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Perfect balance: A novel principle for the construction of musical scales and meters

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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 gravity' 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 patterns 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.
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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<· · · < xK1(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 zCK. 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, . . . , xN1)) 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
K1
X
k=0
z[k] e2πitk/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
K1
X
k=0
z[k] e2πik/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
K1
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
N1
X
n=0
a[n] e2πin/N .(4)
2.3 Relationships Between Evenness and Balance
Theorem 1. Perfect evenness implies perfect balance.
Proof. Under Parseval’s theorem, PK1
t=0 |Fz[t]|2=1
KPK1
k=0 |z[k]|2. By defini-
tion, all |z[k]|= 1, hence PK1
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
NKpitches [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 (NK)-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 NN. 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 MN, 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 (k1)-
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.
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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)
... In recent work, we have investigated three novel mathematical characterizations of musical rhythms and scales: balance [3,4], evenness [5,4], and interonset interval entropy [5]. Balance and evenness are, respectively, linearly related to the magnitudes of the zeroth and first coefficients of the discrete Fourier transform (DFT) of the rhythm when that rhythm is represented as points on a circle using complex numbers. ...
... (an understanding of musical notation is not necessary for this article -numerical representations are used subsequently -but an accessible summary of musical symbols and terminology is available on Wikipedia [21]). The onsets of these beats can be represented numerically as a set of fractions of the period (which is 16 quavers in duration): X = 0, 3 16 , 6 16 , 10 16 , 12 16 . These fractions can then be straightforwardly converted into a sequence of complex numbers, ordered by angular position, by multiplying them by 2⇡i and taking their exponential: z = ⇣ exp 0 16 ⇥ 2⇡i , exp 3 16 ⇥ 2⇡i , exp 6 16 ⇥ 2⇡i , exp 10 16 ⇥ 2⇡i , exp 12 16 ⇥ 2⇡i ⌘ . ...
... It is worth noting that all perfectly even rhythms are also perfectly balanced; crucially, however, perfectly balanced rhythms are not necessarily perfectly even [4]. Unlike balance, evenness does not correspond to any statistical moment of the onsets' circular distribution; but, in circular statistical terms, it can be expressed as the circular concentration of the angular deviations between the rhythm's onsets and a discrete uniform distribution, where those angular deviations are between the kth members of both circularly ordered sets [3]. ...
Article
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There is an uncountable number of different ways of characterizing almost any given real-world stimulus. This necessitates finding stimulus features that are perceptually relevant – that is, they have distinct and independent effects on the perception and cognition of the stimulus. Here, we provide a theoretical framework for empirically testing the perceptual relevance of stimulus features through their association with recognition, memory bias, and aesthetic evaluation. We deploy this framework in the auditory domain to explore the perceptual relevance of three recently developed mathematical characterizations of periodic temporal patterns: balance, evenness, and interonset interval entropy. By modelling recognition responses and liking ratings from 177 participants listening to a total of 1252 different musical rhythms, we obtain very strong evidence that all three features have distinct effects on the memory for, and the liking of, musical rhythms. Interonset interval entropy is a measure of the unpredictability of a rhythm derived from the distribution of its durations. Balance and evenness are both obtained from the discrete Fourier transform (DFT) of periodic patterns represented as points on the unit circle, and we introduce a teleological explanation for their perceptual relevance: the DFT coefficients representing balance and evenness are relatively robust to small random temporal perturbations and hence are coherent in noisy environments. This theory suggests further research to explore the meaning and relevance of robust coefficients such as these to the perception of patterns that are periodic in time and, possibly, space.
... In recent work, we have investigated three novel mathematical characterizations of musical rhythms and scales: balance [3,4], evenness [5,4], and interonset interval entropy [5]. Balance and evenness are, respectively, linearly related to the magnitudes of the zeroth and first coefficients of the discrete Fourier transform (DFT) of the rhythm when that rhythm is represented as points on a circle using complex numbers. ...
... (an understanding of musical notation is not necessary for this article -numerical representations are used subsequently -but an accessible summary of musical symbols and terminology is available on Wikipedia [21]). The onsets of these beats can be represented numerically as a set of fractions of the period (which is 16 quavers in duration): X = 0, 3 16 , 6 16 , 10 16 , 12 16 . These fractions can then be straightforwardly converted into a sequence of complex numbers, ordered by angular position, by multiplying them by 2πi and taking their exponential: z = exp 0 16 × 2πi , exp 3 16 × 2πi , exp 6 16 × 2πi , exp 10 16 × 2πi , exp 12 16 × 2πi . ...
... It is worth noting that all perfectly even rhythms are also perfectly balanced; crucially, however, perfectly balanced rhythms are not necessarily perfectly even [4]. Unlike balance, evenness does not correspond to any statistical moment of the onsets' circular distribution; but, in circular statistical terms, it can be expressed as the circular concentration of the angular deviations between the rhythm's onsets and a discrete uniform distribution, where those angular deviations are between the kth members of both circularly ordered sets [3]. ...
Preprint
Full-text available
There is an uncountable number of different ways of characterizing almost any given real-world stimulus. This necessitates finding stimulus features that are perceptually relevant – that is, they have distinct and independent effects on the perception and cognition of the stimulus. Here, we provide a theoretical framework for empirically testing the perceptual relevance of stimulus features through their association with recognition, memory bias, and aesthetic evaluation. We deploy this framework in the auditory domain to explore the perceptual relevance of three recently developed mathematical characterizations of periodic temporal patterns: balance, evenness, and interonset interval entropy. By modelling recognition responses and liking ratings from 177 participants listening to a total of 1,252 different musical rhythms, we obtain very strong evidence that all three features have distinct effects on the memory for, and the liking of, musical rhythms. Interonset interval entropy is a measure of the unpredictability of a rhythm derived from the distribution of its durations. Balance and evenness are both obtained from the discrete Fourier transform (DFT) of periodic patterns represented as points on the unit circle, and we introduce a teleological explanation for their perceptual relevance: the DFT coefficients representing balance and evenness are relatively robust to small random temporal perturbations and hence are coherent in noisy environments. This theory suggests further research to explore the meaning and relevance of robust coefficients such as these to the perception of patterns that are periodic in time and, possibly, space.
... In so doing, this chapter both draws together and extends recent publications related to perfect balance, well-formedness and XronoMorph (Milne, et al., 2015;Milne & Dean, 2016;Milne, et al., 2016;Milne, 2018;Milne, et al., 2018). For the first time, I combine straightforward explanations of the underlying music-theoretical concepts with substantive discussion of their creative potential (generating traditional rhythms -Western and non-Western -whilst, crucially, generalizing from these into unfamiliar rhythmic territory that would be otherwise hard to compose or to perform manually). ...
... To do this, required developing and parameterizing organizational principles for rhythms that operate across continua of timing values. Perfect balance (introduced in (Milne, et al., 2015)) and well-formedness (first detailed in a rhythmic setting 6 in (Milne & Dean, 2016)) both meet these criteria. ...
... I made the first prototypes of XronoMorph in 2014 -originally called MeanTimes (Milne & Dean, 2016) -to test whether the abstract concept of well-formedness and the newly developed, and similarly abstract, concept of perfect balance (Milne, et al., 2015) were viable principles for generating rhythms. Upon actually hearing these two types rhythms for the first time, it seemed to me quite clear that they opened doors to two remarkably interesting and aesthetically pleasing rhythmic spaces. ...
Chapter
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XronoMorph is a musical loop generator that opens up two huge spaces of unusual and interesting polyphonic rhythms: perfectly balanced rhythms and well-formed rhythms. These are rhythms that would often be hard to create in alternative software applications or with traditional musical notation. In this chapter, I explain the algorithmic principles used to generate the loops and how these principles have been parameterized and visualized to facilitate the exploration of paths within these two rhythmic spaces.
... : evenness quantifies the extent to which interonset intervals are of a similar size (Amiot, 2009;Milne et al., 2017Milne et al., , 2015; rhythms with low evenness will, therefore, tend to have groups (cues clustered in time) and gaps (uncued pulses clustered in time). As such, uneven rhythms should have more obvious changes in their local structure across the progression of the rhythm, which may facilitate position-finding: for example, consider the rhythms (1 1 0) and (1 1 0 0) with respective evennesses of 0.866 and 0.707; the latter is more uneven than the former and the contrast between its cluster and its gap is greater. ...
... : for a rhythm represented as weighted points on a circle, this is the circular (directional) variance of the rhythm (Milne et al., 2017(Milne et al., , 2015. The more unbalanced a rhythm, the more informative of the cues' locations is the cues' circular mean; hence, the lower the balance, the better the cues' mean serves as a good lossy descriptor of the rhythm. ...
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Book
Preface; 1. The purpose of the book; 2. Survey of contents; 3. How to use the book; 4. Notation, terminology and conventions; 5. Acknowledgements; Part I. Introduction: Part II. Descriptive Methods: 2.1. Introduction; 2.2. Data display; 2.3. Simple summary quantities; 2.4. Modifications for axial data; Part III. Models: 3.1. Introduction; 3.2. Notation; trigonometric moments; 3.3. Probability distributions on the circle; Part IV. Analysis of a Single Sample of Data: 4.1. Introduction; 4.2. Exploratory analysis; 4.3. Testing a sample of unit vectors for uniformity; 4.4. Nonparametric methods for unimodal data; 4.5. Statistical analysis of a random sample of unit vectors from a von Mises distribution; 4.6. Statistical analysis of a random sample of unit vectors from a multimodal distribution; 4.7. Other topics; Part V. Analysis of Two or More Samples, and of Other Experimental Layouts: 5.1. Introduction; 5.2. Exploratory analysis; 5.3. Nonparametric methods for analysing two or more samples of unimodal data; 5.4. Analysis of two or more samples from von Mises distributions; 5.5. Analysis of data from more complicated experimental designs; Part VI. Correlation and Regression: 6.1. Introduction; 6.2. Linear-circular association and circular-linear association; 6.3. Circular-circular association; 6.4. Regression models for a circular response variable; Part VII. Analysis of Data with Temporal or Spatial Structure: 7.1. Introduction; 7.2. Analysis of temporal data; 7.3. Spatial analysis; Part VIII. Some Modern Statistical Techniques for Testing and Estimation: 8.1. Introduction; 8.2. Bootstrap methods for confidence intervals and hypothesis tests: general description; 8.3. Bootstrap methods for circular data: confidence regions for the mean direction; 8.4. Bootstrap methods for circular data: hypothesis tests for mean directions; 8.5. Randomisation, or permutation, tests; Appendix A. Tables; Appendix B. Data sets; References; Index.