Content uploaded by Andrew J. Milne

Author content

All content in this area was uploaded by Andrew J. Milne on Jun 19, 2015

Content may be subject to copyright.

Perfect Balance: A Novel Principle for the

Construction of Musical Scales and Meters

Andrew J. Milne1, David Bulger2, Steﬀen A. Herﬀ1, and William A. Sethares3

1MARCS Institute, University of Western Sydney, NSW 2751, Australia

{a.milne,s.herff}@uws.edu.au

2Macquarie University, Sydney, NSW 2109, Australia

david.bulger@mq.edu.au

3University of Wisconsin-Madison, WI 53706, USA

sethares@wisc.edu

Abstract. We identify a class of periodic patterns in musical scales or

meters that are perfectly balanced. Such patterns have elements that are

distributed around the periodic circle such that their ‘centre of grav-

ity’ is precisely at the circle’s centre. Perfect balance is implied by the

well established concept of perfect evenness (e.g., equal step scales or

isochronous meters). However, we identify a less trivial class of perfectly

balanced patterns that have no repetitions within the period. Such pat-

terns can be distinctly uneven. We explore some heuristics for generating

and parameterizing these patterns. We also introduce a theorem that

any perfectly balanced pattern in a discrete universe can be expressed

as a combination of regular polygons. We hope this framework may be

useful for understanding our perception and production of aesthetically

interesting and novel (microtonal) scales and meters, and help to dis-

ambiguate between balance and evenness; two properties that are easily

confused.

Keywords: Music, Scales, Meters, Balance, Evenness, Microtonal, Dis-

crete Fourier transform

1 Introduction

Aperfectly balanced pattern is a set of points on a circle whose mean position, or

centre of gravity, is the centre of the circle (see Fig. 1). A perfectly even pattern

is one in which the elements are equally spaced around the periodic circle (see

Fig. 1(a)). More generally, the balance or evenness of a pattern is a measure of

how closely it conforms, respectively, to perfect evenness or to perfect balance

(formal deﬁnitions are provided in Sect. 2). Evenness has been identiﬁed as an

important principle for the construction and analysis of scales and meters [1–3].

However, much research involving evenness has proceeded seemingly unaware

that, in many common examples (e.g., well-formed scales [4]), it is strongly

associated with balance – indeed, perfect evenness implies perfect balance. For

instance, it may be that the musical utility of well-formed patterns derives from

2 Perfect Balance

them having high balance as well as high evenness. In order to tease apart these

two properties, we will demonstrate a number of interesting patterns that are

perfectly balanced but also distinctly uneven and irreducibly periodic. In this

paper, we do not empirically test the recognizability, utility, likeability, and so

forth, of balance; rather, we lay down some of the mathematical and conceptual

framework around which future empirical work may be conducted.

(a) Perfectly balanced,

perfectly even, sub-

periodic.

(b) Perfectly balanced,

uneven, sub-periodic.

(c) Perfectly balanced,

uneven, irreducibly peri-

odic (no sub-periods).

Fig. 1. Three perfectly balanced periodic patterns exhibiting diﬀerent classes of even-

ness and sub-periodicity. The small circles represent a universe of available pitch classes

or metrical times (in these examples, there are twelve, which might correspond to twelve

chromatic pitch classes or twelve metrical pulses). The ﬁlled circles are the notes or

beats making the pattern under consideration.

Fig. 1 presents some simple patterns to elucidate the above-mentioned prop-

erties; they are all perfectly balanced, but they exhibit diﬀerent classes of even-

ness and reducibility of the period. The pattern in Fig. 1(a) is perfectly even (it

might represent a whole tone scale or a 6

4meter).

The pattern in Fig. 1(b) is diﬀerent because it is uneven (it can represent a

diminished scale or a triplet shuﬄe). However, both (a) and (b) have rotational

symmetries; for example, if (a) is compared with a version that has been rotated

by 60◦, the locations of all the ﬁlled circles will perfectly align; the same follows if

(b) is rotated by 90◦. This is because (a) has a fundamental sub-period subtend-

ing 60◦, while (b) has a fundamental sub-period subtending 90◦. A fundamental

sub-period is fundamental in that it is the smallest-sized period of repetition in

the pattern, and all other periods are multiples of it; it is a sub-period because

it subtends an angle smaller than the full circle. A circular pattern with sub-

periods is described as reducibly periodic, a circular pattern with no sub-periods

is described as irreducibly periodic. Importantly, although both (a) and (b) are

perfectly balanced over the whole circle, neither is perfectly balanced over its

fundamental sub-period (this is explained in greater detail in Sect. 3).

The pattern in Fig. 1(c) is particularly interesting because it is perfectly bal-

anced, uneven, and it has no sub-periods. This means the pattern is perfectly

balanced over its fundamental period and, hence, over all its possible periods.

Perfect Balance 3

We describe such a pattern as having irreducibly periodic perfect balance, and

this is the class of patterns this paper focuses on. Such patterns may form useful

templates for novel microtonal scales and meters. Furthermore, the clear sepa-

ration of evenness and balance may allow the impact of both properties, with

regards to perception and action, to be independently measured.

As previously mentioned, evenness and balance are closely intertwined. The

next section aims to demonstrate some connections and diﬀerences between

them, while the section after investigates balance itself.

2 Evenness and Balance

A way to demonstrate the relationships between evenness and balance is to

express a periodic pattern as a complex vector and take its discrete Fourier

transform. Vector x∈[0,1)Khas Kreal-numbered pitch or time values between

0 and 1, ordered by size so x0< x1<· · · < xK−1(the period has a size of 1).

For instance, for the diatonic scale in a standard 12-tet tuning, the vector

x=0

12 ,2

12 ,4

12 ,5

12 ,7

12 ,9

12 ,11

12 . The elements of this vector are mapped to the

unit circle in the complex plane with z[k]=e2πix[k]∈C, so z∈CK. Each

complex element z[k] of zhas unit magnitude, and its angle represents its time

location or pitch as a proportion of the period (whose angle is 2π). We term this

vector the scale vector.

We will also use an alternative vector representation of a periodic pattern

that is suitable for patterns whose scale vector comprises only rational values.

This indicator vector is denoted a∈ {0,1}Nand is given by a[n] = [n/N ∈x]

(for n= 0, . . . , N −1), where Nis the cardinality of the chromatic universe

(which must be some multiple of 1/gcd(x0, x1, . . . , xN−1)) and the square (Iver-

son) brackets denote an indicator function that is unity when the enclosed

relation is true, otherwise zero. Hence the previous 12-tet diatonic scale is

a= (1,0,1,0,1,1,0,1,0,1,0,1).

The tth coeﬃcient of the discrete Fourier transform of the scale vector is

given by

Fz[t] = 1

K

K−1

X

k=0

z[k] e−2πitk/K .(1)

We will use the zeroth and ﬁrst coeﬃcients to characterize balance and evenness.

2.1 Evenness – the First Coeﬃcient

As ﬁrst shown by Amiot [5], the magnitude of the ﬁrst coeﬃcient gives the

evenness of the pattern:

evenness =|Fz[1]| ∈ [0,1] , where

Fz[1] = 1

K

K−1

X

k=0

z[k] e−2πik/K .(2)

4 Perfect Balance

In statistical terms, evenness is equivalent to unity minus the circular variance

[6] of the circular displacements between each successive term of zand each

successive kth-out-of-Kequal division of the period – if the displacements are

all equal, their circular variance is zero and the pattern is perfectly even.

2.2 Balance – the Zeroth Coeﬃcient

Unity minus the magnitude of the zeroth coeﬃcient gives the balance of the

pattern:

balance = 1 − |F z[0]| ∈ [0,1] , where

Fz[0] = 1

K

K−1

X

k=0

z[k].(3)

In statistical terms, balance is equivalent to the circular variance of the pattern

itself. When a pattern’s balance is 0 (i.e., it is maximally unbalanced), the K

elements all have the same pitch or occur at the same time, so they are maximally

clustered; when the balance is 1 (a condition we term perfect balance), they have

the maximal possible circular variance. Importantly, as we will prove below,

perfect balance does not imply evenness; hence these are two distinct properties.

An equivalent deﬁnition, for rational-valued patterns in an N-fold universe,

can be calculated from the indicator vector, this time using the ﬁrst coeﬃcient:

balance = 1 −N|F a[1]|

K∈[0,1] , where

Fa[1] = 1

N

N−1

X

n=0

a[n] e−2πin/N .(4)

2.3 Relationships Between Evenness and Balance

Theorem 1. Perfect evenness implies perfect balance.

Proof. Under Parseval’s theorem, PK−1

t=0 |Fz[t]|2=1

KPK−1

k=0 |z[k]|2. By deﬁni-

tion, all |z[k]|= 1, hence PK−1

t=0 |Fz[t]|2= 1. When |F z[1]|= 1 (perfect even-

ness), all other coeﬃcients of Fzmust, therefore, be zero. ut

Theorem 2. Maximal imbalance implies maximal unevenness.

Proof. The proof follows the same line of argument as that for Theorem 1 but

using the zeroth coeﬃcient of Fzinstead of the ﬁrst. ut

Theorem 3. Perfect balance does not imply perfect evenness.

Proof. This can be simply proven by example (as shown in Sects. 1 and 3). ut

Theorem 4. In a perfectly even N-fold universe, the complement of a perfectly

balanced pattern is also perfectly balanced.

Perfect Balance 5

Proof. The proof is trivial. ut

Remark 1. This theorem parallels how, in an N-fold chromatic universe, the

complement of a maximally even scale of Kpitches is the maximally even scale of

N−Kpitches [7, Proposition 3.2]. For example, the complement of the diatonic

scale in the 12-fold chromatic scale is the pentatonic – more prosaically, the

piano’s black notes ﬁll in all the gaps between the white notes.

Having established the above relationships between evenness and balance, we

now turn our attention to the principle of balance itself.

3 Balance

Here is a physical analogy of balance. Imagine a vertically oriented bicycle wheel

that can rotate freely about a horizontal axle. The wheel has Nequally spaced

slots around its circumference. We also have Kweights all of the same mass and,

into each slot, a single weight may be placed. Each slot represents a periodic

pitch or time, and each weight represents an event at that pitch or time. In

totality, therefore, they may be thought of as representing a scale or a meter

(as described earlier). When the Kweights are placed in the wheel’s slots, after

any perturbation, the wheel will always rotate into a stable position so that

its ‘heaviest’ part is pointing vertically down. Phrased more mathematically, it

will rotate (under the action of gravity) until the sum of the Kvectors – each

pointing from the wheel’s centre to a weight – is pointing vertically downwards.

However, there is a class of perfectly balanced patterns where the wheel has

no preferred orientation in that, provided it is not spun when released, the wheel

will remain in whatever rotational position it is left at. This arises from a pattern

whose sum of vectors is nil – as shown in (3). An alternative visualization is to

think of a horizontal disk resting on a vertical pole at its centre. As alluded to in

Sect. 1, the disk will balance only if the centre of gravity is at the disk’s centre.

As shown in (4), the balance of any K-element pattern in an N-fold universe

can be calculated from the indicator vector a. Using this method, Lewin [8] de-

scribes a scale where this coeﬃcient is zero as having the ‘exceptional’ property.

This is the property we call perfect balance. Building on Lewin’s insights, Quinn

[9] clearly describes the meaning of this and the other coeﬃcients as representing

diﬀerent types of ‘balance’. However, often the distinction between evenness and

balance has not been adequately explored. For example, Callender [10] describes

this coeﬃcient as ‘a measure of how unevenly a set divides the octave’ which

is correct (as shown in Theorem 2) but does not mention the more interesting

property that, when this value is minimized (i.e., balance is maximized), even-

ness is not implied (as shown in Theorem 3). As we discuss later, Amiot [11] has

recently conducted a search for perfectly balanced patterns in a 30-fold universe.

A simple and graphical way to obtain perfect balance is to place the weights

at the vertices of regular polygons – e.g., a digon, equilateral triangle, square,

regular hexagon, and so forth, as shown in Fig. 2. Clearly, the greater the number

of divisors of N, the greater the number of diﬀerent regular polygons available.

6 Perfect Balance

(a) Digon. (b) Equilateral tri-

angle.

(c) Square. (d) Regular

hexagon.

Fig. 2. Perfectly balanced regular polygons in a twelve-fold period.

But these are rather trivial patterns in that they are perfectly even and

each actually comprises Ksmaller identical patterns of length N/K – in other

words, their fundamental periods are 1/K of the circle (put diﬀerently, they

have rotational symmetry of order K). How might we create more interesting,

less even, irreducibly periodic, but still perfectly balanced structures?

We could take a copy of our polygon, rotate it by a distance less than that

separating its vertices, then add it to the original. For instance, we can take an

equilateral triangle and rotate it by one chromatic step and add it to the original

to give an augmented (hexatonic) scale as illustrated in Fig. 3(a). This appears

to create a perfectly balanced and interestingly uneven pattern; however, the

resulting scale still has sub-periods (of length N/n, where nis the number of

vertices in the repeated polygon). Indeed, the pattern in Fig. 3(a) consists of

a fundamental sub-period that repeats three times within the circle. And if we

stretch out this smaller pattern so it takes up a full circumference, as shown

in Fig. 3(b), we can see how it is actually unbalanced (the sum of vectors is

non-zero) over its fundamental period.

(a) Two displaced equilateral

triangles make the augmented

scale.

(b) The augmented scale over its

fundamental (third-octave) pe-

riod.

Fig. 3. A reducible pattern, which is perfectly balanced over a 12-fold period, but not

over its fundamental 4-fold sub-period (as shown by the resultant vector in (b)).

Perfect Balance 7

Similarly – as shown in Fig. 4 – we can take the (N−K)-element complement

of any of the K-vertex regular polygons in Fig. 2 (Theorem 4). But these patterns

also have sub-periods, also of lengths N/K, as the complement of any of the K-

vertex regular polygons is a simple combination of diﬀerent rotations of the

original polygon.

(a) Complement of

digon.

(b) Complement of

equilateral triangle.

(c) Complement of

square.

(d) Complement of

regular hexagon.

Fig. 4. Perfectly balanced, but sub-periodic, complements of regular polygons in a

twelve-fold period. Because they are sub-periodic, they are ‘modes of limited transpo-

sition’ corresponding to Messiaen’s: (a) seventh mode, (b) third mode, (c) second mode

(diminished scale or triplet shuﬄe), (d) ﬁrst mode (whole tone scale or 6

4meter) [12].

The impact of such sub-periods may diﬀer depending on whether the context

is scalic or metrical. In a scalic context, the octave is an interval over which pe-

riodicity is often perceived (pitches an octave apart are typically heard as being,

in some sense, equivalent). This means that the smaller sub-periods within the

octave may be perceptually subsumed by the periodicity of the larger octave.

For example, even though the augmented scale in Fig. 3(a) has repetition every

quarter-octave, the most dominant perceived period of repetition may still be

heard at the octave. In a metrical context, however, there is no speciﬁc duration

that is perceptually privileged, hence sub-periods may be more easily perceived

as perceptually dominant. This might suggest that irreducible periods are more

obviously related to a metrical rather than a scalic context. However, these pos-

sible diﬀerent impacts of sub-periodicity do not imply that perfect balance –

as a general principle – is not equally applicable to meters and scales. In both,

irreducibly periodic patterns may be more useful due to their greater complexity

and less obvious construction. Furthermore, irreducibly periodic scales have mu-

sically useful properties not found in reducibly periodic scales; for example, they

have Ndistinct transpositions, and every diﬀerent scale degree is surrounded by

a diﬀerent sequence of intervals (Balzano’s property of uniqueness [13]).

So, is there a way to create an uneven and irreducibly periodic pattern that

is also perfectly balanced? In the following subsection we will describe a simple

heuristic method. In the subsection after that, we will describe an extension that

enables us to ﬁnd a diﬀerent class of perfectly balanced structures.

8 Perfect Balance

3.1 Heuristics for Irreducibly Periodic Perfect Balance

Coprime Disjoint Regular Polygons Add regular polygons (each expressed

as an indicator vector) such that no two vertices have the same location (if

their vertices did coincide, the resulting magnitude at that location would be

greater than 1, which is not a ‘legal’ element of the indicator vector adeﬁned

in Sect. 2). This ensures balance, but such patterns may contain sub-periods as

in the augmented scale shown in the previous subsection. To avoid sub-periods,

we must additionally ensure that the numbers of vertices of the polygons used

is coprime (their greatest common divisor is 1). For example, in a twelve-fold

period (e.g., an equally tempered chromatic scale or a twelve-pulse meter), there

are only two such patterns – as illustrated in Fig. 5. The ﬁrst is created by adding

a digon and a triangle; the second by adding two digons and a triangle. Note

that, in a twelve-fold period, a third disjoint digon cannot be added because the

three digons would take the form of a regular hexagon, which is not coprime

with the triangle (the resulting pattern would have a sub-period).

(a) Five-element perfectly bal-

anced pattern comprising a

digon and a triangle.

(b) Seven-element perfectly bal-

anced pattern comprising two

digons and a triangle.

Fig. 5. The only two irreducibly periodic perfectly balanced patterns available in a

twelve-fold period. Note how uneven these patterns are. The seven-element pattern is

equivalent to a scale variously denoted the double harmonic, Arabic, or Byzantine. In

the North Indian tradition this scale is the Bhairav thaat, and in the Carnatic tradition

it is the scale used in the Mayamalavagowla raga.

The resulting patterns are perfectly balanced, whilst also being uneven and

irreducibly periodic. They might be thought of as displaced polyrhythms – take a

standard polyrhythm containing two isochronous beats of diﬀerent and coprime

interonset intervals (e.g., 3 against 2), but displace one of the beats with respect

to the other so they never coincide.

These heuristics imply that, for two regular polygons, the period must com-

prise N=jk` equally tempered chromatic pitches or isochronous pulses, where

j, k, ` are integers all greater than 1 and gcd(k, `) = 1 (a k`-fold universe is the

smallest that can embed two polygons with coprime kand `vertices, but there

must be at least twice as many so one of the polygons can be rotated to make

Perfect Balance 9

it disjoint to the other). The smallest possible Nare, therefore, 2 ×2×3 = 12,

2×3×3 = 18, 2 ×2×5 = 20, 2 ×3×4 = 24, 2 ×2×7 = 28, 2 ×3×5 = 30,

3×3×4 = 36, 2 ×4×5 = 40, and so forth.

Similar to Fig. 4, the complement of such a pattern is irreducibly periodic

and perfectly balanced as well, as it is a combination of polygons in which at

least one polygon is coprime to at least one other. This can be seen in Fig. 5,

where the 5- and 7-element patterns are complementary (if one is rotated 180◦).

Interestingly, perfectly balanced patterns do not have to be derived from the

addition of disjoint regular polygons; this is merely one method to ensure perfect

balance.

Searching Across Dihedral Groups of Order KAs Amiot has demon-

strated, it is feasible to conduct a brute-force search for perfectly balanced pat-

terns of size Kin a cardinality of N[11]. To increase speed, Amiot factored out

the dihedral group; that is, his search did not separately consider rotationally or

reﬂectionally equivalent patterns. The search shows that patterns that are not

the sum of disjoint regular polygons do indeed exist. When N= 30 and K= 7,

there is one such pattern out of a total of 17 perfectly balanced patterns (it

would seem, therefore, that such patterns are comparatively rare in a discrete

universe of relatively low cardinality). Amiot’s scale is illustrated in Fig. 6, and

has elements at 0

30 ,6

30 ,7

30 ,13

30 ,17

30 ,23

30 ,24

30 .

Fig. 6. Amiot’s scale, which is not composed of disjoint polygons.

Integer Combinations of Intersecting Regular Polygons However, con-

trary to ﬁrst appearances, Amiot’s scale is actually composed of regular poly-

gons. But this time it is a linear combination of ten vertex-sharing (non-disjoint

or intersecting) regular polygons where ﬁve have a weight of −1 and ﬁve have a

weight of 1. So long as the sum of weights, at each location, and across all the

polygons is either zero or unity, the resulting pattern is ‘legal’ (all its elements

have a magnitude of 1 and therefore lie on the unit circle) and will be perfectly

balanced.

We will illustrate with some simple examples. First, let us start with a digon

with a weight of −1. We can cancel out both its vertices by adding coprime

unit-weighted polygons that share its vertices – as shown in Fig. 7(a), where we

10 Perfect Balance

5/30

6/30

12/3018/30

24/30

25/30

(a) triangle + pentagon −

digon make a 6-element

pattern in a 30-fold period.

0/30

5/30

6/30

10/30

12/3018/30

20/30

24/30

25/30

(b) 2 triangles+ pentagon−

digon make a 9-element

pattern in a 30-fold period.

0/30

6/30

7/30

13/3017/30

23/30

24/30

(c) 2 digons+3 pentagons−

3 digons −2 triangles make

Amiot’s scale.

Fig. 7. Perfectly balanced integer combinations of intersecting regular polygons in a

thirty-fold period. When the vertex of one positive-weighted polygon coincides with

the vertex of one negative-weighted polygon they cancel out to zero.

add an equilateral triangle and a regular pentagon. Indeed, we can add another

intersecting polygon to one of the digon’s vertices which gives, at that location, a

combined weight of unity – as shown in Fig. 7(b). In Fig. 7(c), we show precisely

how Amiot’s scale can be derived from ten intersecting polygons with positive

and negative unity weights.

In forthcoming work, we will show that any perfectly balanced subset of an

equally tempered (or isochronous) universe can be constructed in the same way;

that is, as an integer-weighted sum of regular polygons.

Theorem 5. Let N∈N. Any perfectly balanced vector a∈ {0,1}Ncan be

expressed as an integer combination of regular polygons; that is,

a=

M

X

m=1

jmpm(5)

for some M∈N, integers jm, and N-fold regular n-gons pmwith n > 1.

This theorem shows that the method described in the next section, which

provides a simple parameterization to connect a variety of perfectly balanced

scales across a continuum, can generate any possible perfectly balanced scale or

rhythm embedded in any N-fold universe.

Smoothly Rotating Polygons To explain the proposed method for navigat-

ing over useful continua of perfectly balanced scales, it may be helpful to ﬁrst

consider an analogous approach for evenness. The approach is to maximize even-

ness under the constraint of a given jand k, where jis the number of large steps

all of size `and kis the number of small steps all of size ssuch that s<l, and

jand kare coprime. The maximally even pattern of these step sizes is irre-

ducibly periodic and, for a given jand k, is invariant over all `and s(the word

with alphabet land sis a conjugate of a Christoﬀel word encoding an integer

Perfect Balance 11

path of j/k). The choice of jand kessentially constrains the space into a one-

dimensional form that can be parameterized by `/s, the ratio of the large and

small step-sizes. Such scales are typically called well-formed and are discussed

in depth in [14].

An analogous constraint can be applied to perfectly balanced scales. We

can specify a small number kof regular polygons (or perfectly balanced integer

combinations of intersecting polygons like the examples in Fig. 7) such that

their numbers of vertices are coprime, and then simply smoothly rotate the

polygons independently between intersections. This results in a bounded (k−1)-

dimensional continuum that can be smoothly navigated.

As demonstrated in [15], we can take a single well-formed scale, characterized

by (j, k), and search for `/s values that give numerous good approximations

of privileged structures (e.g., just intonation intervals, which have low integer

frequency ratios). An analogous process can be applied to the perfectly balanced

scales as parameterized by the relative phases of their constituent polygons. This

may, therefore, be a useful method for determining a novel class of musically

interesting microtonal scales. We intend to identify such scales in future work.

Optimizing against the DFT Another intriguing possibility is to use op-

timization to ﬁnd perfectly balanced patterns in the continuum. This requires

randomly initializing a K-element pattern (the phase values in z), and optimiz-

ing it against a loss function deﬁned as |Fz[0]|, so as to converge to a perfectly

balanced pattern. Early experiments have shown that such patterns take a wide

variety of forms and describe an interesting manifold. We are currently investi-

gating the use of loss functions incorporating additional factors, such as evenness

and symmetry, so as to impose more regularity on this distribution. This tech-

nique provides a natural match for an æsthetic that embraces unpredictability.

4 Conclusion

We have shown that perfect evenness implies perfect balance, but that perfect

balance does not necessarily imply perfect evenness. Creative work and research

that has targeted evenness may, therefore, have inadvertently targeted balance

too. By disentangling these two properties we hope to have opened up a new

method for analysing, constructing, and understanding scales and meters.

We have demonstrated an analytical method, using the discrete Fourier trans-

formation, as well as geometrically driven approaches, using integer combina-

tions of disjoint or intersecting regular polygons, to construct perfectly balanced

rhythms and scales. The methods suggested in this article are a ﬁrst attempt to

give both musicians and researchers the opportunity to create and manipulate

balance within music.

In addition to the points already mentioned, future work could investigate

diﬀerently weighting each scale degree or time event. Using this method, the prin-

ciple of perfect balance could be applied to any conceivable pattern. For example,

12 Perfect Balance

we might weight the events by their probability of occurring in a stochastic pro-

cess (or prevalence in a composition), by their loudness, or by any conceivable

musical parameter.

It might also be of considerable interest to investigate human perception

and production of perfectly balanced but uneven rhythms and scales in order to

further elucidate the impact of balance.

Acknowledgements

The ﬁrst author would like to thank Emmanuel Amiot for invigorating conversa-

tions about evenness and balance, and also for opening his eyes to the possibility

of perfectly balanced patterns not derived from disjoint regular polygons.

References

1. Clough, J., Douthett, J.: Maximally Even Sets. J. of Music Theory 35, 93–173 (1991)

2. Johnson, T.A.: Foundations of Diatonic Theory: A Mathematically Based Approach

to Music Fundamentals. Scarecrow Press, USA (2008)

3. London, J.: Hearing in Time: Psychological Aspects of Musical Meter. Oxford Uni-

versity Press (2004)

4. Carey, N., Clampitt, D.: Aspects of Well-formed Scales. Music Theory Spectrum

11, 187–206 (1989)

5. Amiot, E.: Discrete Fourier Transform and Bach’s Good Temperament. Music The-

ory Online 15(2) (2009)

6. Fisher, N.I.: Statistical Analysis of Circular Data. Cambridge University Press

(1993)

7. Amiot, E.: David Lewin and Maximally Even Sets. J. of Math. Music 1, 157–172

(2007)

8. Lewin, D.: Re: Intervallic Relations Between Two Collections of Notes. J. Music

Theory 3, 298–301 (1959)

9. Quinn, I.: A Uniﬁed Theory of Chord Quality in Equal Temperaments. PhD Dis-

sertation, University of Rochester (2004)

10. Callender, C.: Continuous Harmonic Spaces. J. of Music Theory 51, 277–332 (2007)

11. Amiot, E.: Sommes Nulles de Racines de l’Unit´e. Bulletin de l’Union des Pro-

fesseurs de Sp´eciales, 30–34 (2010)

12. Messiaen, O.: Technique de mon Langage Musical. Leduc, Paris (1944)

13. Balzano, G.J.: The Pitch Set as a Level of Description for Studying Musical Per-

ception. In: Clynes, M. (ed.) Music, Mind, and Brain, Plenum Press, New York

(1982)

14. Milne, A.J., Carl´e, M., Sethares, W.A., Noll, T., Holland, S.: Scratching the Scale

Labyrinth. In: Agon, C., Amiot, E., Andreatta, M., Assayag, G., Bresson, J., Man-

dereau, J., (eds.) MCM 2011. LNCS, vol 6726, pp. 180–195, Springer, Heidelberg

(2011)

15. Milne, A.J., Sethares, W.A., Laney, R., Sharp, D.B.: Modelling the Similarity of

Pitch Collections with Expectation Tensors. J. of Math. Music 5, 1–20 (2011)