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This is an Accepted Manuscript of an article published by Taylor & Francis in

Journal of Mathematics and Music on 10th July 2018, available online:

http://www.tandfonline.com/10.1080/17459737.2017.1395915

Exploring the space of perfectly balanced rhythms and scales

Andrew J. Milnea∗, David Bulgerb, and Steﬀen A. Herﬀa

aThe MARCS Institute for Brain, Behaviour and Development, Western Sydney University,

Penrith, Australia;

bDepartment of Statistics, Macquarie University, Sydney, Australia

(Received 16 December 2016)

Periodic scales and meters typically embody “organizational principles” – their pitches and

onset times are not randomly distributed, but structured by rules or constraints. Identifying

such principles is useful for understanding existing music and for generating novel music. In

this paper, we identify and discuss a novel organizational principle for scales and rhythms that

we feel is of both theoretical interest and of practical utility: perfect balance. When distributed

around the circle, perfectly balanced rhythms and scales have their “centre of gravity” at the

centre of the circle. The present paper serves as a repository of the theorems and deﬁnitions

crucial to perfect balance. It also further explores its mathematical ramiﬁcations by link-

ing the existing theorems to algebraic number theory and computational optimizations. On

the accompanying webpage, http://www.dynamictonality.com/perfect_balance_files/,

we provide audio samples of perfectly balanced rhythmic loops and microtonal scales, com-

putational routines, and video demonstrations of some of the concepts.

Keywords: perfect balance; evenness; scales; rhythm and meter; consonance and

dissonance

2010 Mathematics Subject Classiﬁcation : 00A65; 05B30; 11R18; 97N80; 90C25

2012 Computing Classiﬁcation Scheme: sound and music computing; user interface design;

convex optimization

1. Introduction

Musical scales, meters, grooves, ostinatos, and riﬀs are typically periodic in nature –

they repeat over a ﬁxed pitch interval like an octave, or a ﬁxed time interval like a

measure or a hypermeasure. Periodic patterns such as these can be naturally represented

by “weighted” points distributed on the unit circle: the angle of each point represents a

pitch class or time class; the weight of each point represents the multiplicity, loudness,

probability, or so forth, of the pitch class or time class.

In this representation, certain scales and rhythms exhibit the property of perfect bal-

ance: a perfectly balanced pattern is one whose points have their “centre of gravity” at

the centre of the circle (a real-world demonstration of this, using weights on the rim of

a bicycle wheel, can be viewed on the accompanying website (Milne 2017)).

In this paper, our focus is on patterns whose pitch classes or onset time classes are

binary-weighted; that is, their weight is either 1 or 0. Any such pattern might represent

∗Corresponding author. Email: a.milne@westernsydney.edu.au

(a) The diminished scale (clock-

wise from the top: C, D, E[, F,

F], G], A, B) is perfectly balanced,

non-primitive, and non-minimal. It

is non-primitive (rotationally sym-

metric) because its step pattern re-

peats every 3 semitones (quarter-

circle).

(b) The diminished scale repre-

sented over its fundamental period

of three semitones is now primitive

but it is not perfectly balanced.

(c) The double harmonic scale (e.g.,

C, D[, E, F, G, A[, B) is perfectly

balanced and primitive (hence it is

perfectly balanced over its funda-

mental period, the octave), but non-

minimal.

(d) The augmented triad (e.g., C,

E, A[) is – like all regular poly-

gons with prime numbers of vertices

–perfectly balanced,minimal, but

non-primitive.

(e) This 6-note binary pattern is

perfectly balanced and primitive

(hence it is perfectly balanced over

its fundamental period); it is also

minimal (it is not a sum of smaller

perfectly balanced patterns).

Figure 1. Scales that exemplify all possible combinations of being perfectly balanced or not perfectly balanced,

being primitive or non-primitive, being minimal or non-minimal. The diminished (octatonic) scale pictured in (a)

is non-primitive (rotationally symmetric), so perfectly balanced; but, over its fundamental period, the diminished

scale is not perfectly balanced, as seen in (b). A perfectly balanced scale need not be rotationally symmetric:

the double harmonic scale pictured in (c) is perfectly balanced but primitive (not rotationally symmetric); this

means it is perfectly balanced over its fundamental period. However, the double harmonic scale is non-minimal

– our later Figure 5(b) pictures the double harmonic scale as a union of smaller perfectly balanced scales. It is

trivial to create perfectly balanced scales/rhythms that are minimal and non-primitive – these are simply regular

polygons with prime numbers of vertices, such as the augmented triad shown in (d). Intriguingly, perfectly balanced

scales/rhythms that are both primitive and minimal are also possible, as shown in (e) (see Figure 6 for further

examples of minimal perfectly balanced primitive patterns).

a periodic musical scale or a periodic rhythmical stream. The simultaneous use of two

or more musical scales, or the sounding of two or more rhythmic streams, is represented

by the sum of such binary-weighted patterns.2

With respect to perfectly balanced patterns, our focus is on primitive patterns; these

are perfectly balanced patterns without rotational symmetry.3Patterns with rotational

symmetry (non-primitive patterns) are always perfectly balanced; for example, equal-

1The double harmonic scale, and its various modes, have a number of alternative names including the Hungarian

minor scale and the Gypsy minor scale; its Forte number is 7-22.

2The mathematical representation that allows for a sum of patterns to be understood in this way is provided

in Section 3.

3In earlier work, we used the term irreducibly periodic instead of primitive (Milne et al. 2015).

2

step scales like the chromatic and whole tone, as well as non-equal step scales like the

diminished (octatonic) and augmented (hexatonic), have rotational symmetry and so

are perfectly balanced, as in Figure 1(a). However, none of these patterns are perfectly

balanced over their fundamental period of repetition (the semitone, whole tone, minor

third, and major third, respectively), as in Figure 1(b). In recent work (Milne et al.

2015), we demonstrated the existence of perfectly balanced primitive patterns, as in

Figure 1(c), and methods to construct them. Such patterns are perfectly balanced over

their fundamental period (smallest possible period); something that is not generally true

for non-primitive patterns.

Most of these binary-weighted perfectly balanced patterns can be constructed by sum-

ming smaller binary-weighted perfectly balanced patterns. There are, however, perfectly

balanced patterns that cannot be constructed as sums of smaller perfectly balanced pat-

terns: these are minimal perfectly balanced patterns. Intriguingly some of these minimal

perfectly balanced patterns are also primitive, as in Figure 1(e). We are interested in

minimal perfectly balanced patterns because they form a useful set of building blocks

from which all other perfectly balanced patterns can be constructed: as exempliﬁed by

our free music software application XronoMorph (Milne et al. 2016).

The present paper further explores the mathematical ramiﬁcations of perfect balance

by linking the existing theorems to algebraic number theory and providing a number

of new theorems. In order to navigate the space of perfectly balanced patterns, we also

demonstrate computational methods for ﬁnding minimal perfectly balanced patterns and

searching the manifold of perfectly balanced microtonal scales for musically useful prop-

erties, such as perfectly balanced scales that maximize speciﬁc interval properties (e.g.

“tonal aﬃnity” where, as detailed in Section 6.2.1, we also introduce a novel spectral

entropy model of the overall tonal aﬃnity of a single pitch class set). We introduce a

useful parameterization of balance, realized in XronoMorph, that uses minimal perfectly

balanced patterns to generate complex and groovy rhythms. We ﬁnish by considering

potential aesthetic properties of perfectly balanced scales and rhythms; for example, we

explore how the perfectly balanced rhythms realizable in XronoMorph are a generaliza-

tion of standard polyrhythms. In the accompanying website (Milne 2017), we provide

audio and video examples of these and some of the perfectly balanced microtonal scales

identiﬁed by our computational searches.

2. Previous work related to perfect balance

The notion of “balance” has been previously related to music. Within the familiar 12-

tone-equal-tempered (12-TET ) system, Lewin (1959) described ﬁve sets of musical scales

and their associated properties. He identiﬁed one set whose members exhibit what he

called the “exceptional property” and this corresponds to what we call perfect balance.

In the 12-TET context explored by Lewin, all exceptional property scales are the union

of disjoint equal-interval pitch-class sets – regular polygons – like the whole tone (reg-

ular hexagon), diminished seventh (square), augmented triad (equilateral triangle), and

tritone (digon). All of these scales, if represented in circular form (as in Figure 1), have

their centre of mass at the centre of the circle (Milne et al. 2015), hence our use of the

term perfect balance.

To identify his ﬁve sets of scales, Lewin used an indicator vector (or characteristic

function) to represent any K-tone scale in 12-TET. This is a binary vector with Kones

and 12 −Kzeros, appropriately arranged: for example, the diminished (octatonic) scale

in Figure 1(a) has the indicator vector (1,0,1,1,0,1,1,0,1,1,0,1); the double harmonic

3

scale in Figure 1(c) has (1,1,0,0,1,1,0,1,1,0,0,1). When the ﬁrst coeﬃcient of the

discrete Fourier transform of a scale’s indicator vector uis zero (i.e.

11

P

n=0

une−2πin/12 = 0),

that scale has the exceptional property. Other zero-valued coeﬃcients correspond to

diﬀerent properties.

In an analysis of the various coeﬃcients obtained from Lewin’s approach, Quinn (2004)

illustrated the meaning of diﬀerent zero-valued coeﬃcients with a concrete analogy: “bal-

anced” weighing scales with diﬀering numbers of pans, where the number of pans, and

their allowed content, diﬀers according to the coeﬃcient under consideration. The ﬁrst

coeﬃcient is a weighing scale with twelve pans – one for each 12-TET pitch class.

Nonzero values of these coeﬃcients are also of interest (Amiot 2016). In particular,

across all K-tone scales in a given N-TET, the scale producing the largest K-th coeﬃ-

cient is maximally even (Amiot 2007). However, the magnitude of this coeﬃcient across

diﬀerent scales does not have a monotonic relationship with Thomas Noll’s intuitively

reasonable measure of evenness, which is detailed in Amiot (2009) and Section 3.4. The

magnitude of the discrete Fourier transform’s ﬁrst coeﬃcient has been identiﬁed with

“unevenness” (Callender 2007) and, indeed, when this coeﬃcient is maximal the scale is

maximally uneven (its pitch classes are all clustered together). However, scales that are

not maximally even can have a zero-valued ﬁrst coeﬃcient; so there is a terminological

discrepancy.

Indeed, it was an exploration of precisely how the magnitude of the discrete Fourier

transform’s ﬁrst coeﬃcient is not the same as unevenness that initially motivated our

interest in this coeﬃcient: if it is not a measure of unevenness, what “thing” does it

measure, and might that “thing” – what we now call imbalance – be inherently interesting

or applicable to music?

Amiot (2010) extended Lewin’s exceptional property beyond the 12-TET universe by

conducting an exhaustive search for rythmes p´eriodiques equilibr´es (balanced periodic

rhythms) in a 30-equal universe (a period divided into 30 equally spaced parts). The

rhythmic context provides a useful justiﬁcation for searching a non-12-equal universe

(another justiﬁcation, which we subsequently explore, is for microtonal scale systems).

Intriguingly, this search demonstrated the existence of perfectly balanced patterns that

are not sums of equal-step binary-weighted patterns (i.e. they are not unions of regular

polygons); so, in that respect, they are unlike the exceptional property pitch-class sets

in 12-TET identiﬁed by Lewin. This hinted that perfectly balanced patterns have more

complex constructions than might be superﬁcially supposed.

In Milne et al. (2015), we showed how these intriguing patterns can be generated by

integer combinations of equal step patterns (i.e. subtracting as well as adding patterns),

and fully generalized and explored Lewin’s exceptional property – as perfect balance –

to any K-tone pattern in any N-equal universe and to irrational patterns that do not

ﬁt into any equal universe. We also used Noll’s quantiﬁcation of evenness to clarify the

distinction between balance and evenness. We went on to provide a practical software

implementation of perfect balance, detailed in Milne et al. (2016).

As with many music-theoretical concepts, there is a body of prior pure mathematical

research that strongly intersects with perfect balance. These antecedents in algebraic

number theory include the work of Lam and Leung (2000) on vanishing roots of unity,

which itself built on work by L. R´edei, N.G. de Bruijn, and I.J. Schoenberg (as cited in

the former). This work is discussed in greater detail in Theorem 5.4.

Finally, it is worth mentioning that, in rhythmic contexts, the term “balance” has also

been used by Toussaint (2013) to denote periodic rhythms where the numbers of events

in the two halfs of the circle never diﬀer by more than one, regardless of the angle of the

4

diameter bisecting the circle. This is analogous to the standard deﬁnition of balance in

combinatorial word theory, but not with ours.

3. Basic notions concerning patterns, balance, and evenness

For clarity, we now provide formal deﬁnitions of terms that will be used subsequently.

First, we introduce our mathematical representation of pitch-class and time-class sets as

subsets of the unit circle (equivalently, R/Z). Secondly, we show how these are trans-

formed into patterns which, as elements of a vector space, can be summed. We also

introduce two useful low-dimensional auxiliary vectors – the Argand vector and the in-

dicator vector. Finally, we deﬁne balance as a function of both of these vectors, and

evenness as a function of just the Argand vector.

3.1. Pitch/time-class sets

In music, arrangements of pitches and onset times are often periodic over a given pitch or

time interval. For example, every pitch interval in the diatonic scale repeats every 12 semi-

tones (the octave); every pitch interval in the diminished (octatonic) scale repeats every

3 semitones (the quarter-octave). Similar examples are common in a rhythmic context:

every interonset interval in the clave rhythm . . . ˇ“?ˇ“(>ˇ“>ˇ“ˇ“>ˇ“?ˇ“(>ˇ“>ˇ“ˇ“>. . .

repeats every 8 quarter notes; every interonset interval in the ﬂamenco ﬁgure

. . . ? ? ˇ“(? ? ˇ“(?ˇ“(?ˇ“(?ˇ“(? ? ˇ“(? ? ˇ“(?ˇ“(?ˇ“(?ˇ“(. . . repeats every 12 eighth notes.

Periodic pitches or times are conveniently represented by pitch classes or time classes,

where all pitches or times that diﬀer by a given period are considered equivalent. A

natural representation of a pitch/time class, given suitable rescaling (i.e. dividing its

numerical value by the period), is as a member of R/Z.4The quotient R/Zis equivalent

to the unit circle S1so, like many music theorists, we often use the circle for visualizing

a pitch-class or time-class set.

Deﬁnition 3.1 A pitch/time class is an element of R/Z, represented by a real number

in the interval [0,1).

Deﬁnition 3.2 A weight for a pitch/time class is a real number associated to it. Examples

of a weight of a pitch/time class are its multiplicity, its loudness, its salience, or its

probability.

The Kpitch/time classes in a set are denoted x0, . . . , xK−1∈R/Z, and their associated

weights are denoted w0, . . . , wK−1∈R.

Deﬁnition 3.3 Rotation of a pitch/time-class set by a real number tis the addition of t

to each of its elements; that is, x0+t, . . . , xK−1+t∈R/Z.

Musically, rotation corresponds to a pitch transposition or temporal shift of the entire

pitch/time-class set.

Deﬁnition 3.4 A pitch/time-class set is rational if all its elements are rational; that is,

x0, . . . , xK−1∈Q/Z.

4This is just a rescaling of the R/12Zthat Tymoczko (2006) uses to characterize pitch classes in semitone units.

Using R/Zinstead, makes the representation agnostic with respect to the pitch units (semitones, cents, octaves,

etc.) or time units (quarter-notes, measures, seconds, milliseconds, etc.), and is equally applicable to pitch classes

and time classes.

5

Deﬁnition 3.5 An N-equal pitch/time-class set is {0,1/N, . . . , (N−1)/N} ⊆ R/Z, or

any rotation thereof. The term equal-step is used when Nis unspeciﬁed.

For pitch classes, we sometimes use the familiar term N-TET (for N-tone equal tuning

or temperament). When emphasizing geometrical aspects of an equal-step set, we often

use the term regular polygon or regular N-gon (the locations of the edges do not play a

role in our analysis, but they can be useful for visualizing patterns).

Deﬁnition 3.6 An N-equal universe of a given rational pitch/time-class set is any N-

equal superset of the latter. Its smallest N-equal universe is the N-equal universe with

the smallest possible Nthat contains the given rational pitch/time-class set.

For instance, the smallest N-equal universe of the diminished triad 0,3

12 ,6

12 is the

4-equal scale 0,3

12 ,6

12 ,9

12 . Clearly, all multiples of the smallest N-equal universe are

N-equal universes; for example, the diminished triad also has 8-equal, 12-equal, 16-equal

universes, and so forth.

3.2. Patterns, Argand vectors, and indicator vectors

Given the above deﬁnitions of pitch/time classes as elements of R/Zwith associated

weights (Deﬁnitions 3.1 and 3.2), we can now deﬁne patterns.5

Deﬁnition 3.7 A pattern is a function λfrom R/Zto Rthat maps all but ﬁnitely many

values to 0. The support of a pattern is the set of all points of R/Zat which the pattern

is nonzero.

More concretely, given a pitch/time-class set {x0, . . . , xK−1}with associated nonzero

weights w0, . . . , wK−1, the resulting pattern is

λ(x) = (wkif x=xk

0 if x6∈ {x0, . . . , xK−1}.(1)

The support of the pattern is the given pitch/time-class set {x0, . . . , xK−1}. In this

way, each pattern attaches the Knonzero weights w0, . . . , wK−1to the unit circle, as in

Figure 1. A pattern can, equivalently, be written as w0ex0+·· ·+wK−1exK−1, where exk

is the basis function indexed by xk; that is,

exk(x) = (1 if x=xk

0 if x6=xk.

Patterns are, therefore, elements of a vector space (the direct sum of the one-dimensional

spaces spanned by ex). Importantly, this means that patterns can be linearly combined

(e.g., added and subtracted) to make new patterns; something that is not possible with

pitch/time-class sets. More explicitly, the pattern corresponding to the union of disjoint

5Our deﬁnition of pattern as a certain kind of function is related to Amiot’s (2016) deﬁnition of distribution,

but generalizes it so that the domain is R/Zrather than Z/nZ. Like distributions, the set of patterns equipped

with the operations of addition, scalar multiplication, and convolution (deﬁned as (f∗g)(x) = P

t∈R/Z

f(t)g(x−t))

is an algebra (Amiot 2016). Other formalisms could be used to model patterns as distributions of weights on the

unit circle; for instance, a pattern could be deﬁned as an element of the group ring R[R/Z] (see Section 5.2), or as

a discrete signed measure on R/Z.

6

pitch/time-class sets Xand Yis equal to the sum of patterns corresponding to each:

λX ∪Y =λX+λY.

In this paper, we focus on binary-weighted patterns, whose weights are all 0 or 1,

and on sums of such patterns. Binary-weighted patterns are useful representations of

pitch/time-class sets with no prior weights (all elements of the set have equal status).

Deﬁnition 3.8 A rational pattern is a pattern with support in Q/Z. A pattern is called

N-equal or equal-step if its support is. Similarly, the rotation of a pattern λ(x)by tis the

new pattern κ(x) = λ(x−t); the support of the rotated pattern κis the support of the

original pattern λplus t.

We may refer to a real pattern when we wish to explicitly avoid an assumption of

rationality. We sometimes use the word pattern also to include any of its rotations,

similar to the use of the word scale; the context should make this clear.

Two ﬁnite-dimensional vector representations of pitch/time-class sets are useful in our

analysis: the Argand vector and the indicator vector, which are now deﬁned.

Deﬁnition 3.9 For a binary-weighted pattern with support representatives

{x0, . . . , xK−1}ordered as 0 ≤x0<··· < xK−1<1, we deﬁne the Argand vector

as (e2πix0, . . . , e2πixK−1)∈CK.6If λis a pattern, we denote its Argand vector by zλ.

The support of any pattern has a unique representation satisfying 0 ≤x0<··· <

xK−1<1, and thus the Argand vector of a pattern is well deﬁned. Note that the Argand

vector of the union of two or more non-trivial pitch/time-class sets is not equal to the

sum of their corresponding Argand vectors: for patterns κand λ,zκ+λ6=zκ+zλ. Unlike

patterns, therefore, the sum of two or more Argand vectors does not have an obvious

musical interpretation.

Deﬁnition 3.10 For a rational pattern λwith an N-equal universe (x0, x0+ 1/N, . . . , x0+

(N−1)/N), an indicator vector7u= (u0, . . . , uN−1)∈RNevaluates λat the Nequally

spaced points starting from x0; that is, un=λ(x0+n/N).

This deﬁnition implies that all unevaluated values are zero. Because indicator vectors

are discretised representations of patterns, a union of disjoint pitch/time-class sets within

a common N-universe can be represented as a sum of indicator vectors. If λis a pattern,

we denote its indicator vector by uλ.

Deﬁnition 3.11 A primitive pattern is one without rotational symmetry; that is, for all

non-integer turns, the pattern is never exactly the same as when unrotated.

Proposition 3.12 Deﬁnition 3.11 implies that a pattern λis primitive if any one of

the following conditions holds:

•λ(x−t)6=λ(x)for all t∈(0,1);

•zλand zλe2πitare unequal as sets for all t∈(0,1);

•uλis diﬀerent from all of its nontrivial word rotations.

Deﬁnition 3.13 The j-th coeﬃcient of the discrete Fourier transform of the Argand

6In the ﬁgures, we typically use the rotated and reﬂected mapping ie−2πixk, to correspond with the familiar

clockface orientation (clockwise, starting from the top).

7The term characteristic function is often used for a binary indicator vector; for example, see Lewin (1959);

Amiot (2016).

7

vector is given by (Fz)j=

K−1

P

k=0

zke−2πijk/K . The m-th coeﬃcient of the discrete Fourier

transform of the length-Nindicator vector uis given by (Fu)m=

N−1

P

n=0

une−2πimn/N .

Discrete Fourier transforms of indicator vectors are commonly used in music theory

(e.g., Lewin (1959,1987); Quinn (2004); Callender (2007); Amiot (2007)). For our pur-

poses, the discrete Fourier transform of the Argand vector has the advantage that it is

applicable to irrational as well as rational patterns, and its coeﬃcients give values for

both balance and evenness, as we now show.

3.3. Balance and perfect balance

We now provide deﬁnitions of balance and perfect balance as functions of a pattern. We

describe how balance can be computed from an Argand or indicator vector, when they

exist. We also deﬁne the useful property of minimality for perfectly balanced patterns

(further discussed in Section 5.2.2).

Deﬁnition 3.14 The balance of a pattern λ:R/Z→Ris one minus the magnitude of

the centre of gravity of the pattern; that is,

B(λ) =

1−X

x∈R/Z

λ(x)e2πix.X

x∈R/Z

|λ(x)|if λ(x)6= 0 for some x,

1 if λ(x) = 0 for all x.

If λis a binary-weighted pattern, then λhas an Argand vector z; let Kbe its length.

Then B(λ) = 1−

K−1

P

k=0

zk

/K = 1−|(Fz)0|/K. We will slightly abuse notation by writing

B(z) for this balance.

If the pattern λis rational in an N-equal universe, then it has a length-Nindicator

vector u. Again we abuse notation and write B(u) = B(λ) = 1 − |(Fu)1|/K. This

expression corresponds to the formalism used by Lewin (1959) (as discussed in Section 2),

who deﬁned the exceptional property as (Fu)1= 0.

Balance is equivalent to the weighted circular variance of the angles appearing in the

pattern (Mardia 1972;Fisher 1993). The maximal possible value for circular variance is

1, so perfect balance implies maximal circular variance.

Remark 3.15 The above deﬁnitions imply that balance is insensitive to the rotation

(pitch transposition or time displacement) of the pattern: for all t,B(λ(x)) = B(λ(x−t))

and B(z) = B(ze2πit), while B(u) = B(σ(u)), where σ(u) is some cyclic permutation of

u. They also imply that balance is insensitive to circular direction (musical inversion);

that is, B(z) = B(z∗), where ∗denotes complex conjugation.

Deﬁnition 3.16 A pattern λis perfectly balanced if it has balance 1; that is,

P

x∈R/Z

λ(x)e2πix= 0. Likewise, a pattern λis perfectly imbalanced if it has balance 0;

that is, P

x∈R/Z

λ(x)e2πix

=P

x∈R/Z

|λ(x)|.

8

Remark 3.17 A consequence of Deﬁnition 3.14 is that a pattern comprising a single

pitch/time class is perfectly imbalanced. This means that equal-step patterns are not

balanced over their fundamental (smallest possible) period. For example, consider a

whole-tone scale. Given an octave period, this comprises 6 equally spaced pitch classes

0,2

12 ,4

12 ,6

12 ,8

12 ,10

12 , up to rotation. Its fundamental period is the whole tone of size

2/12 and, over this period, the scale becomes simply {0}, which is perfectly imbalanced.

Deﬁnition 3.18 A minimal perfectly balanced pattern has no proper subset that is

perfectly balanced. Equivalently, a minimal perfectly balanced pattern cannot be con-

structed from the sum of two or more perfectly balanced patterns.

Minimal perfectly balanced rational patterns are considered in depth in Section 5.2.2.

3.4. Evenness

As previously mentioned in Section 2, evenness is a widely discussed aspect of musical

scales and rhythms, and it has an intimate relationship with balance (see Section 4).

Here we provide deﬁnitions of evenness and perfect evenness as functions of the Argand

vector of a binary-weighted pattern.

Deﬁnition 3.19 The evenness of a K-element binary-weighted pattern with Argand

vector zis E(z) = |(Fz)1|/K (Amiot 2009).

Evenness is equivalent to 1 minus the circular variance of the rotational diﬀerences

between each k-th element of the pattern and the k-th element of a K-equal division of

the period (Milne et al. 2015).

The classically deﬁned maximally even pattern, which maximizes |(Fu)K|where u∈

{0,1}N, corresponds to that found by maximizing E(z) across patterns with the same

Kand Nvalues (Amiot 2016, Theorem 5.6).

Remark 3.20 As with balance, the deﬁnition of evenness implies that it is insensitive

to rotation of the pattern: for all t,E(z) = E(ze2πit). It also implies that evenness is

insensitive to circular direction (musical inversion); that is, E(z) = E(z∗).

Deﬁnition 3.21 A binary-weighted pattern is perfectly even if it has evenness 1; that is,

|(Fz)1|=K. Likewise, a pattern is perfectly uneven if it has evenness 0.

Remark 3.22 Deﬁnition 3.19 implies that a pattern comprising a single pitch/time class

is perfectly even (this is because the discrete Fourier transform of a single number is a

periodic repetition of that number).

4. Relationships between balance and evenness

Balance and evenness are independent but related properties of patterns. Across diﬀerent

patterns they have a strong positive relationship (quantiﬁed below); patterns with high

balance often have high evenness, and vice versa. This is important to note, because

evenness is often considered one of the fundamental principles for the construction and

analysis of scales and meters (Clough and Douthett 1991;London 2004). This suggests

that the aesthetic properties often attributed to evenness may also be attributable to

balance. Examples, such as those shown below, where balance and evenness are distinctly

diﬀerent, are, therefore, of particular interest: they may allow a systematic investigation

9

(a) (b) (c)

Figure 2. Three distinctly uneven but perfectly balanced patterns in a 30-equal universe.

of their respective aesthetic properties.

For binary-weighted patterns, we ﬁrst consider some mathematical relationships be-

tween these two properties, and then make some empirical observations.

Theorem 4.1 Perfect evenness implies perfect balance.

Proof. By Parseval’s theorem,

K−1

P

j=0 |(Fz)j|2=K

K−1

P

k=0 |zk|2. By deﬁnition, all |zk|= 1,

hence

K−1

P

j=0 |(Fz)j|2=K2. When |(Fz)1|=K(perfect evenness), all other coeﬃcients of

Fzmust, therefore, be zero.

Theorem 4.2 Perfect imbalance implies perfect unevenness.

Proof. The proof follows the same line of argument as that for Theorem 4.1 but using

the zeroth coeﬃcient of Fzinstead of the ﬁrst.

Remark 4.3 Theorems 4.1 and 4.2 apply when K > 1; that is, when there is more than

one independent discrete Fourier transform coeﬃcient. As mentioned in Remarks 3.17

and 3.22, in the “boundary condition” of a scale or rhythm with just one pitch/time

class, the pattern is both perfectly even and perfectly imbalanced: when K > 1, this is

impossible.

Remark 4.4 Perfect balance does not imply perfect evenness.

This is demonstrated by the counterexamples in Figures 1and 2.

Remark 4.5 We conjecture that perfect unevenness implies perfect imbalance.

Although Theorems 4.1 and 4.2 indicate a strong relationship between these two prop-

erties, Remark 4.4 shows that they are far from identical. Indeed, the patterns illustrated

in Figure 2are all perfectly balanced; they are also distinctly uneven.

Certain aspects of the relationship between balance and evenness can be usefully

gleaned from scatter plots, as shown in Figures 3and 4. Generally, there are more prim-

itive patterns with higher evenness than balance; that is, above the diagonal line in

Figures 3and 4(we refer here to the mathematical quantities of balance and evenness,

not to any supposed perceptual response, which may not be linearly related to the math-

ematical quantities). There is clearly a strong positive relationship between balance and

evenness – the Spearman correlation between them in Figure 3is .94 and in Figure 4

it is .93; these are typical values across diﬀering sets of scales. Despite this, there are a

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Balance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Evenness

(a)

0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Balance

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Evenness

10-note: C D D# E F G G# A A# B

9-note: C D D# F F# G A A# B

11-note: C D D# E F F# G G# A A# B

8-note: C D E F G Ab Bb B

double harmonic

5 note: C E F G# B

harmonic minor/major

melodic minor

diatonic

pentatonic

9-note: C D# E F G G# A# B

8-note: C D# E F G A A# B

(b)

Figure 3. Scatter plots of the balance and evenness of all binary-weighted primitive patterns in a 12-equal universe.

The right-hand ﬁgure is a labelled close-up of the most even and most balanced binary-weighted primitive patterns

in a 12-equal universe. See Deﬁnitions 3.11,3.14, and 3.19 for the notions of primitive patterns, their balance, and

their evenness. The diagonal line, which crosses through points where balance equals evenness, is provided as a

visual reference.

(a) (b)

Figure 4. Scatter plots of the balance and evenness of all binary-weighted primitive 7-pitch/time class patterns in

equal universes from 8 to 30. The right-hand ﬁgure is a close-up of the most even and most balanced 7-pitch/time

class patterns. See Deﬁnitions 3.11,3.14, and 3.19 for the notions of primitive patterns, their balance, and their

evenness. The diagonal line, which crosses through points where balance equals evenness, is provided as a visual

reference.

number of interesting perfectly balanced primitive patterns that are not perfectly even

(audibly so) as shown by the patterns running down the right-hand edges of the graphs.

11

5. Properties of perfect balance

5.1. General properties

Theorem 5.1 The sum or diﬀerence of perfectly balanced patterns is perfectly balanced.

Proof. If κand λare two perfectly balanced patterns, then

X

x∈R/Z

(κ+λ)(x)e2πix=X

x∈R/Z

κ(x)e2πix+X

x∈R/Z

λ(x)e2πix= 0,

so that κ+λis also perfectly balanced.

In terms of pitch/time-class sets, this means that the union of perfectly balanced dis-

joint sets is also perfectly balanced. Furthermore, the complement of a perfectly balanced

set in any of its perfectly balanced universes is also perfectly balanced. In a rational set-

ting, Theorem 5.1 parallels how, in an N-equal chromatic universe, the complement of a

maximally even scale of Kpitches is a maximally even scale of N−Kpitches (Clough

and Douthett 1991;Amiot 2007, Proposition 3.2). For example, the complement of the

diatonic scale in the chromatic scale is the pentatonic – more prosaically, the piano’s black

notes (which are maximally even) ﬁll in all the gaps between the white notes (which are

also maximally even). In a more general setting, it parallels how the complement of any

well-formed scale, with respect to the next higher cardinality well-formed scale of which

it is a proper subset, is also well-formed.8

Lemma 5.2 A binary-weighted pattern has N-fold rotational symmetry if and only if it

is the sum of one or more N-equal patterns (all Nbeing the same).

Proof. In the Argand vector, the k-th pitch/time class has angle arg(zk) radians. A

pattern has N-fold rotational symmetry when Nis the largest number such that λ(x+

1/N) = λ(x). This implies there are pitch/time class with angles 2πm/N + arg(zk)) for

all m∈Z. Geometrically, these are the vertices of a regular N-gon one of whose vertices

is at arg(zk). In this way, the entire pattern can be constructed from K/N appropriately

rotated regular N-gons (N-equal patterns). The same argument can be done in reverse

to show that any pattern constructed from independently rotated regular N-gons has

N-fold rotational symmetry.

Theorem 5.3 Any rotationally symmetric (non-primitive) binary-weighted pattern is

perfectly balanced.

Proof. All N-equal binary-weighted patterns are perfectly balanced (they are, by deﬁni-

tion, perfectly even so their perfect balance follows from Theorem 4.1). Any rotationally

symmetric (non-primitive) pattern is the sum of N-equal patterns (Lemma 5.2) hence,

by Theorem 5.1, such patterns are perfectly balanced.

8We are not aware of a published proof for the complementarity of well-formed scales, but it follows directly from

representing well-formed hierarchies with successive convergents and intermediate convergents (Norman Carey,

personal communication, May 24, 2017). In addition to perfectly balanced rhythms, XronoMorph can also generate

a hierarchy of well-formed rhythms where each higher (faster) rhythmic level is generated by splitting the long

beats of the next lower level. Any level (except the lowest) can be placed into “complementary mode” where only

events that do not occur in any lower level are sounded. A result of the well-formed complementarity property is

that, although no sounded beats from two levels ever coincide, all such levels are well-formed both individually

and in sum (Milne and Dean 2016).

12

Rotational symmetry is, therefore, a trivially easy way to achieve perfect balance. This,

in part, motivates our speciﬁc interest in perfectly balanced primitive patterns (see Sec-

tion 5.2.1). Also, one might wonder whether it makes sense to place any rotationally

symmetrical pattern onto a circle because this does not reﬂect its true periodicity. This

seems a reasonable concern for rhythmic patterns but, for pitch patterns, there is an

important perceptually motivated interval of periodicity: the octave. For example, the

familiar diminished (octatonic) scale (Figure 1(a)) has a periodicity every three semi-

tones (a quarter circle) but there is still an overarching perceptual periodicity that spans

the octave of twelves semitones (the full circle). In pitch patterns, therefore, placing

rotationally symmetrical patterns into the octave can be a sensible choice. Indeed, rota-

tionally symmetric pitch patterns – also known as transpositionally invariant or TINV

pitch class sets – have found frequent use and been met with much theoretical interest:

for example, all of Messiaen’s modes of limited transposition (Messiaen 1944) are rota-

tionally symmetric and, hence, perfectly balanced; Cohn (1991) enumerates and studies

transpositionally invariant pitch class sets, highlighting their frequent use in twentieth

century art music and possible reasons for their aesthetic utility.

5.2. Perfectly balanced rational patterns

Rational patterns are common (though by no means universal) in music, and they are

often embedded within an equal-step universe that is either explicitly sounded or at least

implied. For example, familiar Western scales (diatonic, harmonic minor, etc.) are subsets

of the 12-TET chromatic scale, while many rhythms are subsets of a regular (isochronous)

pulse that may or may not be sounded. Rationality is also a useful constraint for exploring

balance because rational perfectly balanced patterns are highly “structured,” as we now

detail.

Theorem 5.4 (Lam and Leung 2000)Any perfectly balanced integer-weighted rational

pattern can be expressed as an integer combination of n-equal patterns where all nare

prime factors of N, the cardinality of the pattern’s smallest universe; that is,

B(u)=1 ⇐⇒ u=

M

X

m=1

jmp(m)

for some integers Mand jmand some regular n-gons p(m)where each nis a prime

divisor of N. Recall the notion of n-equal from Deﬁnition 3.5.

The leftward implication is trivial. The rightward implication is the substance of the

version of the R´edei-de Bruijn-Schoenberg Theorem appearing in Theorem 2.2 of Lam

and Leung (2000), but the notation in that article is very diﬀerent to ours. Their theorem

states

ker(φ) =

r

X

i=1

ZG·σ(Pi),

where

•φis the linear functional on integer-weighted rational patterns given by φ(λ) =

P

x∈R/Z

λ(x)e2πix; hence, ker(φ) is the space of perfectly balanced integer-weighted ra-

13

tional patterns within an N-equal universe;

•for each of the rprime factors piof N,σ(Pi) is the unrotated regular pi-gon, that is,

the perfectly balanced pattern with Argand vector (1,e2πi/pi,...,e2πi(pi−1)/pi);

•ZG·σ(Pi) is the ideal generated by all integer-weighted regular pi-gons, rotated so

that the vertices align with any powers of e2πi/N .

Note that the image of φis just the N-th cyclotomic integer ring Z[e2πi/N ], which has

been studied extensively in Galois theory. The degree of this extension over Zis given by

Euler’s totient function ϕ(N). Integer-weighted balanced patterns correspond to linear

relations among the Ngenerators 1,e2πi/N ,...,e2πi(N−1)/N , so that the lattice of such

patterns has dimension N−ϕ(N). Lam and Leung’s proof uses the powerful formalism

of group rings to establish that this lattice is generated by the prime n-equal patterns.

Corollary 5.5 The indicator vector of a perfectly balanced integer-weighted rational

pattern in an N-equal universe with N=pa1

1·· ·par

ris a sum of periodic integer vectors

whose periods are N/pifor i∈ {1, . . . , r}.

Proof. Denote the indicator vector by u. According to the theorem, ucan be written as

u=

r

X

i=1

Mi

X

mi=1

ji,mip(i,mi)

where each p(i,mi)is a regular pi-gon and thus has period N/pi. Now the indicator vector

u(i)=

Mi

X

mi=1

ji,mip(i,mi)

has period N/pi, and u=u(1) +·· · +u(r).

We conjecture that, for binary-weighted patterns, the rsummand periodic integer vec-

tors in the corollary can be chosen so that each takes values from {−1,0}or from {0,1}.

An equivalent statement of our conjecture is that each binary-weighted rational pattern

in an N-equal universe with N=pa1

1·· ·par

ris an unweighted sum of a subset of the prime

polygon patterns, with an integer constant (possibly zero) subtracted from all weights.

It is easy to see that this follows for all Nif it holds for square-free N(that is, when

Nhas no repeated prime factors), and we have veriﬁed the conjecture computation-

ally for all two-digit square-free N. We have also tested thousands of perfectly balanced

binary-weighted patterns with N= 105 and N= 210. Unfortunately, a proof has been

elusive.

5.2.1. Primitive perfectly balanced patterns

As mentioned in Remark 3.17, although an n-equal pattern (in the sense of Deﬁnition

3.5) is perfectly balanced, it is not perfectly balanced over its fundamental period of 1/n

(over its fundamental period, an equal-step pattern reduces to a pattern with a single

pitch/time class). This is also the case for many non-primitive patterns (see Figure 1and

associated text). Clearly, all perfectly balanced primitive patterns are perfectly balanced

over their fundamental period, as are non-primitive patterns whose fundamental periods

are perfectly balanced primitive patterns. This is another motivation for our interest in

primitive patterns.

14

(a) 5 pitch/time classes

in 12.

(b) 7 pitch/time classes

in 12.

(c) 5 pitch/time classes in

18.

(d) 7 pitch/time classes

in 18.

Figure 5. Four perfectly balanced primitive patterns derived from sums of n-equal patterns (unions of disjoint

regular polygons). The pattern in (c), and its reﬂection (not shown), are the smallest perfectly balanced rational

patterns in the smallest N-equal universe not having reﬂective symmetry.

From Lemma 5.2, we see that any pattern constructed from sums of n-equal patterns

(all nbeing the same) is non-primitive. This is also true of the sum of equal-step patterns

with p, q, r, . . . steps when their greatest common divisor is greater than 1. This is because

each pattern can be reconstructed from n-equal patterns where nis that greatest common

divisor. However, when p, q, r, . . . are a coprime set (their greatest common divisor is 1),

the pattern resulting from their sum is primitive. This provides a straightforward method

for constructing perfectly balanced primitive patterns, as illustrated in Figure 5.

Theorem 5.6 An N-equal universe contains perfectly balanced primitive patterns if and

only if Nhas either three or more distinct prime factors, or two prime factors with at

least one repeated.

Proof. If N= 1 then clearly only the empty pattern is perfectly balanced.

If Nhas only one prime factor pthen, by Corollary 5.5, its indicator vector has pe-

riod N/p, so it is not primitive.

If N=pq with pand qprime and unequal then, again by Corollary 5.5, the indicator

vector wis the sum of an integer vector u, with period N/p =q, and another, v, with

period N/q =p. By the Chinese Remainder Theorem, for each ujand vk, there is some

`such that w`=uj+vk. In particular, the range max`w`−min`w`is the sum of the

ranges maxjuj−minjujand maxkvk−minkvk. Each range is a nonnegative integer,

and the range of wis at most 1. Thus one of uor vis constant, and whas the same

period as the other.

Lastly, suppose N=pqr, where pand qare prime and unequal, and r > 1. The sum

of a p-equal pattern containing a pitch/time class of 0 and a q-equal pattern containing

a pitch/time class of 1/N is perfectly balanced and primitive.

This theorem implies that the lowest ﬁfty Ncontaining perfectly balanced primitive

patterns are (this corresponds to OEIS A102467 with the ﬁrst member missing, Sloane

2016): 12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72,

75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117,

120, 124, 126, 130, 132, 135, 136, . . . .

The two perfectly balanced primitive patterns in N= 12 and two of the 17 primitives

in N= 18 are illustrated in Figure 5. It is noteworthy that the unique complementary

pair of perfectly balanced primitive patterns in 12-equal (Figures 5(a) and (b)) includes

the representative of the singular pairwise well-formed scale pattern, abacaba (Clampitt

2007).

15

(a) 6-in-30 (b) 7b-in-30 (c) 8b-in-30 (d) 9-in-30

(e) 8-in-42 (f) 9c-in-42 (g) 10a-in-42 (h) 10d-in-42

Figure 6. Eight minimal perfectly balanced primitive patterns. Each pattern can be constructed by summing

positively-weighted equal-step patterns (depicted by regular polygons with solid lines) and negatively weighted

equal-step patterns (regular polygons with dashed lines). The minimality of these patterns means that none can

be composed as a sum of only positively-weighted equal-step patterns. For example, the 6-in-30 pattern can be

constructed from a “positive” regular pentagon, a “positive” regular triangle, and a “negative” digon such that

the latter “cancels out” one vertex from each of the former. In terms of numbers of pitch/time class, this “6-in-30”

pattern is the smallest minimal perfectly balanced primitive rational pattern, hence the smallest rational pattern

that is perfectly balanced over its fundamental period. The “10d-in-42” pattern, and its reﬂection, are the smallest

minimal perfectly balanced primitive rational patterns without reﬂective symmetry.

5.2.2. Minimal perfectly balanced patterns

From Theorem 5.4, it follows that any equal-step pattern with a prime number of

pitch/time classes is a minimal perfectly balanced pattern. Interestingly, in some N,

there are non-equal step perfectly balanced patterns (as would be depicted by irregular

polygons) that are also minimal (hence they are primitive as well). This is because such

patterns can be constructed from summing positively- and negatively-weighted equal-

step patterns (as in Theorem 5.4), but they cannot be constructed from summing only

positively-weighted equal-step patterns. For example, consider the patterns illustrated in

Figure 6.

Minimal perfectly balanced patterns are useful because, for any N, they are a relatively

small subset of all possible perfectly balanced patterns, and this subset can then be used

as “building blocks” to construct all possible perfectly balanced patterns. Indeed this is

the strategy we use in the design of XronoMorph, as detailed in Section 6.1. For example,

up to rotation and reﬂection, there are 17 binary perfectly balanced patterns in N= 12,

but only two minimal perfectly balanced binary-weighted patterns (both equal-step);

there are 3,360 binary perfectly balanced patterns in N= 30, but only nine minimal

perfectly balanced binary-weighted patterns (of which three are equal-step, and six are

not).

Theorem 5.7 An N-equal universe contains minimal perfectly balanced primitive pat-

terns if and only if Nhas three or more distinct prime factors.

Proof. If N=pαqβwith pand qprime, then the indicator vector wis the sum of

16

an integer vector u, with period N/p, and another, v, with period N/q. Consider the

decompositions u=u(0) +· ·· +u(N/pq−1) and v=v(0) +·· · +v(N/pq−1) given by

u(j)

k=(ukif k=jmod N/pq

0 otherwise, v(j)

k=(vkif k=jmod N/pq

0 otherwise.

Each of the vectors u(j)+v(j)is perfectly balanced, and binary because, for each jand

k, we either have u(j)

k+v(j)

k= 0 or we have u(j)

k+v(j)

k=wk. If two or more of these

are not identically zero, then their sum cannot be minimal. If only one is not identically

zero, without loss of generality, assume it is u(0) +v(0); now all of the weights occur at

multiples of 1/pq, so by Theorem 5.6, it is not primitive.

On the other hand, suppose that N=pqrs, where p,qand rare distinct primes. We

will only place weights at multiples of 1/pqr, so we can assume without loss of generality

that s= 1. A q-gon with a vertex at angle 0, minus a p-gon with a vertex at angle 0,

plus r-gons with vertices at each of the angles (in radians) 2π/p, 4π/p, . . . , 2π(p−1)/p,

is a perfectly balanced primitive binary-weighted pattern, and it is easily seen to be

minimal.

This implies that the ﬁfty lowest Ncontaining minimal perfectly balanced primitive

patterns are multiples of the sphenic numbers9(OEIS A000977, Sloane 2016): 30, 42, 60,

66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165,

168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 220, 222, 228, 230, 231, 234, 238,

240, 246, 252, 255, 258, 260, 264, 266, 270, 273, . . . .

All minimal perfectly balanced patterns in any Nnot in the above sequence are easy

to ﬁnd – they are simply all regular n-gons, where nis a prime factor of N. But the task

of ﬁnding the additional minimal primitive patterns in the above-listed Nis far from

trivial. In order to do this, we have used Normaliz, a software application developed by

Winfried Bruns, Robert Koch, and collaborators “for computations in aﬃne monoids,

vector conﬁgurations, lattice polytopes, and rational cones” (Bruns et al. 2016). From

Theorem 5.4, we can produce an integer basis for the set of perfectly balanced binary

patterns. From this, Normaliz eﬃciently calculates a Hilbert basis for the conic lattice of

nonnegative perfectly balanced binary patterns, that is, a minimal set whose nonnegative

integer combinations give all nonnegative perfectly balanced binary patterns.

It has been practicable to calculate them for the sphenic numbers (products of three

distinct primes) up to N= 102. Factoring out rotation, there are 9 minimals in N= 30,

21 minimals in N= 42, 189 in N= 66, 57 in N= 70, 633 in N= 78, and 7713 in

N= 102. The indicator vectors for the minimals in N= 30, N= 42, and N= 70

are listed in Tables A1,A2, and A3 (in the Appendix), where they are arranged ﬁrst

by their total numbers of events/pitches, then in lexicographical order, with each scale

in canonical form (Amiot and Sethares 2011). The minimals for the remaining Nare

available in the accompanying website (Milne 2017).

5.3. Perfectly balanced real patterns

In this section, we explore some properties of real perfectly balanced patterns. First, we

show that perfectly balanced real patterns cannot be approximated arbitrarily closely

by perfectly balanced rational patterns; secondly, we demonstrate a convenient method

9Sphenic numbers are products of three distinct primes.

17

for most closely approximating any unbalanced pattern with a perfectly balanced real

pattern.

5.3.1. Approximating perfectly balanced real patterns with perfectly balanced rational

patterns

Theorem 5.4 tells us that any perfectly balanced rational pattern can be expressed as

an integer combination of equal-step patterns. And certainly every scale can be approx-

imated arbitrarily closely by a rational scale. So it would be easy to suppose that the

rationality condition of Theorem 5.4 can be omitted, but this is not the case.

Consider, for instance, the vector z= (1,0.1 + √0.99i,−0.6 + 0.8i,−0.6−0.8i,0.1−

√0.99i) of ﬁve complex numbers on the unit circle. This vector represents a perfectly

balanced pattern insofar as its ﬁve elements have unit magnitude and sum to 0. This

pattern can be approximated arbitrarily closely by rational patterns, but not by perfectly

balanced rational patterns; it can be shown that no perfectly balanced rational scale

approximates zwith a total absolute error less than about 2% of full rotation.

5.3.2. Approximating real patterns with real perfectly balanced patterns

It can be useful to ﬁnd the closest perfectly balanced approximation of an arbitrary

pattern; for example, we may want to balance an unbalanced scale or rhythm, whilst

doing as little damage as possible to an existing property. Lagrange multipliers provide

a very eﬃcient solution, with an appealing geometrical interpretation.

Let xand ybe K-dimensional column vectors with x2

k+y2

k= 1 for each k. The values

in xand yare, therefore, the Cartesian coordinates of points on a unit circle (or the real

and imaginary components of the Argand vector z). We seek K-dimensional uand vsuch

that

K−1

P

k=0 (uk−xk)2+ (vk−yk)2is minimized subject to the constraints u2

k+v2

k= 1 for

all k, and

K−1

P

k=0

uk=

K−1

P

k=0

vk= 0.The ﬁrst constraint ensures the pattern represented by u

and vis on the unit circle, the next two constraints ensure it is perfectly balanced. The

term to be minimized corresponds to the sum of squares of the chord lengths between

the pitches with coordinates (xk, yk) and those with coordinates (uk, vk). For small pitch

changes, this serves as a close approximation of the sum of squared pitch distances and,

importantly, it allows for the following simpliﬁcation of the problem.

Associating the Lagrange multipliers λk,µand νwith the constraints gives the La-

grangian

L=

K−1

X

k=0 (uk−xk)2+ (vk−yk)2+

K−1

X

k=0

λku2

k+v2

k−1+µ

K−1

X

k=0

uk+ν

K−1

X

k=0

vk,

whose stationary points occur when

∂L

∂uk

= 2((1 + λk)uk−xk) + µ= 0 and

∂L

∂vk

= 2((1 + λk)vk−yk) + ν= 0.

18

Figure 7. A geometric interpretation of the calculation of the closest perfectly balanced pattern. The pattern

shown by triangular markers (a 12-TET major triad) is uniformly translated to give the three square markers.

These are then radially projected back onto the unit circle to give a perfectly balanced triad (augmented triad)

shown by the circular markers.

Solving for ukand vk, we have

uk

vk=1

1 + λkxk

yk−1

2µ

ν.

As illustrated in Figure 7, this states that the Kgiven pitches, as represented by the

Kcoordinate vectors (xk, yk)T, have all been translated by the same, albeit unknown,

vector −(µ, ν)T/2, and then each rescaled by 1/(1 + λk) to bring them back onto the

unit circle. This does not provide an analytical solution for uand v, but it simpliﬁes it

to a bivariate and convex problem: optimizing µand νto minimize the squared balance,

which is K−1

P

k=0

uk2

+K−1

P

k=0

vk2

, of the resulting scale. This can be quickly solved by a

standard unconstrained optimization routine: in the accompanying website (Milne 2017),

we provide a matlab script, rebalance.m, to perform this optimization.

6. Exploring the manifold of binary-weighted perfectly balanced patterns

of Kelements

This section discusses the space of perfectly balanced K-element pitch/time-class sets,

where K≥2 is a ﬁxed integer (and each pitch/time class has a weight of 1). This space

is a manifold. We thank Thomas Fiore for discussion of this topic.

Let AKbe the unordered conﬁguration space of Kdistinct points in R/Z,

AK=(x1, . . . , xK)∈(R/Z)K:xj6=xkfor distinct j, k ∈ {1, . . . , K}/SK;

this is a K-dimensional manifold (Kassel and Turaev 2008, p.29). The space AKis in

bijective correspondence with the set of (unordered) subsets of R/Zwith Kelements,

{{x1, . . . , xK} ⊆ R/Z:xj6=xkfor distinct j, k ∈ {1, . . . , K}},

19

but AKis topologized as the quotient of a subset of a product. The quotient is the

quotient by the symmetric group SK.

We deﬁne BKas the set of perfectly balanced patterns in AK, given by

BK=(x∈AK:

K−1

X

k=0

cos(2πxk) = 0 =

K−1

X

k=0

sin(2πxk)).

Proposition 6.1 The subspace BKof AKis a manifold for every K≥2.

Proof. The space B2of perfectly balanced dyads is homeomorphic to the collection of

antipodal (unordered) pairs {z, −z}in the circle S1, and this quotient space is the one-

dimensional real projective space RP1, known to be homeomorphic to S1. In our setup,

using R/Zin place of S1, the space B2has as its points the unordered pairs {r, r + 1/2}

with r∈R/Z. Clearly, there is one such pair for each r∈[0,1/2), and r= 0 and r= 1/2

make the same pair, so we see that B2is a circle.

Now assume K > 2, and consider the map f:AK→R2given by f(x0, . . . , xK−1) =

PK−1

k=0 cos(2πxk),PK−1

k=0 sin(2πxk). The pre-image f−1{0}is BK, so that, by the Pre-

Image Theorem (Tu 2010, pp.105–106), BKmust be a manifold provided that 0 is a

regular value of f, that is, whenever f(x) = 0, the K×2 Jacobian matrix Dxfhas full

rank, which means it has two linearly independent columns. But if x∈BK, then there

must be two components xiand xjwith xj−xi=ρfor ρ∈(0,1/2), so that

∂f1(x)

∂xi

∂f1(x)

∂xj

∂f2(x)

∂xi

∂f2(x)

∂xj

= 2π−sin(2πxi)−sin(2πxj)

cos(2πxi) cos(2πxj)

= 2πsin(2πρ)6= 0.

The manifold BKis complicated: to our knowledge, there is no closed-form one-to-one

parameterization of the manifold of real perfectly balanced patterns for general K. To the

extent that we wish to use balance as a principle for generating new scales and rhythms,

it is still useful to have parameterizations that at least make subsets of this manifold

easier to navigate, and/or constraints embodying musically useful properties in order to

derive a manageable subset of balanced possibilities.

In the following subsections, we explore three types of parameterizations and/or con-

straints. The ﬁrst is the method of rhythm generation used in the music software appli-

cation XronoMorph, which uses sums of minimal perfectly balanced patterns depicted as

polygons that can be freely rotated. Secondly, we move to the pitch domain and conduct

a search for perfectly balanced high-aﬃnity scales that have numerous good approxima-

tions of high-aﬃnity intervals, where the only constraint (beyond perfect balance) is that

no scale step is less than 20 cents. Finally, using methods inspired by Sethares (1993,

2005); Sethares et al. (2009), we consider spectrally matching tones to some of the inter-

esting rational patterns that contain minimal perfectly balanced primitive patterns (see

Proposition 5.7), notably in N= 30 and N= 42. Musical examples using some of the

resulting rhythms and scales are available in the accompanying website (Milne 2017).

It is worth noting that we tried a fourth approach, which was to apply the XronoMorph

parameterization directly to the pitch domain and conduct a complete search for high-

aﬃnity scales. Although the resulting scales have some useful properties (such as low

entropy interval distributions, as discussed in Section 6.1), they are generally very close

20

Figure 8. In XronoMorph’s user-interface, a small disk rotates clockwise around the circle and when it hits

a polygon vertex a MIDI event is triggered. The speed at which the disk rotates (the length of the period)

is controlled by the long horizontal slider at the top. Each minimal polygon can be assigned a MIDI pitch,

velocity, duration, and channel, and directed to up to three, out of a total of twelve, tracks. Each of these twelve

tracks can be thought of as an “instrumentalist” who plays any polygon being sent to it. Each track produces

sound from a built-in sampler, from a plugin AU or VST synthesizer, or directs the MIDI to a port to drive a

standalone software or hardware synthesizer. In this way an ensemble of twelve instrumentalists can be formed,

and each polygon can be played by up to three of these instrumentalists. XronoMorph can be downloaded from

https://www.dynamictonality.com/xronomorph.htm, additional videos can be viewed at https://www.youtube.

com/c/xronomorph.

to the familiar 12-TET tuning (so not novel in that respect), or they provide poor approx-

imations of just intonation (so not successful in that respect). The “second” approach

performed more successfully, which demonstrates that its additional ﬂexibility was im-

portant for this task. For this reason, we report the results of the “fourth” approach in

the accompanying website (Milne 2017), rather than here.

6.1. Perfectly balanced rhythms in XronoMorph

XronoMorph – illustrated in Figure 8– is a freeware Windows and macOS applica-

tion (built in Max 7 Cycling ’74 (2017)) that we have developed to demonstrate the

musical possibilities of either perfectly balanced or well-formed rhythms.10 The latter

are discussed extensively in Milne and Dean (2016) and the application, as a whole, is

introduced in Milne et al. (2016).

XronoMorph treats minimal perfectly balanced patterns (depicted as, and henceforth

described in this section as, polygons) as the fundamental building blocks from which to

construct more complex patterns. Up to eight polygons can be added to the circle and

independently rotated (by predeﬁned whole-number divisions of the period or by any

real number of turns, hence creating irrational patterns).

10XronoMorph is available from http://www.dynamictonality.com/.

21

In order to accommodate a musically suﬃcient number of possibilities, XronoMorph

allows the following perfectly balanced polygons to be chosen and summed: all regular

n-gons up to n= 12, all regular prime-n-gons up to n= 29, and all 24 minimal primitive

polygons available in N= 30 and N= 42 (see Tables A1 and A2). This allows a huge

variety of composite perfectly balanced rhythms to be produced.

The principal user-parameters for deﬁning perfectly balanced rhythms are, therefore,

the selection of perfectly balanced polygons and their independent rotations. The circle

in Figure 8shows a rhythm that consists of eight minimal perfectly balanced polygons,

all but two of which have been independently rotated to make them distinct. The inde-

pendent pitches played by each polygon can produce harmonies and elicit “half-heard”

hocketed melodies.

XronoMorph automatically calculates the smallest universe (see Deﬁnition 3.6) for the

entire set of polygons used; this is the smallest equal division of the period required for

every vertex to occur at one of those divisions. These divisions can be independently

soniﬁed as an isochronous (perfectly even) pulse. When “snap to pulse” is engaged,

polygon rotation is quantized to the smallest universe (whether or not it is soniﬁed).

This typically produces rhythmic patterns where the sounded events are non-isochronous,

but there is always a relatively fast isochronous pulse connecting all of the pattern’s

events. This is a common (though by no means universal) feature of real-world rhythms

– a simple counter-example is a swing rhythm where the swing ratio is irrational, a

practice commonly employed by jazz drummers and a standard manipulation available

in digital audio workstations. By disengaging “snap to pulse,” subtle – but still perfectly

balanced – deviations from this isochronous grid are easy to make, and in ways that

can be quite diﬀerent to familiar swing ratio adjustments. This is useful given that

micro-timings (small deviations from the regular grid) can be one important means for

creating “groovy” rhythms in some genres, although micro-timing deviations can also be

detrimental to groove (Davies et al. 2013;Fr¨uhauf, Kopiez, and Platz 2013).

Another interesting aspect of this parameterization is that for a pattern comprising

only minimal polygons, even when they have an irrational number of turns (relative to

one another), the resulting pattern has an upper bound of the number of diﬀering interval

sizes that is smaller than when all Ktime classes’ locations were free. This means that

the resulting patterns have distributions of interval sizes with an upper bound on entropy

that is lower than for unparameterized perfectly balanced patterns. Lower interval size

entropy may hold cognitive advantages for such patterns, making them musically useful.

6.2. Perfectly balanced approximations of just intonation

An important trend within modern microtonal practice and theory is the notion of unit-

ing two essentially incommensurable properties of musical scales. The ﬁrst is that scales

should be relatively simple (predictable, low in entropy )11 in terms of their distribution

of interval sizes; the second is that they should contain a preponderance of intervals that

are good approximations of the low integer frequency ratios (just intonation intervals)

that are typically considered as harmonically consonant and melodically ﬁtting. This has

been most rigorously explored (Erlich 2006;Milne, Sethares, and Plamondon 2008) in

terms of the paradigm of the well-formed scales of Carey and Clampitt (1989) or, equiv-

11The entropy of a probability mass function p, where piis the probability of being in state i, is given by

−P

i

pilog pi(Shannon 1948). Entropy can be thought of as a measure of unpredictability. For example, take a

scale in an N-equal universe, and index the Npossible interval sizes by i; let piequal the number of intervals

in the scale corresponding to interval idivided by the total number of intervals in the scale. The entropy of the

resulting distribution is a measure of the unpredictability of the interval sizes in the scale.

22

alently, the moment of symmetry scales (MOS) scales of Wilson (1975). These are scales

that comprise no more than two step sizes distributed in such a way as to ensure the

scale is as even as possible; put diﬀerently, when the two step sizes are represented by

distinct letters, they form a well-formed word, which is some conjugate of a Christoﬀel

word (Dom´ınguez, Clampitt, and Noll 2009). Such scales have a conﬂuence of remarkable

properties; notable here is that they have the lowest entropy distribution of interval sizes

(other than equal step scales). Numerous well-formed scales, with diﬀering numbers of

large and small steps, with diﬀering sizes, have been found that contain good approxi-

mations of just intonation intervals, as detailed in the afore-mentioned Erlich paper, in

online discussion fora (e.g. the “tuning” mailing list and the “Xenharmonic Alliance”

Facebook groups), and the Xenharmonic Wiki.

Analogously, it is interesting to consider if, and to what extent, perfectly balanced

scales might contain numerous good approximations of just intonation intervals; this

would provide an alternative way of structuring consonances in an organized (perfectly

balanced) scalic form. Perfect balance does not, in itself, minimize the entropy of the

distribution of interval sizes, but it does oﬀer an alternative organizational principle that

may be cognitively relevant. To approach this task, we search through scales, with a

given number of pitch classes, to ﬁnd local maxima of a utility function that models the

number and accuracy of high-aﬃnity intervals in the scale. This is now detailed.

6.2.1. The utility function: Negative spectral entropy

A straightforward way to model the aﬃnity (goodness of ﬁt of pitches played successively

or simultaneously) of all tones in a scale is to model the entropy of the Gaussian smoothed

spectrum they would typically produce. The logic behind this approach is now explained.

If two harmonic complex tones (i.e. tones whose partials are all multiples of a single

fundamental frequency) have a frequency ratio of 3/2, the third harmonic of the lower

tone has the same frequency as the second harmonic of the upper tone. Indeed, in this

example, numerous harmonics in both tones have the same frequency, which suggests they

will have a higher aﬃnity than pairs of tones with fewer spectral matches. Measuring

the similarity of two spectra, therefore, seems a reasonable starting point to model the

aﬃnity of two tones played either simultaneously or successively.

However, the limited frequency resolution of human perception allows a certain amount

of “give and take” in that two tones a few cents larger and smaller than a perfectly

tuned 3/2, will be heard as similarly consonant. This occurs despite the physical fact

that, in these “perturbed” frequency ratios, none of the previously mentioned partials

overlap. A way to account for this is to explicitly model uncertainties and inaccuracies

of pitch perception by convolving, in the frequency domain, the partials’ amplitudes

with a Gaussian kernel. This “smears” each partial’s physical frequency over a small

range of frequencies, hence small discrepancies in frequency between two partials (such

as those within the just noticeable frequency diﬀerence) do not grossly impact on the

amount of spectral overlap and, hence, the model’s predictions of aﬃnity. Indeed, this

is precisely the approach detailed in Milne et al. (2011) and, more extensively, in Milne

(2013) Chapter 3, and successfully tested against experimentally obtained data in Milne,

Laney, and Sharp (2015,2016); Milne and Holland (2016).

This model is concerned with the aﬃnity between two tones, or two chords or, more

generally, between two sets of spectra. However, in our utility function we wish to model

the overall aﬃnity of every tone with every other tone, which is a related but diﬀerent

task. One simple way to achieve this is to simply treat the entire spectrum produced by

all scale tones sounding simultaneously as a probability distribution, and calculate its

23

entropy. When the entropy is low, there are numerous overlapping (Gaussian smoothed)

partials, when the entropy is high there are fewer overlapping (Gaussian smoothed)

partials. Negative entropy, therefore, becomes a useful model for the intrinsic aﬃnity of

the scale spectrum.

To be more concrete, we use a spectrum that contains the ﬁrst 12 harmonics of all

tones, each with an amplitude of h−0.6, where his the harmonic number, and smoothed

in the log-frequency domain by a Gaussian kernel with a standard deviation of 6 cents.

Every such partial is treated as a pitch class; that is, all pitches fall within one octave

and the Gaussian convolution is circular over the octave – this is to match the octave

periodicity of the scales under examination. All of these parameters have been chosen

because they correspond with values previously optimized to empirical data in the just

mentioned papers.12 It is also worth mentioning that, when optimizing across 7-tone

scales with no constraint on balance, the “best” scale approximates a meantone diatonic

scale with perfect ﬁfths of approximately 698 cents and major thirds of approximately

388 cents, which provides additional reassurance that the utility function leads to sensible

results (this scale and its spectral entropy are detailed in the caption of Table B1).

6.2.2. Unparameterized search

Because entropy will always be minimized when tones have identical pitch classes, it is

necessary to apply a linear constraint on the scale pitches to ensure no interval drops

below an arbitrary minimum. We have used 20 cents as that minimum because, with

a smoothing kernel with a 6 cents standard deviation, the overlap of partials rapidly

increases when they are closer than approximately 20 cents. Two additional nonlinear

constraints ensure the scale is perfectly balanced: for a scale z, considered as a subset of

the unit circle, these constraints are

K−1

P

k=0

Re(zk) = 0 and

K−1

P

k=0

Im(zk) = 0.

The optimization routine uses negative spectral entropy (deﬁned in Section 6.2.1) as

the utility function, of which we ﬁnd the local maxima. We apply a version of Improving

Hit-and-Run (Zabinsky et al. 1993) augmented by a gradient-based local search. This

algorithm is applied repeatedly, with decisive penalties added to the utility function to

avoid all previously found optima. Therefore, successive iterations ﬁnd the best scale,

the second-best scale, the third-best, and so on. The associated matlab routines are

provided in the accompanying website (Milne 2017). The best scales of each cardinality

are summarized in Table B1, in the Appendix. Corresponding Scala ﬁles are available in

the accompanying website (Milne 2017).

6.3. Matching spectra to perfectly balanced rational scales

A number of interesting perfectly balanced rational patterns occur in N-equal universes

with minimal perfectly balanced primitive patterns (those Ns are listed below Propo-

sition 5.7). Furthermore, rational patterns typically have low entropy interval size dis-

tributions in comparison to the real patterns produced in Section 6.2. Such patterns

would seem, therefore, to be good contenders for musical scales. However, N-TETs with

minimal primitives – like 30-TET, 42-TET, 66-TET, 70-TET, and so forth, typically pro-

vide poor approximations of low-integer frequency ratios in comparison to neighbouring

12It is worth noting also that there is an interesting and successfully tested piano-tuning application Entropy

Piano Tuner, which suggests an optimal tuning for a piano given the inharmonicity of its strings (Hinrichsen

2012). This application also seeks to minimize spectral entropy, although it does not use Gaussian smoothing.

24

N-TETs that do not contain minimal primitives (e.g. 31-TET, 41-TET, 65-TET, and

72-TET).

For this reason, we are currently exploring the use of spectral matching to maxi-

mize the aﬃnity of tones using such scales, as in Milne, Laney, and Sharp (2016).

To do this we use the freeware Dynamic Tonality synthesizers available at http:

//www.dynamictonality.com choosing spectral tunings where all partials fall at some

N-th division of the octave, where Nis the smallest universe of the pattern. Examples

are provided on the accompanying website (Milne 2017).

7. Discussion

At this stage, we can only conjecture that perfect balance, or balance more generally, is

a perceptible or aesthetically meaningful property of periodic patterns of pitch or time.

In future research, we hope to undertake experimental tests of the recognizability of

balance, in particular by focusing on examples where balance and evenness are distinctly

diﬀerent.

Having said that, our experience of perfectly balanced rhythms and scales suggests

they hold aesthetic promise, and XronoMorph has received an enthusiastic reception

from musicians and music producers (e.g. as evidenced by the user comments provided

on http://www.dynamictonality.com/XMreviews.htm). In this section, we discuss one

key characterization of perfect balance that may contribute to its aesthetic appeal. We

also discuss a number of other future avenues for research and practice.

7.1. Generalized polyrhythms

In the rational case, Theorem 5.4 shows that all perfectly balanced patterns are an integer

combination of equal-step patterns; that is, they are sums of positively- and negatively-

weighted isochronous pulses. This allows us to understand the rhythms produced by

XronoMorph (i.e. sums of independently rotated minimal perfectly balanced patterns)

as a generalization of the familiar polyrhythms that form such an important part of

sub-Saharan African music (Arom 1991) and its “diaspora,” including jazz.

Polyrhythms are commonly deﬁned as comprising two or more isochronous voices with

diﬀering speeds (tempos), where those speeds have simple coprime ratios such as 2:3, 3:4,

2:5, 3:5, 3:7, 3:8, and so forth; furthermore, there is one time location where voices sound

simultaneously (one intersection). Being the sum of equal-step patterns, these canonical

polyrhythms are perfectly balanced: a familiar example is the 2:3 cross rhythm, shown

in Figure 9(a).

The XronoMorph parameterization allows for such polyrhythms, but generalizes them

in two ways. Firstly, the isochronous rhythmic levels (regular polygons) can be individu-

ally rotated so they never coincide, as in the rhythmic version of the double harmonic scale

which comprises two digons and an equilateral triangle, as illustrated in Figure 9(b).13

It also means that rhythms can be constructed where there are multiple points of in-

tersection, rather than just one. Rhythms such as these are more akin to the African

polyrhythms detailed by Arom (1991), which are simultaneous, but sometimes intersect-

ing (“partially interweaving” in Arom’s terminology) rhythmic streams. An example of

a perfectly balanced such pattern of accents is found in the Aka pygmy m`o.nz`ola dance

rhythm, observed by Arom and illustrated in Figure 9(c). Here, the two digons represent

13This is a theoretical generalization of polyrhythms perhaps ﬁrst made by Hofstadter (1985), Chapter 9.

25

(a) A standard 2:3

polyrhythm in N= 12

with a single intersection

(ringed disk).

(b) A generalized 2:3

polyrhythm in N= 12,

with the coprime poly-

gons rotated to avoid

intersections: a rhythmic

analog of the double

harmonic scale.

(c) The perfectly bal-

anced m`o.nz`oli 2:3:4

dance rhythm in N= 12

with multiple points

of intersection (ringed

disks).

Figure 9. Three perfectly balanced generalizations of polyrhythms. Soniﬁcations are available in the accompanying

website (Milne 2017).

(a) “Deal With It”:

a generalized 2:4:7

polyrhythm in N= 28

with multiple points

of intersection (ringed

disks).

(b) “Dots. . . ”: a gener-

alized 2:3:5 polyrhythm

in N= 30 using min-

imal primitive perfectly

balanced patterns. It has

no intersections.

Figure 10. Two self-composed generalized polyrhythms. Soniﬁcations are available in the accompanying website

(Milne 2017).

a single rhythmic voice (the ng´u´e played on the wide end of drum), the two triangles rep-

resent a second musical voice (the `end`omb`a played on the narrow end of the drum), while

the square is a third rhythmic voice (the di.kp`akp`a played on the body of the drum).

We have found these types of polyrhythm to provide fertile ground for compositional

purposes – a self-composed 2:4:7 polyrhythmic pattern, which uses multiple intersections

is illustrated in Figure 10(a).

Secondly, the minimal primitive polygons form another intriguing generalization: here,

some of the regular polygons (isochronous beats) are negatively weighted; such beats are

never heard directly, but they have a “ghostly” impact in that they cancel out (silence)

positively weighted beats. Figure 10(b) shows a self-composed rhythm, called “Dots .. .,”

which combines regular and minimal primitive polygons, individually rotated so as to

avoid intersections – what Arom would call “strict interweaving.” Audio examples are

provided in the accompanying website (Milne 2017).

26

7.2. Future directions

In this subsection we brieﬂy outline some possibilities for future investigation.

7.2.1. Maximizing balance under constraints

The focus of this paper has been on perfect balance. Contrarily, with evenness the focus

tends to be on maximization under musically useful constraints that actually prohibit

perfect evenness. The two most commonly used constraints are to maximize the evenness

of a K-pitch/time class pattern in an N-equal universe; when Kand Nare coprime

(i.e. their greatest common divisor is 1), the resulting pattern has two step-sizes and is

commonly called “Euclidean” (Toussaint 2013). A more general constraint is to maximize

evenness given a prescribed number of large steps and a prescribed number of small

steps. This results in the family of well-formed (or MOS) scales and rhythms, which

are a superset of Euclidean and contain irrational patterns. In both cases, we see how

the use of constraints turns perfect evenness – a rather bland property that results only

in equal-step scales and isochronous rhythms – into the more interesting, and deeply

structured, maximally even patterns.

With perfect balance, however, this is not so obviously the case. Firstly, perfectly

balanced patterns already take a wide variety of diﬀering forms (unlike perfectly even

patterns). Secondly, deep structure seems to arise, most obviously at least, in the case

of perfectly balanced rational patterns which, as shown in Theorem 5.4, are all integer

combinations of equal-step patterns. This is not to say that maximizing balance under

constraints where perfect balance is impossible (e.g. choosing a prime Nand K < N) is

of no interest. It is simply an aspect of balance that we have not explored.

7.2.2. Perfecting balance with diﬀering real-valued weights

We have also focused on binary-weighted patterns and on sums of binary patterns as

used in conventional polyrhythms. However, balance is a meaningful property of, and is

deﬁned for, patterns with any weights (see Deﬁnition 3.14). This allows us to balance

musical scales and rhythms whilst also taking into account their pitch/time classes’

individual loudnesses, durations, prevalences, and so forth. Having said that, under such

an approach, the deep structure that arises from binary-weighted rational patterns would

no longer hold.

7.2.3. Multi-voiced scales

In non-minimal perfectly balanced patterns, perfectly balanced subsets of the whole

pattern can each be assigned to a diﬀerent voice. This means that the scale used by each

voice is perfectly balanced as well as their combination. This is how XronoMorph operates

with rhythms (when in “PB” mode). Extending this to scales presents an interesting

extension, which can be exempliﬁed by the perfectly balanced double harmonic scale {C,

D[, E, F, G, A[, B}. This scale can be broken into the perfectly balanced subsets {C, E,

A[},{D[, G}, and {F, B}, with each of these subsets assigned to a diﬀerent voice. An

analogous procedure could also be used for well-formed patterns, which can be broken

into distinct well-formed subsets (Milne and Dean 2016).14

14For example, the white-note diatonic scale can be broken into the well-formed subsets {C, F, G},{D, A}and

{E, B}.

27

7.2.4. Minimal primitives with height greater than one

Intriguingly, for certain N-equal universes, there are minimal perfectly balanced primitive

rational patterns that are not uniformly weighted. Steinberger (2008) connects minimal

perfectly balanced patterns to cyclotomic polynomials, and deﬁnes the height as the

greatest weight of a pattern. He provides a construction (ibid., Theorem 1.1) of a minimal

perfectly balanced pattern of arbitrary height cin an N-even universe where N= 3pq

and pand qare the two smallest primes greater than 2c. Accordingly, such minimal

patterns are seen only in relatively high N; the ﬁrst example is a minimal perfectly

balanced pattern in N= 105, with a height of 2, shown in Figure 1 (ibid.).

7.2.5. Perceptual/cognitive tests

Finally, we hope in due time to experimentally test whether balance is a recognizable

property of musical scales and rhythms. Despite the strong positive relationship between

balance and evenness, Figures 3and 4show that it is possible to select patterns across

which balance and evenness are almost completely independent (e.g. by selecting patterns

that fall within narrow horizontal or vertical stripes of the scatter plots). These may

be useful for testing the perceptual and cognitive properties arising from balance and

evenness.

8. Conclusion

We have provided a number of key results to characterize and analyse balance in relation

to evenness (4.1–4.4), and the properties of perfect balance for real patterns (5.1–5.3)

and for rational patterns (5.4–5.7). We have explored the relationship of perfectly bal-

anced real patterns to perfectly balanced rational patterns and to general real patterns

(Section 5.3). We have developed musically meaningful methods for choosing subsets of

the perfectly balanced manifold and parameterizing them (Section 6). In this way, we

have shown how balance can inform music theory, analysis, and practice, and we have

demonstrated some musical uses of perfect balance with concrete examples (Section 7).

Acknowledgements

We thank Emmanuel Amiot for pointing us towards Lam and Leung’s seminal work

and whose brief but fascinating report (Amiot 2010) inspired some of this research. We

thank Winfried Bruns for advising us on the use of Normaliz and taking an enthusiastic

interest in our pattern search problem. We also thank Gareth Hearne for pointing out

the anomalous properties of the “boundary condition” pattern with only one pitch/time

class, as detailed in Remark 4.3.

We would also like to thank the two anonymous reviewers and the co-Editors-in-Chief

Thomas Fiore and Clifton Callender for helpful comments, suggestions, and detailed

scrutiny of the manuscript, which led to signiﬁcant improvements in its clarity.

Funding

Dr Andrew Milne is the recipient of an Australian Research Council Discovery Early

Career Researcher Award (project number DE170100353) funded by the Australian Gov-

28

ernment.

Supplemental online material

Supplemental online material for this article can be accessed at http://www.

dynamictonality.com/perfect_balance_files/. On this website, we provide a video

demonstration of balance using a bicycle wheel, some videos of XronoMorph gen-

erating perfectly balanced rhythms, some musical examples, and many of the com-

putational routines used in the paper. XronoMorph can be downloaded from http:

//www.dynamictonality.com/xronomorph.htm, and additional related videos can be

viewed at https://www.youtube.com/c/xronomorph/.

Disclosure statement

No potential conﬂict of interest was reported by the authors.

ORCID

Do not change this. Production will take care of it if the paper is accepted.

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31

Appendix A. Minimal perfectly balanced binary-weighted patterns

Table A1. All 9 minimal perfectly balanced patterns, up to rotation, in

N= 30.

Pattern’s indicator vector (for readability, grouped in sixes) Name

000000 000000 001000 000000 000001 2-gon

000000 000100 000000 010000 000001 3-gon

000001 000001 000001 000001 000001 5-gon

000000 000110 000010 000010 000011 6-in-30-gon

000000 000110 000011 000000 000111 7a-in-30-gon

000001 000001 100000 100010 000011 7b-in-30-gon

000000 000111 000001 000100 000111 8a-in-30-gon

000001 000101 000100 000110 000011 8b-in-30-gon

000001 000101 000101 000100 000111 9-in-30-gon

Table A2. All 21 minimal perfectly balanced patterns, up to rotation, in

N= 42.

Pattern’s indicator vector (for readability, grouped in sixes) Name

000000 000000 000000 001000 000000 000000 000001 2-gon

000000 000000 010000 000000 000100 000000 000001 3-gon

000001 000001 000001 000001 000001 000001 000001 7-gon

000000 000001 100001 000001 000001 000001 000011 8-in-42-gon

000000 000001 100001 000011 000000 000001 100011 9a-in-42-gon

000000 000011 000010 000010 000110 000000 000111 9b-in-42-gon

000000 010000 110000 100000 100000 100001 100001 9c-in-42-gon

000000 000001 100011 000010 000000 100001 100011 10a-in-42-gon

000000 000011 000010 000110 000100 000001 000111 10b-in-42-gon

000000 000011 000110 000000 000111 000000 000111 10c-in-42-gon

000000 000011 100010 000000 100001 100001 000011 10d-in-42-gon

000001 000011 000010 000010 000110 000100 000101 10e-in-42-gon

000000 000011 000110 000100 000101 000001 000111 11a-in-42-gon

000000 000011 100010 000010 100000 100001 100011 11b-in-42-gon

000000 000111 000000 000111 000100 000001 000111 11c-in-42-gon

000000 010000 110001 100001 000000 010001 110001 11d-in-42-gon

000001 000011 000010 000110 000100 000101 000101 11e-in-42-gon

000000 000111 000100 000101 000101 000001 000111 12a-in-42-gon

000000 010001 110001 000001 010000 010001 110001 12b-in-42-gon

000001 000011 000110 000100 000101 000101 000101 12c-in-42-gon

000001 000111 000100 000101 000101 000101 000101 13-in-42-gon

32

Table A3. All 57 minimal perfectly balanced patterns, up to rotation, in N= 70.

Pattern’s indicator vector (for readability, grouped in ﬁves) Name

00000 00000 00000 00000 00000 00000 00001 00000 00000 00000 00000 00000 00000 00001 2-gon

00000 00000 00010 00000 00000 00100 00000 00000 01000 00000 00000 10000 00000 00001 5-gon

00000 00001 00000 00001 00000 00001 00000 00001 00000 00001 00000 00001 00000 00001 7-gon

00000 00000 00010 01000 00000 01100 00000 01000 00000 01000 00000 11000 00000 01001 10-in-70

00000 00000 00010 01000 00000 11100 00000 01001 00000 00000 00010 11000 00000 01101 13a-in-70

00000 00001 00010 00000 00010 01100 00000 01100 00000 01000 00000 11000 00000 11001 13b-in-70

00000 00001 00100 00010 00000 01001 00100 00000 10000 01001 00000 00001 10000 00011 13c-in-70

00000 00001 00100 10000 00000 11001 00000 10001 00000 10001 00000 10001 00000 10011 15a-in-70

00000 00010 00000 01010 00000 01110 00000 01010 00000 01000 00000 11010 00000 01011 15b-in-70

00000 00000 00000 11010 00000 01111 00000 00010 00000 01000 00000 11110 00000 01011 16a-in-70

00000 00000 00010 11000 00000 11101 00000 00001 00010 00000 00010 11100 00000 01101 16b-in-70

00000 00001 00010 10000 00010 01101 00000 00100 00010 01000 00000 11100 00000 11001 16c-in-70

00000 00001 00100 10000 00000 11001 00000 10011 00000 00001 00100 10000 00000 11011 16d-in-70

00000 00001 00100 10010 00000 01001 00100 10000 00000 11001 00000 10001 00000 10011 16e-in-70

00000 00001 00110 00000 00110 01000 00100 01000 00100 01000 00100 11000 00000 11001 16f-in-70

00000 00001 00110 00000 01110 00000 00100 10000 01000 00001 01100 10000 00010 10001 16g-in-70

00000 00001 10100 00010 00000 01001 00100 00010 10000 01001 00100 00000 10000 01011 16h-in-70

00000 00010 00000 01110 00000 01110 00000 01000 00000 11000 00000 11011 00000 00011 16i-in-70

00000 00011 00000 00110 00000 00110 01000 00100 01000 10000 01000 10001 00000 10011 16j-in-70

00000 00100 10000 01001 00000 00101 10000 00011 00000 00101 00000 00110 00000 01101 16k-in-70

00000 00000 00000 11000 00000 11111 00000 00011 00000 00000 00000 11110 00000 01111 17a-in-70

00000 00001 00000 10010 00000 01111 00000 00110 00000 01000 00000 11100 00000 11011 17b-in-70

00000 00001 00100 10000 00000 11011 00000 00011 00100 00000 00100 11000 00000 11011 17c-in-70

00000 00001 00100 10010 00000 01001 00100 10010 00000 01001 00100 10000 00000 11011 17d-in-70

00000 00001 00100 10010 00000 01011 00100 00000 00100 11000 00000 11001 00000 10011 17e-in-70

00000 00001 00110 10000 00010 01001 00100 00000 00110 01000 00100 11000 00000 11001 17f-in-70

00000 00001 10110 00000 01110 00000 00100 00000 11000 00001 11100 00000 10010 00001 17g-in-70

00000 00011 00000 00110 00000 01110 00000 01100 00000 11000 00000 11001 00000 10011 17h-in-70

00000 00011 00100 00010 00100 00010 01100 00000 01100 10000 01000 10001 00000 10011 17i-in-70

00000 00100 00000 11010 00000 01101 00000 10010 00000 01001 00000 10110 00000 01011 17j-in-70

00000 00000 00000 11100 00000 11111 00000 00001 00000 10000 00000 11111 00000 00111 18a-in-70

00000 00001 00000 10110 00000 01111 00000 00100 00000 11000 00000 11101 00000 10011 18b-in-70

00000 00001 00100 10010 00000 01011 00100 00010 00100 01000 00100 11000 00000 11011 18c-in-70

00000 00001 00110 00000 00110 11000 00000 11001 00000 00001 00110 10000 00010 11001 18d-in-70

00000 00001 00110 10000 00010 01001 00100 10000 00010 01001 00100 10000 00010 11001 18e-in-70

00000 00001 10010 00001 01110 00000 00110 00000 01000 00001 11100 00000 11010 00001 18f-in-70

00000 00001 10110 00000 00110 01000 00100 01000 10000 01001 10100 00000 10010 01001 18g-in-70

00000 00011 00100 00010 00100 01010 00100 01000 00100 11000 00000 11001 00000 10011 18h-in-70

00000 00100 10000 01010 00000 01101 10000 00010 10000 01001 00000 00110 10000 01011 18i-in-70

00000 00110 00000 01100 00000 11100 00000 11001 00000 10001 00000 10011 00000 00111 18j-in-70

00000 00001 00000 10100 00000 11111 00000 00101 00000 10000 00000 11101 00000 10111 19a-in-70

00000 00001 00010 10000 00010 11101 00000 00101 00010 00000 00010 11100 00000 11101 19b-in-70

00000 00001 00110 10000 00010 11001 00000 10001 00010 00001 00110 10000 00010 11001 19c-in-70

00000 00011 00100 00010 00100 01010 00100 01010 00100 01000 00100 11000 00000 11011 19d-in-70

00000 00011 01000 00100 00000 10110 01000 00101 01000 10000 01000 10001 00000 10111 19e-in-70

00000 00011 10100 00010 00100 01000 00100 01010 10000 01001 10100 00000 10000 01011 19f-in-70

00000 00100 00000 11100 00000 11101 00000 10001 00000 10001 00000 10111 00000 00111 19g-in-70

00000 00100 10010 01001 00000 00101 10010 00001 00010 00101 00010 00100 00010 01101 19h-in-70

00000 00100 11010 00001 01000 00100 10010 00001 01000 00101 10010 00000 01010 00101 19i-in-70

00000 00101 00000 10110 00000 01101 00000 10100 00000 11001 00000 10101 00000 10011 19j-in-70

00000 00110 01000 00100 01000 10100 01000 10001 01000 10001 00000 10011 00000 00111 19k-in-70

00000 00101 00000 10100 00000 11101 00000 10101 00000 10001 00000 10101 00000 10111 20a-in-70

00000 00101 10010 00001 00010 00101 00010 00101 00010 00101 00010 00100 00010 01101 20b-in-70

00000 00101 00010 10000 00010 11101 00000 10101 00010 00001 00010 10100 00010 11101 22a-in-70

00000 00111 01000 00100 01000 10100 01000 10101 01000 10001 01000 10001 00000 10111 22b-in-70

00000 10011 01000 00101 00000 10110 01000 00101 01000 10100 01000 10001 01000 10101 22c-in-70

00000 10101 00010 10001 00010 10101 00010 10101 00010 00101 00010 10100 00010 11101 25-in-70

33

Appendix B. Perfectly balanced approximations of just intonation

Table B1. Perfectly balanced scales with a 1200 cent period with minimal spectral entropy. The ﬁrst section contains 5-tone

scales, the second contains 6-tone scales, and so on. Within each section, scales are ordered by negative spectral entropy

– i.e. the “best” scale is ﬁrst. For comparison, the “best” general (not necessarily perfectly balanced) 7-tone scale is the

meantone-like 0.0,113.2,312.0,502.7,698.8,812.8,1009.8, which has spectral entropy of −1.18. Each graphic in the “interval

content” column shows the number of each interval in the scale when each scale pitch is smeared (convolved) with a Gaussian

kernel with a standard deviation of 6 cents. The kernels are “normalized” to ensure that when the scale has nidentically

sized intervals, the peak at that interval’s size will be precisely n; when there are nintervals with slightly diﬀering sizes the

peak will be less than n.

Pitch classes (cents) Spectral

entropy Interval content

0.0 73.8 501.0 576.0 887.5 −1.28

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 164.6 493.1 658.4 928.7 −1.24

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 205.0 472.5 702.7 932.1 −1.22

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 111.9 498.8 611.2 905.3 −1.21

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 103.9 495.9 600.0 703.9 1095.9 −1.18

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

0.0 76.3 385.2 495.4 781.5 889.5 −1.16

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 83.7 381.7 580.9 701.0 971.9 −1.14

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 68.1 383.7 562.8 701.1 964.5 −1.14

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 110.6 312.1 499.0 656.6 810.2 999.9 −1.11

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

0.0 112.5 308.1 422.1 694.3 810.8 926.5 −1.07

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

0.0 76.3 392.3 500.0 579.7 887.5 993.3 −1.07

0 1 2 3 4 5 6 7 8 9 10 11

0

2

0.0 103.5 389.9 494.5 600.0 703.5 989.9 1094.5 −1.02

0 1 2 3 4 5 6 7 8 9 10 11

0

4

8

0.0 91.1 389.2 480.8 590.9 704.1 976.1 1090.2 −1.01

0 1 2 3 4 5 6 7 8 9 10 11

0

4

0.0 104.0 281.4 495.8 600.0 704.0 881.4 1095.8 −0.99

0 1 2 3 4 5 6 7 8 9 10 11

0

4

8

0.0 76.4 188.2 393.3 504.2 582.3 781.4 890.9 999.5 −0.95

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

0.0 56.5 262.8 441.1 554.2 644.6 758.4 936.2 1142.1 −0.94

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

0.0 78.6 304.9 385.5 501.1 580.0 696.1 887.5 1005.4 1084.4 −0.91

0 1 2 3 4 5 6 7 8 9 10 11

0

4

8

0.0 103.4 284.5 389.0 495.1 600.0 703.4 884.5 989.0 1095.1 −0.89

0 1 2 3 4 5 6 7 8 9 10 11

0

4

8

0.0 42.3 120.4 314.8 427.7 625.1 700.1 743.3 816.1 1127.1 −0.88

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

0.0 50.4 113.3 317.6 435.2 498.6 549.4 757.0 817.8 934.0 1047.0 −0.82

0 1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

0.0 81.6 195.9 284.9 390.4 499.2 583.8 698.8 781.0 892.3 993.8 1087.1 −0.83

0 1 2 3 4 5 6 7 8 9 10 11

0

4

8

12

34