terms of cardinality and scale-step content. In order to present our results we introduce the following definitions.
Definition 2.7 Let S be a two-octave scale represented by a distance sequence S = (p 1 , p 2 , . . . , Definition 2.8 Two scales are said to be step equivalent if they contain the same number of specific scale-step intervals. The class of all step-equivalent Pressing scales is denoted by SC(x, y, z), where x, y, z are the number of ones, twos, and threes in the scale, respectively.
As a matter of fact, for any class SC(x, y, z), the relations x + 2y + 3z = 24 and d = x + y + z always hold, where d is the cardinality of the scale.
Definition 2.9 Two scales are said to be D-equivalent if one can be obtained from the other by replacing a pair of distances by other pairs with the same sum complying at the same time with the PSR.
Due to the DT constraint which limits the spectrum of a diatonic third to < 2 >= {3, 4}, a PS can be characterized as a set of consecutive distance pairs belonging to D 1 and D 2 . In the following argumentation, we use D-equivalence to define the cardinality of TOPS by representing all scales as sequences of interval pairs 22− for all distance pairs belonging to D 2 − and 12− for all distance pairs belonging to D 1 . While the definition of D-equivalence theoretically allows for any substitution within D 1 and D 2 , respectively, this is not pertinent to our argumentation, which merely relies on distance pairs 22 and 12. Notice that all arrangements of distance pairs 22 and 12 necessarily comply with the PSR. Notice also that D-equivalent scales are not necessarily step equivalent by Definition 2.8. Whilst substituting 21 −→ 12, or vice-versa, merely alters the sequence of step-wise distances within a scale, 13 −→ 22, or vice-versa, alters the scale's specific scale-step interval content. The latter substitution will be important for defining the scale-step content of TOPS later on. 8 As stated previously, all TOPS, up to a D-equivalent scale, can be portrayed as a succession of distance pairs 22 and 12 in which the total sum of distances equals 24. In the case of uneven cardinalities, all scales up to a D-equivalent scale can be portrayed as a succession of distance pairs 22 and 12 with an additional unpaired distance at the end, provided the total sum of distances equals 24. This is useful in order to characterize TOPS in terms of cardinality and scale-step content. For this purpose, the order of the distance pairs themselves is unimportant, since the amount of distance pairs for each cardinality and within each class of step-equivalence will remain constant. For instance, all 14-note TOPS, up to a D-equivalent scale, will invariably consist of an arrangement of the following sequence of distance pairs SD = (22 22 22 12 12 12 12), provided the PSR are fulfilled -i.e. (22 12 22 12 22 12 12), (22 12 22 22 12 12 12), etc.
The following theorem characterizes TOPS in terms of cardinality.  Proof Let S be a scale of d notes with a two-octave modular interval span and SD a class representative of all scales, up to a D-equivalent scale, that may be assembled by rearranging the same set of distance pairs while fulfilling the PSR. By Theorem 2.6, it is enough to consider pairs of consecutive distances. Those distances are taken from sets D 1 = {12, 21}, D 2 = {22, 13, 31}.
Considering D-equivalence, it is enough to consider the distance pair 12 for D 1 and 22 for D 2 . Therefore, up to a D-equivalent scale, all PS can be expressed in distance pairs 12 and 22, the sum of said distances being either 3 or 4. Let x, y be the number of elements in D 1 and D 2 that are found in S, respectively. In the case of scales of an uneven cardinality, one distance will remain unpaired. Let the unpaired distance be 1. This argument leads to two Diophantine equations from which the distance pairs which constitute the PS of each cardinality will be derived.
Since the greatest common divisor of 3 and 4 is 1, all the previous equations have integer solutions, which can be found by using standard algebra methods (Euclid's algorithm and Bezout's theorem). We next analyse the relevant solutions to our problem (solutions giving non-negative values for x and y) and the scales obtained in each case.
TOPS are only possible in the case of d = 13, 14, 15. The only 12-note scales within SD 1 that comply with the PSR are (22 22 22 22 22 22) and the D-equivalent (13 13 13 13 13 13). Both scales are octave-repeating by Definition 2.7 and are therefore not TOPS. They correspond to the whole-tone and hexatonic scale, respectively. If d = 16, then (12 12 12 12 12 12 12 12) is the only viable PS and is also an octave-repeating scale, namely, the octatonic.

Two-octave pentatonic-equivalent scales
With a view to including scales that are intervallically similar to the usual pentatonic scale [02479], easily the most widely employed complementary scale of the Pressing scale collection, we now define two-octave pentatonic-equivalent scales (TOPES). 9 Definition 2.11 TOPES are those non-octave repeating scales consisting of scale steps < 1 >= {2, 3} that avoid interval sequence 33 and have at least a 2 and a 3. The class of all equivalent TOPES will be denoted by TOPES(y, z), where y, z are the number of twos and threes in the scale, respectively. We will refer to these intervallic constraints as pentatonic-equivalent scale restrictions (PESR).
Scale-steps are limited to cl2 and cl3 in order to resemble the usual pentatonic scale, which does not contain semitones or scale-steps larger than cl3. To ensure the scale progresses in stepwise fashion, interval sequence 33 -also not present in the usual pentatonic -is not allowed. The complements of those TOPS containing scale-steps of cl3 will necessarily contain scale-steps of cl1 and therefore do not qualify as TOPES. 10 Scales adhering to the PESR are viable only in the case of cardinalities 10 and 11, as we can see in the following theorem.
Proof A TOPES is a sequence composed of elements in {2, 3} with the restriction that the sequence 33 cannot occur. Therefore, the number of threes cannot be greater than the number of twos plus one. Let y, z be the number of twos and threes in S, respectively. The associated Diophantine equation is: 2y + 3z = 24 whose solution is y = 6 + 3n, z = 4 − 2n, n ∈ Z. The solutions such that y, z ≥ 0 are given by the pairs (0, 8), (3, 6), (6, 4), (9, 2), (12, 0). Because the sequence 33 is not allowed, the pair (3, 6) does not provide a TOPES and therefore is discarded. The pair (0, 8) has no twos and by definition it is not a TOPES either. The solution (12, 0) is all composed of twos and is also discarded. Only the pairs (6, 4), (9, 2) lead to TOPES. The pair (6, 4) corresponds to the class scale given by SD − TOPES 1 = (23 23 23 23 22), which contains 6 twos and 4 threes. Amongst other options, this allows for the usual pentatonic scale in a two-octave span, which is naturally discarded since TOPES are non-octave-repeating by Definition 2.11. The pair (9, 2) yields the class scale given by TOPES 2 = (23 23 22 22 22 2). Given that the cardinality of these scales is an uneven number, all are non-octave-repeating.
The sum of scale-steps, which are the elements in the distance sequence, must equal the scale's modular interval-span (the range of the scale), 24 in our case. Let x, y, z be the number of scalesteps of size 1 (cl1), size 2 (cl2), and size 3 (cl3), respectively. Since for TOPS the spectrum of scale-steps is < 1 >= {1, 2, 3}, the Diophantine equation associated is 24 = x + 2y + 3z. For a TOPES, since they are combination of elements in < 1 >= {2, 3}, its Diophantine equation is 24 = 2y + 3z, so the scale-step content of a given group of TOPES can be expressed as TOPES(y, z), denoting the TOPES given by 2y + 3z = 24. In Theorem 2.12, it was proved that the general solution of the TOPES equation was x = 6 + 3n, y = 4 − 2n, n ∈ N, where only the pairs (6, 4), (9, 2) generated TOPES. As for the TOPS, the general solution of the Diophantine Values for k and m giving scales complying with PSR and PESR are shown in Table 1.

Complete list of TOPS and TOPES
Naturally, not all arrangements of a given scale-step-content comply with our scale restrictions. Table 2 shows the total number of TOPS and TOPES arranged by cardinality and scale class. The total number of modes shows the complete number of possible modes for each option of cardinality/scale class, of which the total number of Pressing modes are those that adhere to the PSR or PESR. The total number of modes is equal to the multinomial coefficient 24 x,y,z = 24! x!·y!·z! . In the column labelled distinct Pressing scales, rotationally equivalent scales are grouped together, thereby showing the total number of distinct Pressing scale collections. Appendix 1 contains all 141 two-octave Pressing scales and pentatonic equivalents in standard notation, indicating the number of pcs and the interval-class vector (icv) for each scale. As opposed to the standard icv considering interval classes (ics) 1-6, the icv-mod24 considers ics 1-12 and will be discussed in the following section. An important consideration regarding the sonority of a TOPS is chromatic density, which can be easily measured by the amount of pcs in the scale's two-octave span, referred to as pc-cardinality. The possible number of pcs in TOPS ranges from 8-12. Table 3 shows the amount of TOPS of a certain pc-cardinality within each scale class. Most TOPS have between 9 and 12 pcs, with scales of 10 pcs being the most frequent. Only eight contain the full pc-set. The pc-content of the five SC(2, 11, 0) scales ranges neatly from 8 to 12, the highest corresponding to the maximally even scale of this cardinality. 11 This scale will be studied in depth in the next section.
The two scales of cardinality d = 10 are the only TOPES of this cardinality that comply with the PESR and are the complementary scales of scales number 54 and 55 in Appendix 1 belonging to class SC(0, 4, 10). The four TOPES of cardinality d = 11, scales number 138-141 in Appendix 1, are the complementary scales of scales 5, 4, 3 and 2, belonging to SC (2, 11, 0).
The resulting modes of the PS can be divided into tonic-repeating and non-tonic-repeating modes, depending on whether or not the tonic is repeated in the second octave or not. Tonicrepeating PS-modes may or not be combinations of two standard octave-repeating Pressing scales. For a classification of PS-modes, please refer to Appendix 2.

Scale properties
The structural properties of musical scales and of the major scale in particular have been extensively studied by researchers in the field of what is often referred to as "diatonic set theory " or "diatonic theory." Many noteworthy properties of the diatonic collection have been identified by researchers such as Clough and Myerson (1985), Clough and Douthett (1991), Carey and Clampitt (1989) or Gamer (1967), often with a view to understanding what has made this collection so appealing, as well as establishing a model for the discovery of alternate scale systems, generally microtonal, that share the same properties. 12 In "Scales, Sets, and Interval Cycles: A Taxonomy, " Clough, Engebretsen, and Kochavi (1999) study the relationship between eight scale features all found in the diatonic collection: (1) Generated (G): The scale tones can be "generated" by a single interval, e.g. a fifth (ic7) in the case of the diatonic collection.
(2) Well-formed (WF): Identified by Carey and Clampitt (1989), the property implies that the symmetry of the generative interval is preserved by scale ordering (p. 189).
In the case of the major scale, each scale step is equivalent to two steps within the generative interval cycle of a fifth.
(3) Myhill property (MP): Each generic interval (e.g. second) comes in two specific sizes (e.g. minor and major second). This property was identified by Clough and Myerson (1985) and further examined in Clough, Engebretsen, and Kochavi (1999). (4) Distributionally even (DE): Each scale step comes in either one or two specific sizes identified by Clough, Engebretsen, and Kochavi (1999). (5) Maximally even (ME): Scale tones are distributed as evenly as possible within the total number of available chromatic pitches. This property was identified by Clough and Douthett (1991) in the context of pitch; later, it was studied by other authors from an algorithmic and rhythmic standpoint (Gómez, Talaskian, and Toussaint 2009). (6) Deep (DP): In a chromatic universe of c available pitches, a deep scale is that of cardinality d = c 2 or d = c 2 + 1, in which every interval has a unique number of occurrences, referred to as unique-multiplicity property. Deep scale property was identified by Gamer (1967). (7) Diatonic (DTO): Diatonic, in the sense employed by Clough and Douthett (1991), refers to a maximally even scale in which c = 0mod4 and c = 2(d − 1). These scales display a single ambiguity corresponding to the interval c/2. "Ambiguity," defined by Rahn (1991), occurs when two different generic intervals within a given scale have the same specific interval 12 For an excellent overview of diatonic theory, the reader can refer to chapter 1 of Carey (1998). size, as in the augmented fourth and diminished fifth in the diatonic collection, both corresponding to ic6. Scale-steps come in two specific sizes, two of which are "half-steps" (ic1) and, subsequently, the remaining scale-steps are "whole-steps" (ic2). Clough and Douthett (1991) use the term diatonic to refer to the usual diatonic collection (i.e. the major scale) and hyperdiatonic for larger sets displaying the same characteristic. We will adopt the latter term, which better suits our purposes. (8) Balzano (BZ): Scales with intervals of size k and k + 1 such that d = 2k + 1, the sum of the two, and c = k(k + 1), the product of the two. It was first identified by Balzano (1980). Harasim, Schmidt, and Rohrmeier (2020) proposed an axiomatic framework for scale theory. In it, they unify the study of several of the previous properties, including maximal evenness and well-formedness.
A natural question arises, namely, is it possible to find these properties in non-octave-repeating scales? At first glance this may appear impossible for the following reason. The aforementioned properties have in common the existence of a generator (Clough, Engebretsen, and Kochavi 1999, p. 74), meaning the scale's pcs can be generated by a single interval and then rearranged in stepwise fashion in order to form the scale. In the case of an octave-repeating scale, the pcs are arranged within an octave-span. In the case of a non-octave-repeating scale, however, this poses two important problems. Firstly, how should one go about arranging the generated pcs if not in an arbitrary manner? That is to say, how can one determine which pc should be placed in which octave and for what reason? Secondly, if a non-octave-repeating scale can potentially contain the full pc aggregate, what is the role of the generator? If all pcs are present, does this not render the generator irrelevant? The fact is one can find these properties in non-octave-repeating scales by considering the scale's modular interval-span (24 semitones in the case of a two-octave scale) as the interval of periodicity, that is, the interval at which pitches are functionally equivalent.

Generated two-octave scales -Mod24 and icv-mod24
Without intending to question the by now surely undisputed reality of octave-equivalence, in order to understand certain structural relationships between pitches in two-octave scales we must think in modulo 24 (mod24), as opposed to modulo 12 (mod12), that is, in a cycle of 24 semitones. Figure 8 shows a 24 semitone cycle fixing each pc in a specific octave with the use of the superscripts 1 and 2. 13 Thus, pitches obtained by a generative interval cycle may be unequivocally arranged in scalar order.
Naturally, standard interval equivalencies must be adapted, since the sum of a given interval and its inversion must equal 24; interval equivalences are therefore 1-23, 2-22, 3-21, etc., as opposed to 1-11, 2-10, 3-7, etc. Hence, an icv considering a modular interval-span of twooctaves -referred to from now on as icv-mod24 -will contain 12 digits, as opposed to 6. The relevance of this as an analytical tool can be observed in the two scales in Figure 9.
The graph above the scales in standard notation shows the scales expressed in integer numbers, the number of distinct pcs 14 and the scale's icv-mod24. Both scales contain the full chromaticset, meaning that the standard six-digit icv would be equal in both cases to that of the chromatic scale. Yet the icv-mod24 shows that the interval content of both scales is actually quite different. Take, for instance, the variation as far as the number of different interval occurrences. The greater  the number of interval multiplicities, the greater the number of possible distances between different transpositions of a given scale. In the major scale, for instance, its icv 2,5,4,3,6,1 shows that each interval occurs a unique number of times. Therefore, the common tones between a given key and its possible transpositions will be distinct, coinciding with the number of occurrences of the interval by which the scale is transposed; transposition by a semitone results in two common tones, by a whole tone in five common tones, and so on and so forth. 15 Since the number of interval occurrences in the major scale are a permutation of numbers one to six, a distance hierarchy can be established between possible transpositions, which is naturally interesting from a compositional viewpoint. Whilst in scale (a) each ic occurs a unique amount of times, similarly to the major scale, in scale (b) ics 2, 4, 5, 6, 8 and 10 each occur eight times, implying, less variety in the amount of common tones under transposition than in scale (a). 16 Thus, the icv-mod24 proves to be a valuable tool for analysing two-octave scales. This concept can easily be applied to scales of any given octave-range by considering an icv of c/2 ics (recall that c is the size of the chromatic universe).

Structural properties of the hyperdiatonic two-octave scale
Having established a way of applying the concept of a generator to two-octave scales we will proceed to search for a two-octave scale that shares the properties of the diatonic collection listed in Clough, Engebretsen, and Kochavi (1999), with the exception of BZ. This property rests "on an interval k and the next larger interval k + 1, whose sum is equal to the generator and whose product is equal to the size of the chromatic universe" (p. 77). When considering 12 pitches to the octave, as is the case in this study, this requirement cannot be met in non-octave-repeating scales spanning less than 6 octaves.
Let us commence by searching for a two-octave scale with DTO property. Provided that the chromatic universe c is congruent to 0 (mod 4), it will contain a single scale with this property (Clough and Douthett 1991, p. 141). This scale will be the ME-set in which c = 2(d − 1), with d being the cardinality of the scale. Hence, a two-octave scale with this property must be the ME-set for cardinality d = 13. By applying Clough and Douthett's algorithm to find the ME set for d = 13 and c = 24 as follows, . . . , 12 = (0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22) we arrive at the scale (described by its note sequence) shown in Figure 9(a). Its distance sequence is (1222221222222). Clough and Douthett (1991) show that DTO scales can also be defined as ME sets which present a single ambiguity with clen c/2. In the case of the usual diatonic scale, this ambiguity is the tritone or clen 6. In the case of the hyperdiatonic two-octave scale, as we shall refer to this scale from now on, 17 this ambiguity is the octave or clen 12. This can be easily verified by observing Table 4 below showing the generic and specific intervals in the hyperdiatonic two-octave scale. Clen 12 is the only specific interval corresponding to two consecutive generic intervals, has an interval size of c/2 and appears only once, as does the tritone in the usual diatonic scale. Thus, the hyperdiatonic two-octave scale shows DTO property.
A diatonic set, in the sense employed here, can be generated by a single interval. It is common knowledge that in the case of both the pentatonic and diatonic collections the generative interval i is equal to the scale's cardinality, 5 and 7, respectively. Clough and Douthett (1991) prove this to be the case for all hyperdiatonic scales (p. 164). Hence, the generating interval of the scale in Figure 9(a) should be 13, a minor ninth. Figure 10 shows a cycle of minor ninths in a modular interval-span of 24 semitones. Since i and c are coprime, the cycle covers all 24 semitones. It may appear at first glance to be a semitone cycle, yet the numbers fixing each pc in a specific octave show that it is not. A sequence of 13 pitches counting clockwise from E 1 arranged in scalar order renders the hyperdiatonic two-octave scale shown in Figure 9(a).  The interrelation between scale properties has been extensively studied in the field of diatonic theory. All ME scales are also DE; DTO scales are endowed with MP and DP; MP implies WF (Clough, Engebretsen, and Kochavi 1999, p. 78). All DP scales show unique-multiplicity property, which, in the case of the hyperdiatonic two-octave scale, can be verified by regarding the scale's icv-mod24, a permutation of intervals 1-12 (see Figure 9(a)). The scale's "two-element spectrum" (Clough and Myerson 1985, p. 262) -characteristic of MP, meaning that each generic interval comes in two specific sizes -is shown in Table 4. In WF scales the symmetry of the generator cycle is preserved by scale ordering, referred to by Carey and Clampitt (1989) as the symmetry condition. In the case of the major scale, each scale-step spans two steps within the generator cycle, congruent to modulo 7, the scale's cardinality. This can be easily observed in the regular polygon obtained by connecting the pitches within a sequence of fifths -the generating interval -by scale order, shown in Figure 11(a). The symmetry of the generator cycle is also preserved by scale ordering in the case of the two-octave hyperdiatonic scale, where each scale-step also spans two steps within the generator cycle, congruent modulo 13, as shown in Figure 11(b).
In conclusion, the hyperdiatonic two-octave scale is endowed with all scale properties of the diatonic collection listed in Clough, Engebretsen, and Kochavi (1999) -G, WF, MP, DE, ME, DP and DTO -with the exception of BZ. Figure 11. (a) Scale-step symmetry within the major scale, taken from Carey and Clampitt (1989); (b) Scale-step symmetry within the hyperdiatonic two-octave scale.

Complementary scale
When the generating interval of a scale is coprime to the number of pitches in the chromatic universe it runs through the full collection of pcs, meaning that the complement of said scale can also be generated by the same interval, as in the case of the diatonic collection and the usual pentatonic scale [02479], both generated by ic5. The pentatonic scale has unique multiplicity, as well as WF, ME and MP, since these properties always hold under complementation (Clough, Engebretsen, and Kochavi 1999, p. 101). Given the properties of the hyperdiatonic two-octave scale described above, it is to be expected that it would bare the same relation to its complement as the diatonic to the pentatonic. Figure 12 shows that this is indeed the case. The complement of the hyperdiatonic two-octave scale is a ME scale and its icv-mod24 is a permutation of integers 0-11 (Figure 12(a)), meaning the scale has unique multiplicity. Figure 12(b) reveals that the symmetry of the generating interval is maintained in scale ordering, and thus the scale is WF. Table 5 shows the number of occurrences of each generic/specific interval. Since each generic interval comes in two specific sizes the scale is also endowed with MP.

Maximally even sets
For every cardinality d within a chromatic universe of c pitches there is a single maximally even set ME(d, c) which can be arrived at by applying Clough and Douthett's algorithm. As shown above, the hyperdiatonic two-octave scale is the ME set for cardinality d = 13 and its complement is the ME set for cardinality d = 11. When c and d are not coprime, as in the case of d = 10, d = 14 and d = 15, the ME set will be a symmetric scale consisting of intervalically similar scale fragments. ME (15,24) [0-1-3-4-6-8-9-11-12-14-16-17-19-20-22], scale no.129 in Appendix 1, belonging to SC(6,9,0), is a symmetrical scale consisting of three five-note scale fragments spanning eight semitones each and corresponds to scale no. 707 in Slonimsky's (1947) Thesaurus. In fact, it is the only symmetric scale of the TOPS collection and the only possible ME set, other than the hyperdiatonic two-octave scale ME(13,24) and its complement ME(11,24). Given that mcd = 2 in the case of both ME(10,24) and ME(14,24), the ME sets for cardinalities d = 10 and d = 14 each consist of two similar scale fragments spanning 12 semitones and are  thus in reality octave-repeating scales in a two-octave span, namely the usual pentatonic and the diatonic collection, respectively.

Conclusions
The subject of non-octave-repeating scales has not as of yet received significant attention in peerreviewed journals. Our paper starts by providing historical context of the use of these structures from the middle ages to their application in contemporary composition and jazz improvisation. Whilst non-octave-repeating scales have been dealt with in specific contexts, such as the Znamenny Rospev, a broad overview showing the existence of non-octave-repeating scales as a way of organizing pitch content in a wide range of musical cultures and idioms has until now been missing.
In this paper we isolate and study a particular group of non-octave-repeating scales for the first time which share intervallic characteristics found in the most commonly employed scales in Western music. We provide a comprehensive list of such scales, being, to our knowledge, the first comprehensive list of non-octave-repeating scales of any nature. We hope that this may prove to be useful for composers and improvisors seeking to expand their musical vocabulary through scales that offer an array of unexplored musical possibilities, whilst being structurally similar on an essential level to those scales towards which most musicians show a natural preference.
By considering a scale's modular-interval span as the interval of periodicity, we identify a twooctave scale endowed with structural properties shared by widely used one-octave scales, such as the major diatonic, within the field of diatonic theory. Such properties in non-octave-repeating scales are discussed and successfully identified in the present study. We hereby bring forth a scale that shares important structural properties with the diatonic collection and at the same time contains the full pc aggregate. Whilst it is true that these formal properties do not automatically imply that a certain scale necessarily has more musical potential than others, as argued for example by Tymoczko (1997, p. 135-136) regarding the diatonic collection, the existence of these properties in a two-octave scale containing the full pc aggregate is, nevertheless, noteworthy. We hope that this both chromatic and diatonic scale, along with the complete collection of TOPS, will kindle the imagination and curiosity amongst the many musicians who employ both tonal and atonal harmonic procedures in their work.
Given the relatively small body of literature dedicated to octave-repeating-scales and the variety of potential musical applications -as well as the huge number of theoretically feasible constructs -, the possibilities for further research are considerable. While this paper provides historical context, a more in-depth study focusing on the application of non-octave-repeating scales in a variety of musical settings would be helpful towards understanding and further developing the musical potential inherent in these scales. This study is limited to two-octave scales of specific intervallic characteristics, yet these could be easily modified or expanded to accommodate any attributes musicians may deem interesting for whichever purpose, such as lifting the consecutive semitone constraint in order to consider scales containing a certain number of successive semitones. This paper also excludes scales with a modular interval-span of three or more octaves, as well as those where the interval of repetition is smaller than an octave -with the exception of scale no. 707 in Slonimsky's Thesaurus -or larger, as in the works of Joel Hoffman or Gao Weijie. Further properties studied in the field of diatonic theory may also be found in non-octaverepeating-scales of varying range. The possible musical applications of the scales isolated in this study both in composition and improvisation, as well as of other non-octave-repeating scales, is also a viable and potentially fruitful line of investigation.

Glossary of acronyms and abbreviations
pc: Pitch-class. ic: Interval-class. icv: Interval-class vector. dlen: Diatonic length or generic interval size (as in a minor second), as defined by Clough and Douthett (1991). clen: Chromatic length or specific interval size (as in a semitone), as defined by Clough and Douthett (1991). cl: A specific clen (cl1 = one semitone, cl2 = two semitones, etc.). d: Cardinality d. The cardinality of a given scale. c: Chromatic universe c. The total collection of available pitches. DS: Diatonic seconds. Intervallic constraint defined by Tymoczko (2004) whereby the specific size of one scale-step is to be either one or two semitones. NCS: Non-consecutive semitones. Intervallic constraint defined by Tymoczko (2004) whereby a scale cannot contain successive one-semitone steps.
DT: Diatonic thirds. Intervallic constraint defined by Tymoczko (2004) whereby the specific size of two scale-steps is to be either three or four semitones. PSR: Pressing scale restrictions. Restrictions whereby the NCS and DT constraints are fulfilled. PS: Pressing scale. A scale in which the PSR are fulfilled. TOPS: Two-octave Pressing scale. TOPES: Two-octave pentatonic-equivalent scale. PESR: Pentatonic-equivalent scale restrictions. Restrictions that define TOPES. Proof =⇒) Assume that mode (p 1 , . . . , p d , q 1 , . . . , q d ) is a TOPS mode. Then, every pair of distances belongs to set D = {12, 21, 22, 13, 31}. In particular, that is true for p d q 1 and q d p 1 .
⇐=) By hypothesis, S 1 and S 2 are PS-modes. Their distance pairs already belong to set D = {12, 21, 22, 13, 31} in Definition 2.5 and comply with restrictions (1)-(5) in Theorem 2.6, thus adhering to the PSR. When combining S 1 and S 2 to form a two-octave mode, two additional distance sequences occur between the adjacent pitches on either side of the tonic in both octaves, namely, p d q 1 and q d p 1 . By the hypothesis of this part of the proof, these two distances already belong to set D. Therefore, when combining S 1 and S 2 , the resulting mode complies with the PSR. Therefore, mode (p 1 , . . . , p d , q 1 , . . . , q d ) is a TOPS mode.
The scale-step content of each scale class determines what possible combinations of one-octave PS-modes occur in each case. Scale classes including distances of one and two may include combinations of modes of the diatonic, acoustic, whole-tone and octatonic scale. Scale classes including distance 3 may include combinations of modes of the harmonic minor and harmonic major collection. Scale classes with three distances of three may allow for combinations including the hexatonic scale modes. The description of the different possibilities within each scale class is as follows: Cardinality 13 • The five scales belonging to SC(2,11,0) include combinations of the whole-tone scale and modes of the diatonic and acoustic scale, respectively. • The eight scales belonging to SC(3,9,1) include combinations of the modes of the harmonic minor scale with the whole-tone scale and combinations of the modes of the harmonic major scale with the whole-tone scale. • The nine scales belonging to SC(4,7,2) do not include any combinations of one-octave PS-modes. Since the cardinality of a one-octave PS ranges from 6-8, a tonic-repeating TOPS mode of cardinality d = 13 must be comprised of two one-octave modes of cardinalities d = 7 and d = 6, respectively. In SC(4,7,2) there are two distances of three. A seven note PS with two distances of three is not possible and the only six note PS that allows for distance three is the hexatonic which requires for three distances of three (131313). • The eight scales belonging to SC(5,5,3) include combinations of the hexatonic and modes of the diatonic or acoustic collection, respectively. • The five scales belonging to SC(6,3,4) include combinations of the hexatonic and modes of the harmonic minor or harmonic major collection, respectively.

Cardinality 14
• The 20 scales belonging to SC(4, 10, 0) contain combinations of modes of the diatonic and acoustic collection as well as combinations of the two modes of the octatonic and the whole-tone scale. • The 35 scales belonging to SC(5, 8, 1) contain combinations of modes composed of either the diatonic or acoustic collection in one octave and modes of the harmonic minor or harmonic major collection in the next. Given that there are not enough distances of three to form a hexatonic scale and the octatonic and whole-tone scales do not allow for distance three, there are no possible combinations of six and eight note PS modes. • The 24 scales belonging to SC(6, 6, 2) contain combinations of modes of the harmonic minor and/or harmonic major collection. Given the number threes, combinations of six and eight note PS modes are not possible for this scale-class. • Given that heptatonic PS with more than one distance of three are not possible, the five scales belonging to SC (7,4,3) contain only combinations of the hexatonic and octatonic.
Cardinality 15 Given the cardinality of one-octave PS, tonic-repeating TOPS modes of cardinality d = 15 may be combinations of the octatonic and the four heptatonic PS. • The 10 scales of SC(6, 9, 0) contain combinations of the octatonic with modes of the diatonic or acoustic collection, respectively. • The six scales of SC(7, 7, 1) contain combinations of the octatonic with modes of the harmonic minor or harmonic major collection, respectively. Table A1 shows the total number of tonic-repeating TOPS modes and of those that are comprised of one-octave PS-mode combinations within each scale class.