Fishing for complements with chord, scale, and rhythm nets

The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.


Complementation as a basic operation in music theory
Transposition and inversion are fundamental operations in music theory.They act on sets such as chords that are embedded into a comprising space such as a (diatonic) scale, the pitch-class cycle, or the line of fifths.For example, a transposition by two units applied to a C major triad embedded into the C major scale yields an E minor triad, and a transposition by two units applied to the C major triad embedded directly into the chromatic pitch-class cycle yields a D major triad.These two transpositions are, of course, instances of diatonic and chromatic transpositions, respectively, and the general concept applies to arbitrary comprising spaces.
Analogously, transposition and inversion can be applied to rhythms represented as beat patterns (also called onset patterns) embedded into a metric space.The formalization of rhythm used throughout the paper models rhythms as Boolean patterns on a cyclic stream of pulses, without an explicit representation of metrical weights.Note that this formalization, recently also used by Yust (2021), is different from the conceptualization of rhythm as the interaction of meter and grouping structure as proposed in the Generative Theory of Tonal Music (GTTM Lerdahl and Jackendoff 1983).In contrast, metrical qualities are introduced into our model in Section 6 using the discrete Fourier transform.
The main topic of this paper is complementation, another operation which -from a theoretical perspective -is similarly basic as transposition and inversion.This operation maps a chord embedded in a comprising space to its complement in that space.For example, the complement of a C major triad embedded into the C major scale is a B half-diminished seventh chord, and the complement of an A minor triad is a G dominant seventh chord.Complementation is similar to transposition and inversion in that it acts on a set embedded in a space.In contrast to transposition and inversion, however, complementation changes the tonal character of the family it acts on.Intuitively speaking, it maps its arguments into a different world as exchanging the foreground with the background in one of M.C.Escher's tiled drawings.In the example above, the world of C major's diatonic triads is mapped to the world of C major's diatonic seventh chords.Notably, complementation preserves voice leadings in that it only involves a change in direction (Harasim, Schmidt, and Rohrmeier 2016), the transition from C to Am leads G to A and the transition from Bm 7 5 to G 7 leads A to G, while all other notes are preserved.In particular, both pairs of chords are connected through a single voice's move. 1 While transpositions can be seen as automorphisms of voice leading graphs, complementations can be seen as graph antihomomorphisms because of the direction change.
The hexatonic pole is a particular example of complementation (Cohn 1996), it maps major and minor triads to their complements in symmetric hexatonic scales.For instance, a C major triad embedded in the scale {C, E , E, G, A , B} is mapped to an A minor triad.The listening impression of transitioning into a different world -in terms of chordal areas -can be observed directly for the hexatonic pole as used in Wagner's Parsifal or in the the score of the film Lord of the Rings, because in contrast to the diatonic example above, the hexatonic pole stays in the realm of major and minor triads.In general, however, complementation is not directly observable in a piece of music, but acts as a remarkably fruitful bridging element between preexisting theoretical realities and as a source for creative thinking.
In contrast to the two examples described above that consider complements in diatonic and hexatonic scales, the focus of this paper is complementation in the chromatic pitch-class cycle.Concretely, we consider diatonic seventh chords and interpret their chromatic complements both as 8-note scales and 8-beat rhythms on an equidistant 12-beat grid.This approach turns out to be particularly useful as a tool to organize and understand the respective 8-note scales, which were popularized by the jazz theorist Barry Harris (García-Valdecasas Vaticón 2020).
In the general discourse on scale concepts in connection with voice leading, our approach puts considerable emphasis on the diatonic origin of the chords that are mapped under the complement operation -even though these complements are not necessarily diatonic themselves.The diatonic origin is modeled using 2nd-order Clough-Myerson scales (see Section 3.1).In the case of triads and seventh chords, it turns out that their complements obey coverings through chords from closely related families (e.g.Cube Dance triads and Power Tower seventh chords).Future investigations might also look into complementations of other voice-leading graphs, such as nearly even n-note chords (as studied in Section 3.11 in Tymoczko (2011), p. 103).

Measuring voice-leading distances
The preservation of voice leadings under complementation is at the core of the work presented in this paper.We use it to explore and to compare geometric spaces consisting of categorically different musical objects such as chords, scales, or rhythms.Note that parts of the literature on voice-leading geometry use a mathematically more general setting in which voice-leading distances are not necessarily invariant under complementation (e.g.see Tymoczko 2006, for alternative voice-leading measurements and Hall and Tymoczko 1998, for the discussion of specific L p metrics).In order to explain this in greater detail, we first present preliminary definitions and propositions.
Following -and slightly simplifying2 - Harasim, Schmidt, and Rohrmeier (2016), let the cyclic group Z n (here of order n = 12) denote optionally a cycle of chromatic scale degrees or a cycle of abstract rhythmic pulses.The elements of Z n are classes of integers that we refer to by their smallest non-negative representatives.The chromatic (or temporal) distance between two tones (or pulses) x, y ∈ Z n is defined as the length of the shorter one of the two paths from x to y and from y to x on the chromatic circle (or the circle of pulses) in ascending order, ( 1 ) For example when n = 12, then d(3, 5) = 2 and d(1, 10) = 3.This concept of distance is also known as Lee distance.
One-to-one voice leadings are modeled as bijective functions between two equally sized sets of tones.Let X , Y ⊆ Z n denote two of such sets and let m = |X | = |Y |.Using the notation of the set of bijections (i.e.one-to-one voice leadings) from X to Y as Bij(X , Y ), the distance between X and Y can be measured based on the L p norm of the voices' movements given by d(x, f (x)), where p ≥ 1 and Z n m = {X ⊆ Z n | |X | = m} denotes the set of subsets of Z n of size m.Any bijection that realizes the minimum is called a minimal voice leading from X to Y, and in general there can be more than one of those.For instace when n = 12 and p = 1, then D p VL (Dm 7 , G 7 ) = 3 and the minimal voice leading (which is unique in this example) moves the tones C to B, A to G, and fixes D as well as F. Harasim, Schmidt, and Rohrmeier (2016) considered p = 1 and showed that In particular, there is always a minimal voice leading that does not move tones that belong to both X and Y. Furthermore, the complement mapping Since the present paper is based on complementation as an isometric isomorphism and since complementation does not preserve voice-leading distances for p > 1, we solely consider D VL := D 1 VL and we refer to this distance simply as the metric of minimal voice leading.This might seem to be a strong limitation of our approach at first, but the opposite is the case.In fact, one of our main conclusions is that the metric D 1 VL is of outstanding importance among all D p VL , because it leads to strong characterizations and insights by connecting musical phenomena that otherwise would appear unrelated.

Power towers and clapping-music patterns' phylogenetic graph
The initial motivation for the research presented in this paper was the observation that the "Power Towers" voice-leading graph (Douthett and Steinbach 1998) is directly linked to the "phylogenetic graph" of the clapping-music patterns as constructed by Colannino, Gómez, and Toussaint (2009).Both graphs are shown side-by-side in Figure 1.This observation was preceded by the finding that the chord analogue of the complement of Steve Reich's signature rhythm, the clapping-music pattern , is a minor seventh chord (Yust 2021).The challenge to compare the two graphs and the respective concepts of distance then followed naturally from the theorem that complementation preserves L p voice-leading distances (Harasim, Schmidt, and Rohrmeier 2016).
It is insightful to study the rotations of the clapping pattern as well as the prominent 7-beat patterns that they contain.The clapping pattern extends the Yoruba pattern by one extra beat.The Yoruba pattern is the rhythmic analogue of the Mixolydian mode, and it is a rotation of the African standard pattern or bell pattern , which in turn is the rhythmic analogue of the Ionian mode.Recent psychological research indicates that there might be a cognitive correlate for the pitch-rhythm correspondence (Wen and Krumhansl 2019), but that connection is still fuzzy and requires further research.
The eight-note pitch analogue of the clapping pattern is a mode of the Barry Harris major (6th-diminished) scale (García-Valdecasas Vaticón 2020): To be more precise, the Barry Harris major scale extends the Ionian mode by the flattened sixth degree 6.And up to rotation, this is analogous to the clapping pattern being an one-beat extension of the Yoruba pattern.The literal rhythmic analogue of the Barry Harris major scale has been mentioned in Colannino, Gómez, and Toussaint (2009) as a resultant clapping pattern of the Lala People, which in turn extends the African standard pattern by one beat.

Organization of this paper
The main part of this paper is organized in five sections.Section 2 starts with reviewing preliminaries such as the concept of minimal voice-leading distance.We then show the equivalence of this distance to the swap distance between rhythms and explicate the correspondence of the graphs shown in Figure 1.Section 3 then presents a novel construction of the power-tower graph based on diatonic seventh chords using the betweenness relation.Section 4 conceptualizes the chromatic complements of diatonic seventh chords as families of Barry Harris scales.Using the graph construction presented in Section 3, we translate diatonic chord sequences and sequences of alterations into the realm of Barry Harris scales, which transforms diatonic transitions into chromatic alterations and vice versa.Section 5 is devoted to the question how the bijective voice-leading distance generalizes to chords of different cardinality and how such generalizations behave under complementation.We propose three generalizations based on injective functions, surjective functions, and total relations.While the injective voice-leading distance is always invariant under complementation, the surjective voice-leading distance is almost never invariant.Since relational voice-leading distances are also not invariant in general, we characterize a subclass that is invariant.Finally, core aspects of the previous sections are combined in Section 6 under the perspective of the discrete Fourier transform.In particular, we show the usefulness of Fourier phases to approach a musically interpretable distance measure between rhythms.

Minimal voice leading and swap distance
The goal of this section is to describe explicitly how the phylogenetic graph of rhythms is embedded in a complementary version of the Power Towers.First, we present a proof of the equivalence of the two distance concepts, and then discuss the rhythmic Power Towers.

Decomposition of minimal crossing-free voice leadings
In what follows, we show that there is always a minimal voice leading that is also crossingfree.However, some of the crossing-free voice leadings that are minimal move tones in from the intersection of X and Y.We then construct a decomposition of minimal crossing-free voice leadings into what Colannino, Gómez, and Toussaint (2009) call swaps in order to explicitly describe the equivalence of the swap distance and the metric of minimal voice leading.
n of the elements of a set X ∈ Z n m shall be called a voicing of X.Let 0 < a 1 , . . ., a m < n denote natural numbers that satisfy x i + a i mod n = x i+1 for i ∈ {1, . . ., m}.The associated voicing is then said to be in closed position if m i=1 a i = n.Furthermore, we say that a voice leading f : Note that the definition of crossing-free still allows the paths of voices to overlap in the pitch-class circle.What is excluded are crossings of voices in opposite directions and passing maneuvers between voices in the same direction.The excluded voice leadings are those that look like crossings in staff notation.Tymoczko (2006) showed the existence of a minimal, crossing-free voice leading between two chords, in the context of a generalized class of measures for voice-leading.For our particular metric, we propose two lemmata which address the previous two situations, visualized in Figure 2. Together, these lemmata show that there is always a crossing-free voice leading that is also minimal.
Lemma 2.2 A voice leading f : X → Y in which two voices cross each other in opposite direction is not minimal.Lemma 2.3 A voice leading f : X → Y in which one voice overtakes another one in the same direction is equivalent to a voice leading f : X → Y (with regard to distance) where these two voices exchange their destination tones.
Proof Suppose f (x) = x + k mod n and f (x + a) = x + (a + l) mod n, with natural numbers 1 ≤ a, and a + l < k.Without loss of generality, let k ≤ n 2 .The contribution of x and x + a to (see right side of Figure 2 for an illustration).
Proposition 2.4 For given sets X , Y ∈ Z n m there exists a minimal voice leading f : X → Y that is crossing-free.
Proof As shown by Harasim, Schmidt, and Rohrmeier (2016), there exists a minimal voice leading f : X → Y in which common tones are fixed.Lemma 2.2 implies that f cannot contain crossing voices in opposite directions.In accordance with Lemma 2.3 we may successively remove all cases where one voice overtakes another one in the same direction without violating the minimality of the voice leading.By induction on the distance D VL (X , Y ) = k.For k = 1, f itself is parsimonious.For k > 1 it is sufficient to show that there is at least one note x of X which can move one step x x = x ± 1 towards f (x).This would imply that the intermediate chord X satisfies D VL (X , X ) = 1 and that the remaining voice leading f : X → Y (with f (x ) = f (x) and f equal to f elsewhere) satisfies the induction hypothesis with D VL (X , Y ) = k − 1.So let us choose any note x ∈ X .If it can move one step ±1 in the direction of its destination f (x), we are done.If it has a direct neighbor x ± 1 ∈ X in its way (i.e. in the direction of f (x)) then the obstacle note x ± 1 must also move in that same direction, otherwise the f would not be crossing-free.The same argument applies to potential further obstacle notes x ± 2, etc. Eventually there is a note x ± r, which can freely move to x ± (r + 1) in the direction of f (x ± r), as desired.
These propositions hold because distance is preserved under complementation precisely as a consequence of restricting the voice-leading metric to L 1 .Otherwise, the parsimonious voice leadings introduced as workarounds for crossings contribute in L 1 more to the aggregate L p distance than the crossing version of the voice leading.This can be seen easily in an example in the chromatic universe Z 4 with X = {0}, Y = {2}, and a distance measurement based on L p for p ≥ 2: The minimal voice leadings are f : X → Y with 0 → 1 and g : X → Y with (1, 2, 3) → (0, 1, 3).

Isometry between chords and rhythms
In Colannino, Gómez, and Toussaint (2009) rhythms are not described in terms subsets X ⊆ Z n , but rather in terms of their characteristic functions χ X : Z n → {0, 1} (with χ X (k) = 1 for k ∈ X and χ X (k) = 0 elsewhere).The set Z n m of m-element subsets thus corresponds to the set χ Z n m ⊂ {0, 1} n of vectors that have m instances of 1 and n − m instances of 0. The distance between any two such vectors is measured in terms of the minimal number of swaps needed to transfer one into the other, as formalized by the following definition.
Definition 2.7 For any index k ∈ {1, . . ., n}, consider the map that swaps the entries k and k + 1 of a ∈ {0, 1} n .The (undirected) swap graph S n m consists of the characteristic functions χ X ∈ χ Z n m as nodes, and it has an edge between two distinct functions χ X and χ Y whenever there is an index k ∈ Z n such that swap k (χ X ) = χ Y .In particular, loops are not included.The (undirected) swap distance σ (χ X , χ Y ) between χ X and χ Y is the length of the shortest path in S n m that connects them.
Proposition 2.8 The transition χ : Z n m → χ Z n m from subsets X to their characteristic functions is an isometry with respect to D VL and σ .In particular, for any X , Y ∈ Z n m their minimal voice-leading distance coincides with the swap distance of their characteristic functions, Proof This is evident for D VL (X , Y ) = 1 and apart from that it follows from Corollary 2.6 because swaps are parsimonious voice leadings.
Proposition 2.8 substantiates also the following remark in Colannino, Gómez, and Toussaint (2009, 118): This [i.e.computing the swap distance] is equivalent to computing the L 1 norm between the index vectors of the onsets' time coordinates of [. . . ] two rhythms, and corresponds to the distance measure preferred in voice-leading.
However, there is a subtle deviation between Definition 2.7 and the concept of swap distance as used by Colannino, Gómez, and Toussaint (2009) who exclude swap n .That is, they do not consider rhythms cyclic entities.In contrast, we do consider rhythms cyclic entities because they are commonly repeated multiple times in compositions and improvisations.

Power towers of complements
Applying the isometry of the complement mapping as well as the isometry χ between chords and rhythms, Figure 3 shows the embedding of the phylogenetic graph (Colannino, Gómez, and Toussaint 2009), displayed in the right half of Figure 1, into the "Power Towers" graph of the 8-note-complements of the seventh chords from the original Power Towers graph (Douthett and Steinbach 1998), displayed in the left half of Figure 1.The ancestor rhythm A = 101101101101 (Colannino, Gómez, and Toussaint 2009, p. 119) and its two transpositions 110110110110 and 011011011011 (interpreted as the three octatonic scales in pitch space) are characteristic functions of the complements of the three diminished-seventh chords {1, 4, 7, 10}, {2, 5, 8, 11} and {0, 3, 6, 9}, respectively.The 12 transpositions of Reich's Clapping Music pattern (interpreted as the Barry Harris major scales in pitch space) are the complements of the 12 minor-seventh chords Figure 3.In this variant of the Power Towers graph from Douthett and Steinbach (1998), every seventh chord has been replaced by its 8-note complement.The phylogentic graph of the clapping patterns (Colannino, Gómez, and Toussaint 2009) is the subgraph indicated by the filled nodes and thick edges (the erroneous edges connecting W 5 and U 8 with V 11 have been ignored).In the present figure, the graphical edge lengths vary and therefore do not faithfully represent the elementary swap distance 1. and are designated with V k for k ∈ {0, . . ., 11} (as in Figure 1, right).The remaining two kinds of rhythmic patterns correspond to complements of dominant-seventh and half-diminished-seventh chords, and are designated with U k and W k for k ∈ {0, . . ., 11}, respectively.The Yoruba bell patterns, the 12 transpositions of 101011010110 which we designate by B k , are not included directly in the rhythmic Power Towers because they contain 7 instead of 8 beats.However, they are contained in the transpositions of the clapping pattern, B k = V k • χ Z 12 \{1−k} , and will be discussed further in Appendix 1.
The rhythms included in the complement version of Power Towers can also be interpreted as 8-note scales in pitch-class space.This enables a new approach to the systematic study of such scales, by investigating them as complements of seventh chords.The complements of fully diminished chords are symmetric octatonic scales (the ancestral rhythm A and its transpositions), the complements of minor-seventh chords are Barry Harris major scales (clapping patterns V k ), the complements of dominant-seventh chords are Barry Harris minor scales (U k ), and the complements of half-diminished seventh chords could perhaps be called "Barry Harris negative minor scales" (W k ) because they are inversions of such minor scales. 3Potential musical consequences of the Power-Tower structure of these 8-note scales are explored in Section 4.

Constructing power towers from betweenness in Douthett graphs
This section concerns the principled construction and musical interpretation of chord and rhythm graphs such as the Power Towers of seventh-chord complements described in the previous section.The goal is to gain a deeper understanding of the nature of those graphs.Colannino, Gómez, and Toussaint (2009) had the idea to interpret the swap distance that defines such graphs as a similarity measure for rhythms.Later they abandoned their approach in the light of counterexamples, they found categorically different rhythms that are close with respect to the swap distance. 4This can be understood in the light of the pitch-rhythm analogy.Parsimonious voice leadings between chords typically entail significant changes in their harmonic quality.In other words, in the domain of voice leading one would abstain from interpreting the underlying concept of distance as a similarity measure.A better interpretable approach to rhythmic similarity is discussed below in Section 6.
The presence of seventh chords (as complements) enables an interpretation of Figure 3 through investigations in the structure and voice-leading behaviour of diatonic seventh chords.Each transition between contiguous chords falls into one of two classes, inner-diatonic transitions and chromatic alterations, as described below.Our aim is to investigate to what extent these classes can be translated to the complement world.The results are of particular importance for Barry Harris scales as discussed in Section 4. For the start, the fully diminished seventh chords are left aside and some of the remarkable findings from Clough and Myerson (1985), Clough andDouthett (1991), andDouthett (2008) used in the present argument are recapitulated.Thereby, the concepts of Douthett graphs and betweenness graphs are used as defined by Harasim, Noll, and Rohrmeier (2019).

Second order Clough-Myerson scales and douthett graphs
The family of the 48 diatonic seventh chords is a special instance of a family of secondorder Clough-Myerson scales whose musical meaning is the following.Diatonic seventh chords possess two kinds of elementary relations to other diatonic sevenths chords.First, there are the inner-diatonic contiguities between third-related seventh chords within each diatonic scale.Each contiguous pair in the cycle shares three common notes, while the remaining note differs by a diatonic step interval, a major second M 2 or a minor second m2.Secondly, there are chromatic alterations such as the one between C 7 and Cm 7 .The 48 diatonic seventh chords form one long alteration cycle of length 48: The mathematical definition of second-order Clough-Myerson scales is based on the so called , where d and c are positive integers (0 < d < c) that denote the cardinalities of generalized diatonic and chromatic collections (Clough and Douthett 1991). 5epending on the mode index m ∈ Z c , the J-function assigns to every diatonic scale degree k ∈ Z d a chromatic scale degree J m c,d (k) using the floor function, The images J m c,d (Z d ) ⊂ Z c are known to be maximally even in Z c .In the special case where d and c are coprime, these sets have further musically relevant properties initially studied by Clough and Myerson (1985), which motivates the term Clough-Myerson scales (or sets).
For any three integers c, d, e with 0 < e < d < c, a J-function J n d,e : Z e → Z d can be composed with a J-function J m c,d : Clough and Douthett (1991) call the images J m,n c,d,e (Z e ) ⊂ Z c second-order maximally even scales (or sets).In the special case in which e and d as well as d and c are coprime, we call these images secondorder Clough-Myerson scales and denote the set of all of them by CM c,d,e .

Betweenness moderates between the Douthett graph and the minimal-voice-leading distance
The betweenness graph is a means of studying the correspondence between Douthett graphs and the metric of minimal voice leading (Harasim, Noll, and Rohrmeier 2019).Its edges constitute a neighbor relation constructable for any distance mapping.The betweenness graph for the family of the 48 diatonic seventh chords in Z 12 is displayed in Figure 5.The Douthett graph and the betweenness graph of the diatonic seventh chords CM 12,7,4 shown in Figures 4 and 5, respectively, have 84 edges in common but each of them also contains edges not included in the other one.For example, the connection between the minor seventh chord Em 7 = 247e and the major seventh chord CM 7 = 047e contiguous in the Douthett graph is mediated by the half-diminished-seventh chord C m 7 5 = 147e in the betweenness graph.Such chords are called distant neighbors.Conversely, there is an edge in the betweenness graph between the dominant-seventh chord C 7 = 047t and the half-diminished-seventh chord B m 7 5 = 148t because the voice-leading distance is D VL (047t, 148t) = 2 and there is no mediating diatonicseventh chord between the two.However, this edge is not present in the Douthett graph because the chords neither form an inner-diatonic contiguity nor an alteration.Edges of the betweenness graph that do not appear in the Douthett graph are called interscalar contiguities (Harasim, Noll, and Rohrmeier 2019).
This setup evokes three questions in the context of the present article.While the first two questions are approached in Section 4, the third one is answered below.
(1) What is the meaning of the Douthett graph when all seventh chords are replaced by their 8-note complements?(2) Does the distinction between inner-diatonic contiguities and alterations have any musical meaning in the complementary world as well?(3) How do the constructions of the Douthett and the betweenness graph relate to Power Towers? Surprisingly, Power Towers can be obtained through a general construction applied to the family of second-order 11,4 .In this family, the generic seventh chords are the images J n 11,4 (Z 4 ) ⊂ Z 11 .They have the generic interval cycle (3, 3, 3, 2), which is analogous to the cycle (2, 2, 2, 1) in Z 7 before.The J-function J m 12,11 yields four different specifications of this cycle, namely (4, 3, 3, 2) (dominant-seventh chords), (3, 4, 3, 2) (minor-seventh chords), (3, 3, 4, 2) (half-diminished chords), and (3, 3, 3, 3) (diminished-seventh chords).Hence, CM 12,11,4 coincides with the 39 chords of Power Towers.In other words, the seventh chords are second-order Clough Myerson scales in a less familiar setup.Figure 6 shows the associated Douthett graph and illustrates its construction starting from the inner-"diatonic" contiguities of a single 11-note scale.In the second and the third step, all inner-"diatonic" contiguities and all alterations are shown, respectively.Their union constitutes the complete Douthett graph in the final step.Note that all edges that connect diminished chords with dominant-seventh or halfdiminished-seventh chords represent both inner-"diatonic" contiguities and chromatic alterations at the same time.This results from the symmetry of diminished chords and does not happen for the family CM 12,7,4 .7   On a final note for this section, the 36 diatonic triads (12 major, 12 minor, 12 diminished) coincide with the family CM 12,7,3 (Harasim, Noll, and Rohrmeier 2019) in the same way as the diatonic seventh-chords coincide with CM 12,7,4 .Furthermore, the Cube Dance graph (28 triads; 12 major, 12 minor, and 4 augmented) from Douthett and Steinbach (1998) coincides with the betweenness graph of the family CM 12,11,3 .In contrast to Power Towers, the Cube Dance graph also coincides with the betweenness graph of the family of second-order maximally even sets of configuration (c, d, e) = (12, 10, 3).This is interesting because the generic interval pattern (3, 3, 4) over Z 10 represents the diatonic pattern (2, 2, 3) over Z 7 more faithfully than the interval pattern (4, 4, 3) over Z 11 does.Although 12 and 10 are not coprime, the parsimony arises in the case of CM 12,11,3 because none of the involved triads (major, minor, or augmented) contains a tritone.In contrast, the family {J m,n 12,10,4 (Z 4 ) | m ∈ Z 12 , n ∈ Z 10 } contains the 6 French augmentedsixth chords and the 3 diminished-seventh chords.They contain tritones and their voice-leading connections are not parsimonious.

Complementary Douthett graphs and the voice leading among Barry Harris scales
In jazz theory, and in particular in the theoretical approaches to Bebop, some 8-note scales gained particular interest.They play a central role in the teaching of the jazz pianist Barry Harris.Following García-Valdecasas Vaticón (2020, pp. 118-121), we distinguish four types of Barry Harris scales as shown in the upper four staffs of Figure 8.
The first three of these scales are the complements of the minor-seventh chord Em 7 , the dominant-seventh chord F 7 , and the half-diminished-seventh chord D m 7 5 , respectively.Together with the three octatonic scales, the transpositions of these scales thus form precisely the complements of the Power Tower graph.The fourth scale, the 7th 5 diminished scale, is the complement of the French augmented sixth chord {E , G, A, C } and sticks out from the family of complements of diatonic seventh chords.In view of the theoretical challenge to understand the connections among the complements, we put this fourth scale aside for this paper.Another scale is included instead, the complement of major-seventh chords shown in the fifth staff of Figure 8.Despite of its rather bumpy step-interval pattern, it shares a characteristic feature of the other three scales.As any other 8-note scale {x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 }, it is tiled into two disjoint chains of thirds (mod 8), namely the evenly numbered and the oddly numbered notes {x 0 , x 2 , x 4 , x 6 } and {x 1 , x 3 , x 5 , x 7 }.In the case of the complement of major-seventh The investigation of this section is driven by the following question: Since the Douthett graph of the 48 diatonic seventh chords carries music-theoretical meaning, do its diatonic and chromatic pathways also have a musically relevant counterpart once seventh chords are replaced by their complements?In particular, we inspect: (1) the cycle of complements of a diatonic thirds-cycle (2) the cycle of complements of the long chromatic alteration cycle Closer inspection reveals a surprising complementarity between the voice-leading behavior of the original seventh chords on the one hand and their complements on the other.Power Towers as a betweenness graph is a subgraph of the chromatic saturation of the Douthett graph of the diatonic seventh chords (Harasim, Noll, and Rohrmeier 2019).Thus, up to occurrences of some diminished-seventh chords we may still interpret the voice leading among the respective third-chain tiles of the Barry Harris scales with respect to the distinction between diatonic voice leadings and chromatic alterations.While in the diatonic-thirds cycle, the voice leading proceeds in single-voiced diatonic steps, we find that the tiles of the complements proceed along three fifth-related local chunks of the long alteration cycle.After B 7 − B m 7 − B m 7 5 we reach the diminished-seventh chord B •7 (instead of AM 7 ) as the switch point to the next chunk along the long cycle: E 7 − E m 7 − E m 7 5 .The diminished chord E •7 then serves as the switch point to the third chunk A 7 − A m 7 − A m 7 5 .Via A •7 = F •7 , we reach back to B 7 .This also shows that the tiles together form one cycle of length 14 (see upper half of Figure 9).
An analogous complementarity can be observed between the long alteration cycle of all 48 diatonic seventh chords and the cycle of their complements.While the former chords proceed by single-voiced alterations, we find that the third-chain tiles of their complements proceed along local diatonic pathways.In this case the tiles do not form one cycle but two.The only chords occurring in both cycles are the diminished seventh chords.They serve as switch points between whole-tone-related chunks like B 7 − Gm 7 − Em 7 5 (see lower half of Figure 9).
In sum, the chromatic complement mapping translates alteration sequences into generalized diatonic sequences and vice versa when applied to diatonic seventh chords and Barry Harris scales.This is a remarkable property of the harmonic pitch space that leaves plenty of open questions for future research.We conclude this section by suggesting an ad hoc explanation of this property.In the discussion, we refer to Figure 10 which displays the content of Figure 9 in a different way.
To begin with, we mention that there is a different type of voice leading between the complements of the white note seventh chords, which is also minimal.It leaves the five black notes fixed and connects those white-note triads that are the respective C-major-scale complements to the given white-note seventh chords.But as we can see in Figure 10, none of these white-note triads is completely contained in one of the tiling seventh-chords.(They always contain either 2 orange and 1 blue note, or vice versa.)So instead of having D (blue) moving to C (orange) between layers 7 and 6 we have D D (both blue) and C C (both orange) as indicated by the two little arrows connecting layers 7 and 6.How can we know that these are alterations?To answer this question, we need to inspect the 7-generic interval cycles of the tiling seventh chords.Note that their 8-generic interval patterns (2, 2, 2, 2) have no beginnings nor ends.But diatonic voice leadings can be localized with respect to the 7-generic interval pattern (2, 2, 2, 1) as an exchange of the root and the top note, while alterations take place inside of the chain of thirds.
The closing diatonic step intervals (from top notes to roots) of the tiles are indicated by orange and blue dotted arcs in Figure 10.In the upper subfigure there is no case, where the voice leading connects a top voice with the root of the following chord (for the diminished seventh chords we could choose the root to follow through with this).By contrast, in the lower subfigure the voice leading always connects the top note of one seventh chord with the root of the next.
An analogous investigation can be made for the 9-note complements of diatonic triads.We find that the relation between the triadic covering of the diatonic triads' complements in terms of the the Cube Dance triads replicate our finding about the relationship between the seventhchord coverings of the diatonic-seventh chords' complements in terms of the the power-tower seventh chords.Tymoczko (2016) provides instructive musical examples in Shostakowich and Stravinsky with complements of major, minor and augmented chords (i.e. the complements of the Cube Dance graph).
For a future study this observation suggests a systematic investigation of modal tilings of a (generalized) chromatic scale of cardinality n = k • m into k tiles of cardinality m, where one of the tiles is a second-order Clough-Myerson mode, while the remaining ones form its complement and hypothetically covered by the nodes of a generalized Cube-Dance/Power-Towers graph.

Voice-leading distances between chords of different cardinality
The metric of minimal voice leading presented in Section 2 considers equally sized sets of tones only.Hence, it is also called bijective voice-leading distance.In this section, we propose and study concepts of distance between any two subsets of Z n .

Relational voice leadings
So far, we have regarded voice leadings as correspondences or mappings between two sets.However, we can also define a more general class which allows relations between sets, with certain constraints.For each of such voice leadings, we can define the concept of distance analogously to the bijective case, and examine the values and relations which minimize it among a collection determined by two chords, regardless of their cardinality.
Definition 5.1 Let Rel(X , Y ) denote the set of all relations (called relational voice leadings) between X ⊆ Z n (as the origin set) and Y ⊆ Z n (as the destination set) that are left-total, righttotal, and free from redundant pairs.A redundant pair is defined as a pair (x, x) ∈ R that can be removed without violating the totality of R. The (minimal) relational voice-leading distance is defined as where P(Z n ) denotes the power set of Z n .A minimizing relation is also called a8 minimal relational voice leading from X to Y.In general, it is possible to determine a relational voice leading in polynomial time (Tymoczko 2006).Two additional distance measures are defined using injections and surjections.Let |} denote the sets of chord pairs in which the first chord has not more (not less) notes than the second.In addition, let Inj(X , Y ) and Sur(X , Y ) denote the sets of injective and surjective functions f : X → Y , respectively.The (minimal) injective voice-leading distance and the (minimal) surjective voice-leading distance are defined as: Since all distances are defined via minimization, the word "minimal" is commonly omitted in what follows.
In contrast to the injective and the surjective distances, which are studied later in this section, the relational voice-leading distance is not a metric, see also Genuys (2019).It is indeed positive definite and symmetric, but the triangle inequality is violated, as the counterexample X = {0}, Y = {6}, and Z = {5, 7} shows.For each pair, there exists only one relation, R(X , Y ) = {{(0, 6)}}, R(Y , Z) = {{(6, 5), (6, 7)}}, and R(X , Z) = {{(0, 5), (0, 7)}}.Therefore, Despite this drawback, it is worthwhile to study these distances in the light of the importance of split and join voice leadings (Callender 1998).For example, Figure 11 shows a network of whole-tone, acoustic, and octatonic scales with minimal split or join voice leadings of distance D RVL (X , Y ) = 2. Definition 5.2 The connected components of a given relation R ∈ Rel(X , Y ) are defined as the edge sets of the connected components of the (undirected) bipartite graph induced by R with the disjoint union of X and Y as vertex set.Among all possible connected components of R, we distinguish three elementary types of simple components: (1) A connected component that consists of a single pair (x, y) ∈ R is called a link.In this case, the word link is also used to refer to the pair directly.(2) A connected component that is a set of distinct pairs {(x, y 1 ), . . ., (x, y k )} ⊆ R for k > 1 is called a split with source x and targets y 1 , . . ., called a join with sources x 1 , . . ., x k and target y.
A relational voice leading is called reduced if it only consists of simple components.
Proposition 5.3 Every relational voice leading contains a reduced one, (i.e. a voice leading in which each connected component is a link, a split, or a join).
Proof If a connected component of R contains three pairs (x 1 , y 1 ), (x 2 , y 2 ), and (x 2 , y 1 ), then one of them, namely (x 2 , y 1 ), can be removed without violating the right-and left-totality of R.This operation can be repeated until there are no more such triples of pairs.
From Proposition 5.3, it follows directly that every minimal relational voice leading is reduced, because redundant links are excluded from all voice leadings by definition.The following definition classifies the notes connected in a relational voice leading into four classes, which are used in the next section to define a subset of relational voice leadings that are invariant under complementation.
Definition 5.4 For a reduced relational voice leading R ∈ Rel(X , Y ), denote the sets of origins x and of destinations y of links (x, y) ∈ R by R X and R Y , respectively.Let further R ⊂ X ∪ Y denote the root set, which consists of R's split sources and join targets, and let R denote the leaf set, which consists of R's split targets and join sources.
Proof Essentially, the following three equalities allow us to redistribute the voice leading into three (or two, in the last case) relational voice leadings which contain the set of links from (R X , R Y ), joins, and splits, respectively.The new distributions are jointly total (they involve every required note) and the total distance remains at most D RVL (X , Y ), as the repetition of nonconstant edges is avoided.This is necessarily true, because links are included in the first addend, and the other two addends are either the remaining connections or extra constant pairs of the form (x, x) ∈ R × R (respectively, (x, x) ∈ R × R) -the third equality is just a reformulation of the first two), which groups together joins and splits.
Additionally, if a more optimal set of connections were to be found in any of the three factors, it would correspond to a collection of connected components from X → Y of the same kind (links, joins, or splits).In that case, such a sub-relation could replace the original in X → Y due to the totality of the combined new voice leadings, which would contradict its presupposed minimality.
Notice how the fact that R and R encompass elements from both X and Y simplifies the amount of addends in the expression.The third expression allows us to define intermediate chords between any two X, Y in a way that the relational distance between them can be expressed purely as a sum of exhaustive distances.

Relational voice leadings and complementation
A natural question to ask is whether it is possible to establish some equivalence between a minimal relational voice leading and a minimal counterpart on the complement sets, in a similar fashion to that of bijective voice leadings.However, surjective voice leadings rarely provide useful insights about their inner structure in general.For instance, the only isolated links which preserve distance in their complements are those of distance not greater than 2: the possibility to include an element of a set in more than one pair of the relations allows for links (x, x + k), with k > 2, to be optimized in the complement sets by including the pairs Before showing further limitations of complementation for relational voice leadings, let us consider the following example: fix the chords X = {1, 3, 5, 6, 8, 9, 11} and Y = {0, 2, 3, 5, 7, 9, 11}.Note that the distance determined by R = {(3, 3), (5, 5), (9, 9), (11,11), (1, 0), (1, 2), (6, 7), (8, 7)}.coincides with that between X c and Y c (determined by R * = {(4, 4), (10, 10), (0, 1), (2, 1), (7, 6), (7, 8)}).In this case, the connected components do not span more than a step interval, and they are perfectly mirrored in the complement relation.Hence, it is natural to introduce some notation as a means of simplifying further exploration.
Definition 5.6 Consider a relational voice leading R ∈ Rel(X , Y ) and one of its simple connected components C ⊆ R that is either a join or a split.The set of C's leaves C ∩ R is segmented into nonempty subsets Z i , each of which is contiguous on the pitch-class cycle and maximally large.We call these subsets the leaf piles of C. For a single leaf pile Z = {z, z + 1, . . ., z + k} ⊆ Z n , we denote any leaf in the closed interval [z It should come as no surprise that symmetrical distributions between sets and their complements are paramount for preservation of minimal voice leadings.In fact, these reflections act on individual components which are contained in usually disjoint intervals where complement voice leadings are constructed similarly.This is illustrated in Figure 12, in particular by examples where "shortcuts" disallow such preservation.
Nevertheless, certain kinds of relational voice leadings that satisfy the restrictions discussed so far are invariant under complementation.Denote by m 1 and m 2 the amount of notes adjacent to the outermost leaf piles {z 1 , . . ., z 1 + k 1 } and {z 2 , . . ., z 2 + k 2 }, respectively.That is: In Figure 12, Cases 1 and 3 prevent outer shortcuts as a result of m 1 and m 2 being large enough.Definition 5.7 A minimal relational voice leading is in Callender form if each connected component is either a link contained in a step-link cluster, a balanced join, or a balanced split.A step link is a link (x, y) ∈ R such that the distance between x and y is at most two (generalized) semitones, d(x, y) ≤ 2. Step-link clusters are collections (possibly singletons) of intertwined step links spanning no more than two semitones.Balanced splits and balanced joins are defined analogously.A balanced split has either one or two leaf piles.If it has one leaf pile Z 1 , then the root of the split must be central to Z 1 , and it must not have surrounding notes, i.e. m 1 = m 2 = 0.If otherwise it has two leaf piles  m 1 , k 1 , k 2 , m 2 ) = (1, 1, 1, 1), whereas in case 3 (m 1 , k 1 , k 2 , m 2 ) = (2, 2, 1, 1).one element (i.e.z 1 + k 1 + 2 ≡ z 2 mod n), surrounding notes must satisfy m 1 ≥ k 1 , m 2 ≥ k 2 , and at least one of the following statements must hold, where x denotes the root of the join: • The root is the gap between Z 1 and Z 2 (x ≡ z 1 + k 1 + 1 mod n) • The first leaf pile Z 1 is exactly one element larger than Z 2 and the root is the last element of Z 1 (k 1 = k 2 + 1 and x ≡ z 1 + k 1 mod n).• The second leaf pile Z 2 is exactly one element larger than Z 1 and the root is the first element of Z 2 (k 1 + 1 = k 2 and x = z 2 ).
Even if smaller links do not cross longer ones, they can still overlap (such as {(0, 1), (1, 3)} or {(0, 2), (2, 4)}), and thus not preserve relational distances over complementation.The additional restriction in Callender voice leadings dismisses this possibility and gives rise to the following lemma, paramount to the central result of this subsection.
Lemma 5.8 Every voice leading in Callender form can be decomposed into non-overlapping segments of Z n which contain a single step-link cluster, balanced split, or balanced join.

Proof
Step-link clusters are, by definition, contained individually in segments of length no greater than 2. Consider, without loss of generality, a balanced split R 0 spanning an interval [a, b] ∈ Z n , with R 0 = {r 0 } and a note z 1 ∈ [a, b] from another component R 1 .Since R 0 and R 1 are not connected, whenever z 1 ∈ R 0 , the edge (r 0 , z 1 ) ∈ R 0 can be replaced by a distance 0 pair as a component on its own or as an additional edge in R 1 (depending on its typology).This is well defined as r 0 is still covered, but it leads to a more optimal voice leading and thus contradicts the minimality of Callender voice leadings.
Otherwise, if z 1 = r 0 and z 1 / ∈ R 0 , the split must have two leaf piles with {r 0 } = R 0 ⊂ R 0 (i.e. the second or third case).Regardless of the distribution, z 1 fills the "empty space" of the split, and a contradiction is reached as follows.If z 1 is an element of the voice leading's destination set Y, then there are no longer two distinct leaf piles.Otherwise, if z 1 ∈ X then it is closer than r 0 to one of R 0 's leaves.
Proposition 5.9 If there is a minimal relational voice leading from X to Y in Callender form, then their distance is invariant under complementation, Proof Let R be a minimal relational voice leading between X and Y.We start by noticing that all minimal relational voice leadings between any sets A and B can be constructed in three steps, by (1) connecting each element of A to one of its closest elements in B, (2) connecting each element of B to one of its closest elements in A, and (3) removing all redundant constant links.The proof examines this construction for a voice leading between the complements of X and Y by considering each of the segments from the previous lemma individually and showing that the total distance of the so constructed voice leading equals the distance of R.
We already discussed step links and larger ones, except for the cluster {(z ∓ 1, z), (z, z ± 1)}, which preserves distance on the complement as well.In what follows, we break down the other cases, assuming without loss of generality that the component is a split with root x.
If the considered split has a single leaf pile Z 1 , the root x being central is a sufficient condition for invariance because the relation covers the considered part of (R) c in an optimal way, contributing the same distance as the original voice leading (as seen in Figure 12.2).If it otherwise has two leaf piles, the following voice leadings, respective to the three cases in Definition 5.7, preserve the inner structure and ensure an equal distance: , implementing the double reflection shown in Figure 12.3.
Knowing that complementation behaves well when there are no outer notes involved, it remains to be proven that the remaining conditions are sufficient to ensure the non-existence of shortcuts derived from said outer notes.For a single leaf pile Z 1 , finding a shorter connection (see Figure 12.4) would mean that one of the two notes adjacent to the root x is closer to the opposite exterior of the leaf pile than to the direct exterior.That is, Since we consider a local region of Z n , this can without loss of generality be rewritten as: which contradicts that x is centered.Otherwise, if the split comprises two different leaf piles, it is only necessary to check whether the outermost notes are closer to the gap between the piles because every other non-root leaf (i.e.value in R ∩ (R) c ) is even closer to the gap.Thus, a shortcut takes place when one of the following inequalities holds: As above, this is But these contradict the hypotheses k i ≤ m i , with i ∈ {1, 2}.Now recall the example from the beginning of the subsection, i.e. the pair of tritone-related acoustic scales (also known as instances of melodic minor) X = {1, 3, 5, 6, 8, 9, 11} and Y = {0, 2, 3, 5, 7, 9, 11} with R = {(3, 3), (5, 5), (9, 9), (11,11), (1, 0), (1, 2), (6, 7), (8, 7)}.
In fact, the balanced and spread-out distribution of the notes within each scale in Figure 11 facilitates checking the existence of Callender voice leadings between each of the neighboring modes, hence the network of the complements (whole-tone scales, seventh-ninth chords, and   2013) is closely related in that it is constituted solely by acoustic and diatonic scales, which correspond to the Yoruba bell pattern.
diminished chords) preserves the distances.It is possible to expand this new-found graph by considering all of the chords in Power Towers and linking them to the closest seventh-ninth chords, in a way that the original central row is replaced by a plane with the Power Towers graph.As Figure 13 shows,9 new connections are established, such as D9 → Cdim7 → (F9 B9 A 9), D9 Dm7 → B 9, D9 → F m7dim5 or D9 → D7.All of them have distance 2 (the last pair, in particular, being inclusions), which along with the number of added connections reinforces the structural significance of diminished seventh chords, and of minor seventh chords to a lesser degree.
In any case, the condition of Callender voice leading is not a complete characterization of relations which preserve distance under complementation: we are still leaving aside cases like {(0, 1), (0, 2), (5, 3), (5, 4)} or {(1, 1), (1, 3), (5, 4), (5, 5), (5, 6), (9, 7), (9, 9)} ⊆ Z 12 × Z 12 , or other layouts regarding multiple-pile splits with particular distributions.Moreover, the existence of Callender voice leadings between sets is not guaranteed by their complements admitting relational voice leadings in Callender form.However, by taking k 1 = k 2 = 0, the conditions of Callender voice leadings for components with two piles generalize the requirement of central roots featured previously in single leaf pile distributions; unlike the latter, they also allow for components to be strictly contiguous while preserving their Callender condition.Colannino, Gómez, and Toussaint (2009) suggest a generalization of the swap distance to allow a comparison between rhythms of different cardinality, the directed swap distance.

Unidirectional distances (or functional distances)
Let P have more onsets than Q.The directed-swap distance is the minimum number of swaps required to convert P to Q with the constraints that each onset in P must move to some occupied position of Q, and all occupied positions of Q must receive at least one onset from P.
The definition immediately prompts an extended notion of the concept of swap, which now may comprise displacements that "merge" two voices and leave an open spot for an additional 0 to be added, i.e. (a 1 , . . ., 1, 1, . . ., a n ) → (a 1 , . . ., 0, 1, . . ., a n ).This less restrictive version of standard swaps in 2.7 may be denoted as directed swaps.We are now able to explore the distances associated with this type of transformation and relate them -by virtue of Proposition 2.8 -to the aforementioned injective and surjective voice-leading distances to handle sets of different cardinality by means of swaps.
, be sets of possibly different cardinality.The injective distance D ≤ (X , Y ) (see Definition 5.1) coincides with the least amount of directed swaps needed to map X into a subset of Y with cardinality m 1 .Analogously, the surjective distance D ≥ (Y , X ) coincides with the directed-swap distance, in Colannino, Gómez, and Toussaint (2009)), namely the minimum amount of directed swaps which, upon composition, give rise to an exhaustive mapping Y → X .
In general, D ≤ and D ≥ are positive-definite and satisfy the triangle inequality, but the lack of symmetry prevents them from being metrics.One could also suggest a two-part expression of the form D , 2, 5} invalidates the proposal.Intuitively, the (bijective) minimal voice leading between X and a subset of Y establishes some constraints regarding injectivity which are not needed when computing the surjective distance, where the remaining notes are covered by other voices: in this case, the the minimal leading 3 → 5 defined by the bijection X → Y ⊆ Y can be replaced by 3 → 2, since 8 → 5 is the only optimal shift and already handles the requirements of surjectivity.Instead, we may write: Lemma 5.11 For every voice leading Y → X which defines the surjective distance, a subset of Y is induced a bijection with X: this allows for an alternative expression of D ≥ as Similarly, the irregularities of surjective distances can also be spotted while extending sets, adding another contrast with the proper behaviour of injective distances: given any sets Additional notes in the destination set Y 1 act as references to which X 1 can be linked more optimally, whereas adding notes to X 1 necessarily contributes to a more costly voice leading.However, dealing with surjective relations, expanding the origin or the destination sets can lead to more or less optimal distributions: {0, 2} ∪ {6} → {0, 6} or {0, 2} ∪ {11} → {0, 6}, and On the other hand, the inclusion inequalities can provide some insight regarding optimal chord connections, such as the completion of a half cadence I6 4 →V as I6 4 →V 7 ↔V, where We can reinterpret this connection in terms of rhythm as well by considering their complements thought of as pulse patterns, which induce different metric configurations: χ X c 1 features a beat the first of every three pulses 3 8 , whileregarding subdivisions of 4 notes 4 8 -this only happens at the beginning of the pattern; χ X c 3 sets out a radically different distribution, including every 4-beat accent and only the first and the last of the 3-beat ones.But the inclusion of χ X c 2 can serve as a bridging element between the two previous sequences, since (despite having the same count of accents as χ X c 3 ) the last 3-note accent now reinforces the halfway subdivision 6 8 , somewhat more influential to the quaternary metric configuration than the previous case: Proof Consider the characteristic functions χ X , χ Y , for which there exists a sequence of functions swap k j with j ∈ {1, . . .
for every a i ∈ Z n (as defined in 2.7).This translates the notion of a subset into the realm of characteristic functions.Trivially, , so s induces an equinumerous sequence of swaps which act on the complementary sets in a way that Knowing that these swaps find a voice-leading counterpart in An analogous argument for the complement yields the reciprocal inequality D ≤ (X , Y ) ≤ D ≤ (Y c , X c ), hence proving the proposition.
One could also proceed by means of a proof similar to that in Harasim, Schmidt, and Rohrmeier (2016), from the viewpoint of voice-leading distances: every minimal bijection can be reshaped to obtain another one which preserves minimality and moreover acts as the identity on the intersection of the sets.This ensures , which implicitly defines a bijective voice leading that, in turn, is an injective funcion Y c ⊂ Y c 0 → X c with at most the same weight.Analogously, the inequality D ≤ (X , Y ) ≤ D ≤ (Y , X ) is obtained and completes the proof. 10evertheless, a key counterexample disproves the validity of an equivalent proposition on complementation for the surjective distance.Recall (from Subsection 2.3) that the symbols B i (i = 0, . . ., 11) denote the Yoruba patterns (the rhythmic analogues of the diatonic scales), and the symbols V i (i = 0, . . ., 11) Reich's Clapping Music patterns.Then we find D ≥ (V i , B i ) = 1 for every i ∈ {0, . . ., 11}, but D ≥ (B c i , V c i ) = 2, due to the "density" of the pattern.This coincides with the previous observations regarding relational voice leadings, which involve several features from surjectivity.
Corollary 5.13 Determining the injective distances between the transpositions of the Yoruba bell pattern [x.x.xx.x.xx.] and those of the Clapping Music pattern [xxx.xx.x.xx.] is equivalent to computing the injective distances from minor seventh chords [x..x. . . x..x.] to major pentatonic scales [x.x.x..x.x..], up to adjusting transpositions.
Further investigation through the perspective of Fourier analysis will provide, in Section 6, additional insight about the relation between the Clapping Music and Yoruba bell patterns.Furthermore, the appendix contains related remarks regarding the analysis of distances between both patterns under transposition, as introduced in Colannino, Gómez, and Toussaint (2009) (see Case study: Yoruba Bell pattern and Reich's signature rhythm).

Inspecting rhythm embeddings and voice-leading graphs through the Fourier lens
While it took more than four decades to rouse the pioneering ideas in Lewin (1959) from academic slumber, there is now a considerable body of work on the application of finite Fourier transform to the study of pitch class sets, most notably Amiot (2016) and several articles by Jason Yust on applications to music analysis.At least three aspects of this approach are of special interest for the present article: (1) Recently, Yust (2021) offers an auspicious transfer from the pc-set related applications of Fourier analysis to the domain of rhythm particularly dedicated to the study of Steve Reich's signature rhythm, (i.e. the clapping pattern) and related rhythms.His paper offers a valuable new contribution to the driving questions in Colannino, Gómez, and Toussaint (2009).11(2) Based on insights from Amiot (2013), Yust (2018) studies the images of musical networks (Tonnetze, voice-leading graphs) in the tori, which are spanned by the phases of two (or more) phases of selected Fourier coefficients.
(3) The Fourier transforms of a pitch class set and its complement are very closely related.
Except the 0th coefficient, the magnitudes of all Fourier coefficients are equal and all phases are mirror-symmetric, i.e. they differ only by a minus sign.

Definition 6.1 For any function
The values a k := f (k) are called the Fourier coefficients of f.
In the case of the Yoruba and the clapping pattern, n = 12 and the function f is a characteristic function taking the values 0 and 1 only.An instructive musical illustration of the summation formula for the Fourier coefficients in this concrete situation can be found in Yust (2021).The following table lists the first seven Fourier coefficients of the two characteristic  functions of the clapping pattern and the Yoruba pattern.The coefficients a 7 , . . ., a 11 are omitted as they are conjugate to a 5 , . . ., a 1 , respectively.The 0th coefficients are proportional to the number of beats in the rhythms.Leaving them aside, the 5th coefficients have the largest magnitudes for both rhythms, and notably, the phases of the coefficients a 3 , a 4 , and a 5 coincide.This is also illustrated in Figure 15. 13The Fourier coefficients of the other four 8-note rhythms, which contain the Yoruba pattern, are shown in Figure 16.Their comparison with the coefficients of the Yoruba pattern reveals that the phase incidences in Figure 15 constitute a special configuration.Fourier coefficients can be interpreted as measures of certain rhythmic qualities (Yust 2021).In this sense, the Yoruba and the clapping pattern share three rhythmic qualities.A high magnitude |a 4 | indicates that the rhythm is in relatively good solidarity with precisely one of the three meters of four beats.
Analogously, a high magnitude |a 3 | indicates that the rhythm is in relatively good solidarity with precisely one of four meters of three beats, two of them shown below.
Yust does such comparisons regarding the magnitudes.As shown in Figure 15, it is also insightful to take the phases into account.The clapping pattern has indeed a slightly weaker 5th coefficient and slightly stronger 3rd and 4th coefficients than the Yoruba pattern, but despite of these differences in magnitude, the respective phases coincide precisely.This means that both rhythms fit best to the same three metric patterns.They only differ with regard to the respective strengths of these fits.According to the Fourier transform, the Yoruba pattern is therefore more similar to the clapping pattern than to the other four 8-note rhythms.
A very practical application of the Fourier-approach to the present paper is the geometrical realization of voice-leading graphs and their complements as tori of phases.In Figures 4, 5 By looking at these graphs we gain the basic insight that they can be embedded into Fourier space without edge crossings.Hence, these graphs are toroidal.Although the Fourier approach as such does not prescribe a concrete realisation of the abstract edges, it seems natural to realise them as geodesics on the torus of phases.
The upper part of Figure 18 shows the same two graphs again with the transpositions of the fume fume pattern 2-2-3-2-3 (left side) and the Yoruba pattern (right side) added.In the lower part of this figure, the same two graphs are shown on the torus of phases φ 4 and φ 5 .
In this section we only touched upon the connections between the voice-leading graphs and the Fourier approach.There are auspicious perspectives for further investigations.For example, symmetries of the set of nodes of a given graph in a torus of phases, may give rise to transformations of the associated set of paths.A vertical mirroring of the Power Towers graph yields a new type of voice-leading pathway: it is symmetric to -but different from -the complementary alteration cycle (compare the directions of the diagonal pathways in the lower subfigure in 10 with those in Figure 19).

Summary and conclusion
This paper assembles and explores the interdependence of a number of music-theoretical concepts around the set-complement mapping, voice-leading distances, and across the pitch-rhythm correspondence.The motivating aha-moment is depicted in Figures 1 and 3: a voice-leading graph of seventh-chords from Douthett and Steinbach (1998) almost coincides with a "phylogenetic" graph of 8-note rhythms from Colannino, Gómez, and Toussaint (2009).Apart from examplifying the pitch-rhythm analogy, this coincidence is caused by the invariance of voiceleading distances under complementation.To explicate the relation between the two graphs in a more general setting, we revisited findings from Harasim, Schmidt, and Rohrmeier (2016) in Section 2 and established a connection to the (undirected) swap distance in Colannino, Gómez, and Toussaint 2009.As it turned out, the observation that motivated this research is an example of a much richer and more far reaching mathematical theory that also has the potential to systematize classes of scales used in jazz improvisations.
In Section 3, we revisited and compared two approaches to musical geometry, the connectivity of 2nd-order Clough-Myerson scales described in Douthett graphs, on the one hand, and their voice-leading connectivity described in betweenness graphs (Harasim, Noll, and Rohrmeier 2019)  In Section 4, we reported explorations of interpreting the diatonic-seventh-chord complements not as rhythmic patterns but as scales of pitches.These scales and their complements can be thought of intuitively as analogous to Maurits Cornelis Escher's drawings on the basis of tessellations of the plane.Concretely, the Douthett graph CM 12,7,4 of diatonic seventh chords is a prominent and well-understood music-theoretical structure.The associated graph of complements is isomorphic and its nodes are the Barry Harris scales (García-Valdecasas Vaticón 2020), which consist of 8 notes each and are prominent in jazz theory.Since there is yet, to the best of our knowledge, no mathematical theory for the structural properties of these scales, the aim of our explorations is to study how the complement mapping translates voice-leading properties of seventh chords to Barry Harris scales.We focused on the complement mapping acting on two voice-leading pathways, a diatonic cycle of thirds and a part of the alteration cycle.Surprisingly, we found that the complement mapping transforms diatonic relations into alteration relations and vice versa.In addition to our selected examples, readers are invited to explore other pathways in this manner, as suggested in Figure 19, and the precise description of voice-leading geometries of Barry Harris scales constitutes a promising topic for future research.
Section 5 extended the investigations of voice-leading distances and their behavior under complementation to a wider domain of relational voice leadings.In particular, such voice leadings can split and merge voices and are thus able to connect two chords of different size.condition for complementational invariance of relational voice-leading distances.In particular, this preservation always occurs when dealing with injective distances.
Finally, the discrete Fourier transform applied to chords, rhythms, and their complements constitutes a productive approach supplementary to the other methods of the paper, as described in Section 6.The motivation for this consideration was at least threefold: (1) the exploration of the pitch-rhythm analogy under the Fourier perspective by Yust (2021), (2) the invariance of most Fourier magnitudes under complementation (except for a 0 ) and the mirror symmetry for the Fourier phases, and (3) the practical aspect of showing Douthett graphs, betweenness graphs, and other graphs within torii of phases.
The results described in this paper are merely theoretical and could provide a basis for potential transdisciplinary work in theory and cognition.The rejection of the swap distance as a measure of musical similarity by the authors Colannino, Gómez, and Toussaint (2009) poses the challenge to consider other musical interpretations of this distance concept and to examine them alongside the pitch-rhythm correspondence.And it asks for alternative approaches to the theorization of scalar or rhythmic similarity, which will necessarily involve a discussion around how distributing mappings between chords or patterns in a different amount of voices shapes the perception of distance (and, subsequently, around the validity of certain metrics in results which take into account the psychological sensing of such transformations).In this sense, our research exemplifies the relevance of the metric D 1 VL , as it allows for strong results by bypassing distribution incongruities.
With regard to the first challenge, we should remember that Douthett's own interest in the consideration of the simple and the embedded J-functions J m c,d and J m,n c,d,e was the dynamics of chord progressions in connection with voice leading (Douthett 2008).Proximity between two chords means in this context, that they have a potential to follow in succession, despite (or even because) of their differences in character and tonal function.We are optimistic that this type of musical contiguity principle remains effective when we pass from a Douthett graph of chords to the graph of their complements or that of their rhythmic analogues.It remains an open task, for future musical experiments to substantiate this claim.
With regard to the second challenge, our observations in Section 6 (Table 1 and Figures 15 & 16) suggest that phase coincidences between Fourier coefficients with comparatively high magnitudes reflect similarity with respect to certain rhythmic or harmonic qualities.A promising basis for such empirical work is the recent paper Wen and Krumhansl 2019 which establishes pitchclass and beat-class profiles experimentally for the diatonic modes and the analogous rhythmic modal varieties of the bell pattern, respectively.It would be interesting to inspect their results in a future study using the discrete Fourier transform.

Figure 2 .
Figure 2. Illustration of the situations in Lemma 2.2 (left side) and Lemma 2.3 (right side).

Definition 2. 5
Minimal voice leadings of distance D VL (X , Y ) = 1 are called parsimonious.In such a voice leading, one single voice moves by a single chromatic step.Corollary 2.6 Each minimal crossing-free voice leading of distance D VL (X , Y ) = k can be decomposed into precisely k parsimonious voice leadings.

Figure 5 .
Figure 5. Betweenness graph of the 48 diatonic seventh chords with respect to the minimal-voice-leading distance.

Figure 7 .
Figure 7.The betweenness graph of the family CM 12,11,4 coincides with the Power Towers graph.

Figure 8 .
Figure 8. Top: Barry Harris scales according toGarcía-Valdecasas Vaticón (2020).Bottom: 8-note scale complementary to major-seventh chords (not included in the "original" Barry Harris scales).All scales are partitioned into two seventh chord as indicated by the note heads.

Figure 9
Figure 9 displays the voice leading between the Barry Harris scales of type (1-3) and the alien type (5) along these two pathways.The upper half of the figure shows the complements of the diatonic-thirds cycle, and the lower half the complements of the chromatic alteration cycle.The three systems of the respective staff notations show the original diatonic seventh chords on top and the associated complements in the two systems below.Each 8-note complement is split in the tiling seventh chords, which are shown on top of each other.The respective trajectories are shown in the Power Towers betweenness graph.Closer inspection reveals a surprising complementarity between the voice-leading behavior of the original seventh chords on the one hand and their complements on the other.Power Towers as a betweenness graph is a subgraph of the chromatic saturation of the Douthett graph of the diatonic seventh chords(Harasim, Noll, and Rohrmeier 2019).Thus, up to occurrences of some diminished-seventh chords we may still interpret the voice leading among the respective third-chain tiles of the Barry Harris scales with respect to the distinction between diatonic voice leadings and chromatic alterations.

Figure 9 .
Figure 9. Voice-leading progressions of diatonic seventh chords and their complementary Barry Harris scales through the Douthett graph.Top: diatonic chord cycle of C major.Bottom: part of the alteration cycle.

Figure 10 .
Figure 10.Alternative representation of the voice-leading progressions in Figure 9.Each horizontal layer corresponds to a Barry Harris scale (including the exotic types in layers 0 ( = 7) and 2).The voice leadings between adjacent scales are indicated by arrows.The 4-note tiles of each scale are colored in blue and orange, respectively.The chord names are anchored at the roots and the dotted arcs indicate the position of the closing interval between top note and root.

Figure 11 .
Figure 11.Split and join voice leadings with a relational voice-leading distance of 2 in pitch-class space.From top to bottom, each edge represents a split voice leading (upper half of the figure) or a join voice leading (lower half of the figure).

Figure
Figure 13. Figure 11's smooth transition to the complements makes it possible to extend the network with the Power Towers graph.The family CM 12,18,7 in Yust (2013) is closely related in that it is constituted solely by acoustic and diatonic scales, which correspond to the Yoruba bell pattern.
Figure 13. Figure 11's smooth transition to the complements makes it possible to extend the network with the Power Towers graph.The family CM 12,18,7 in Yust (2013) is closely related in that it is constituted solely by acoustic and diatonic scales, which correspond to the Yoruba bell pattern.

Figure 14 .
Figure 14.Smoothing the metric feel change: the layout of the beats in χ X c 1 induces a layout for the rhythmic pattern

Figure 17 .
Figure 17.Betweenness-graphs of the seventh chords (left) and their complements (right) on the torus of Fourier phases 4 and 3.
, 6, and 7, the nodes are represented on the torus 2π • R/Z × 2π • R/Z through the phases φ 4 and φ 3 of the Fourier coefficients a 4 = |a 4 | exp(iφ 4 ) and a 3 = |a 3 | exp(iφ 3 ) of the characteristic functions of the chords associated with these nodes.While these figures are drawn on a plain fundamental domain for the torus, we show on the left side of Figure 17 the same betweenness graph as in figure 5. Taking advantage of the invariance of these two coefficients under chordcomplementation (up to the sign of the phases), we show on the right side of Figure 17 the associated graph of complements.

Figure 18 .
Figure 18.Top: The same torii as shown in Figure 17 together with transpositions of the fume fume rhythm (left) and the Yoruba pattern (right).Bottom: the same graphs as at the top but in the space of Fourier phases 4 and 5.The phase incidences of the clapping and Yoruba patterns imply that their nodes on the tori of phases (4 and 3 on top right, 4 and 5 on the bottom right) coincide.The same is true for the fume fume rhythm (the pentatonic scale in pitch space) and the associated 4-note rhythms on the left side.
on the other hand.On the basis of examples, we encouraged a systematic study of complementation alongside this comparison: we looked at parametrizations of families of complements in terms of suitably chosen 2nd-order Clough-Myerson scales CM c,d,e .An interesting example, which is worth examining in a future study, is Jason Yust's (2013) collection CM 12,18,7 = {J m,n 12,18,7 (Z 7 ) | m ∈ Z 12 , n ∈ Z 18 }, which unifies diatonic and acoustic scales within one Douthett graph.The family corresponding to CM 12,18,5 comprises their complements, which are pentatonic collections and dominant-ninth chords, and the associated Douthett graphs are isomorphic.
Figure 19.A vertical flip of the torus of phases 4 and 3 (i.e. a mirroring about the 3rd phase) maps the alteration cycle CM 7 → C m 7 5 → C m 7 → C 7 → C M 7 → • • • in the Douthett graph to the alternative long cycle CM 7 → C m 7 5 → Em 7 → G 7 → GM 7 → • • • (see upper musical system).And this cycle induces associated pathways along the tiles in the Power Towers graph (see lower two musical systems and the graph below).

Table 1 .
Fourier coefficients for the Clapping pattern and the Yoruba pattern.