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Exploring the space of perfectly balanced rhythms and scales

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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 linking 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, computational routines, and video demonstrations of some of the concepts.
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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
une2πin/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>ˇˇ>ˇ?ˇ(>ˇ>ˇˇ>. . .
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, . . . , xK1R/Z, and their associated
weights are denoted w0, . . . , wK1R.
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, . . . , xK1+tR/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, . . . , xK1Q/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, . . . , (N1)/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, . . . , xK1}with associated nonzero
weights w0, . . . , wK1, the resulting pattern is
λ(x) = (wkif x=xk
0 if x6∈ {x0, . . . , xK1}.(1)
The support of the pattern is the given pitch/time-class set {x0, . . . , xK1}. In this
way, each pattern attaches the Knonzero weights w0, . . . , wK1to the unit circle, as in
Figure 1. A pattern can, equivalently, be written as w0ex0+·· ·+wK1exK1, 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 (fg)(x) = P
tR/Z
f(t)g(xt))
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(xt); 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, . . . , xK1}ordered as 0 x0<··· < xK1<1, we define the Argand vector
as (e2πix0, . . . , e2πixK1)CK.6If λis a pattern, we denote its Argand vector by zλ.
The support of any pattern has a unique representation satisfying 0 x0<··· <
xK1<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+
(N1)/N), an indicator vector7u= (u0, . . . , uN1)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(xt)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 ie2πixk, 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=
K1
P
k=0
zke2πijk/K . The m-th coefficient of the discrete Fourier
transform of the length-Nindicator vector uis given by (Fu)m=
N1
P
n=0
une2πimn/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/ZRis one minus the magnitude of
the centre of gravity of the pattern; that is,
B(λ) =
1X
xR/Z
λ(x)e2πix.X
xR/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
K1
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(xt))
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
xR/Z
λ(x)e2πix= 0. Likewise, a pattern λis perfectly imbalanced if it has balance 0;
that is, P
xR/Z
λ(x)e2πix
=P
xR/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,
K1
P
j=0 |(Fz)j|2=K
K1
P
k=0 |zk|2. By definition, all |zk|= 1,
hence
K1
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
xR/Z
(κ+λ)(x)e2πix=X
xR/Z
κ(x)e2πix+X
xR/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 NKpitches (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 mZ. 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
xR/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(pi1)/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(N1)/N , so that the lattice of such
patterns has dimension Nϕ(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 maxjujminjujand maxkvkminkvk. 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/pq1) and v=v(0) +·· · +v(N/pq1) 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(p1)/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,0.60.8i,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
K1
P
k=0 (ukxk)2+ (vkyk)2is minimized subject to the constraints u2
k+v2
k= 1 for
all k, and
K1
P
k=0
uk=
K1
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=
K1
X
k=0 (ukxk)2+ (vkyk)2+
K1
X
k=0
λku2
k+v2
k1+µ
K1
X
k=0
uk+ν
K1
X
k=0
vk,
whose stationary points occur when
L
∂uk
= 2((1 + λk)ukxk) + µ= 0 and
L
∂vk
= 2((1 + λk)vkyk) + 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 + λkxk
yk1
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 K1
P
k=0
uk2
+K1
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 K2 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=(xAK:
K1
X
k=0
cos(2πxk) = 0 =
K1
X
k=0
sin(2πxk)).
Proposition 6.1 The subspace BKof AKis a manifold for every K2.
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 rR/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:AKR2given by f(x0, . . . , xK1) =
PK1
k=0 cos(2πxk),PK1
k=0 sin(2πxk). The pre-image f1{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 xBK, then there
must be two components xiand xjwith xjxi=ρfor ρ(0,1/2), so that
∂f1(x)
∂xi
∂f1(x)
∂xj
∂f2(x)
∂xi
∂f2(x)
∂xj
= 2πsin(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 h0.6, 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
K1
P
k=0
Re(zk) = 0 and
K1
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´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 (4.14.4), and the properties of perfect balance for real patterns (5.15.3)
and for rational patterns (5.45.7). 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
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