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Computational Creation and

Morphing of Multilevel

Rhythms by Control

of Evenness

Andrew J. Milne and Roger T. Dean

MARCS Institute

University of Western Sydney

Locked Bag 1797

Penrith, NSW 2751, Australia

andymilne@tonalcentre.org

Roger.Dean@westernsydney.edu.au

Abstract: We present an algorithm, instantiated in a freeware application called MeanTimes, that permits the

parameterized production and transformation of a hierarchy of well-formed rhythms. Each “higher” rhythmic level

fills in the gaps of all “lower” levels, and up to six such levels can be simultaneously sounded. MeanTimes has a

slider enabling continuous variation of the ratios of the intervals between the beats (onsets) of the lowest level. This

consequently changes—in a straightforward manner—the evenness of this level; it also changes—in a more complex,

but still highly patterned manner—the evennesses of all higher levels. This specific parameter, and others used in

MeanTimes, are novel: We describe their mathematical formulation, demonstrate their utility for generating rhythms,

and show how they differ from those typically used for pitch-based scales. Some of the compositional possibilities

continue the tradition of Cowell and Nancarrow, proceeding further into metahuman performance, and have perceptual

and cognitive implications that deserve further attention.

For most of human history, musical rhythm—as

with all forms of musical performance—has been a

fundamentally embodied enterprise, close to dance:

for example, limbs making beats by striking ob-

jects together. In all periods, however, the music

involving temporal repetitions evoking dance has

coexisted with “unmeasured” music, largely devoid

of such impulse, such as medieval plainsong, or

the unmeasured music of Couperin. Since the ad-

vent of electronic and computer technology, many

creators of classical, jazz, and popular music have

explored both human and artificial (nonhuman,

often computer-aided) generation of rhythms, for

performance and composition. Notable among these

explorations in Western classical music are (1) the

evasion of rhythmic regularity in some works of

Iannis Xenakis and Karlheinz Stockhausen; (2) the

development of serialized rhythmic structures, of-

ten more readily realized by computational than

human performance; (3) irrational rhythms (such

as those in instrumental composition by Brian

Ferneyhough and the new complexity composers,

some of which parallel those of the roughly con-

temporaneous free jazz and free improvisation, from

John Coltrane and Cecil Taylor to Derek Bailey

and Evan Parker); and (4) the establishment of

noise music and its relatives, in which the surface

Computer Music Journal, 40:1, pp. 35–53, Spring 2016

doi:10.1162/COMJ a00343

c

⃝2016 Massachusetts Institute of Technology.

of the music betrays few discrete (and hence few

rhythmic) features. In the popular music tradition,

analog and digital (computational) sequencers, loop

sequencers (e.g., Acid), and step sequencers have

often been used to produce many of the charac-

teristic sounds of pop. These sounds include the

disco beats of the 1970s, “synth groups” of the

1980s, house music and hip-hop in the 1990s, and

the “glitchy” beats of more recent electronic dance

music.

Several current computer music platforms (e.g.,

Max and Gibber) make it convenient to generate

and control freely definable sequences of events

and the number of times they are exactly repeated.

In contrast to Max and Gibber, which encourage

pulse flexibility and variation, many pattern-based

languages (e.g., the live-coding languages ixi lang

and Tidal) are primarily based on the idea of

fixed isochrony of pulses, and commonly oper-

ate on bars as unitary objects. (Isochrony means

evenly spaced in time—all inter-onset intervals

are identical.) There are also numerous compu-

tational “rhythm machines” in use, published

and unpublished (e.g., Dean 2003; Sioros et al.

2014). We have not, however, found a rhythm gen-

erator like MeanTimes (illustrated in Figure 1),

which is capable of producing (1) a nested hierar-

chy of rhythmic streams that can be simultane-

ously systematically morphed between isochrony

and nonisochrony, and (2) differing levels of

complexity.

A. J. Milne and R. T. Dean 35

Figure 1. The user interface

of MeanTimes. The

hierarchy of well-formed

rhythms is depicted by

polygons inscribed in a

circle. The large horizontal

slider at the bottom is

used to control beat-size

ratio. Above that are

useful preset values for the

slider, then settings for

each of six rhythmic levels

in the well-formed

hierarchy, including their

pitch and velocity values.

At the top is a slider to

control the length of the

period of rhythmic

repetition. Immediately to

the right of the polygons

are tracks, which can host

plug-ins to sonify the

polygons or send MIDI to

external synthesizers. To

the far right are presets,

which can be saved and

allow sequences of

patterns to be arranged

into rows or columns.

Background

In this section, we outline the main theoretical

concepts used to develop the algorithm (which is

described in the subsequent section). These concepts

are drawn from both musical rhythm theory and

scale theory, and are applicable to the analysis

of rhythm and meter in the symbolic realm, and

to its physical realization (though, for the most

part, we gloss over the fluctuations in tempo that

typically occur in real-world performance). We also

discuss some of their limitations when applied to

certain familiar rhythms and suggest some possible

solutions.

Rhythm and Meter

Given that our purpose is to discuss computational

production and morphing of rhythm, we need to

delineate the concepts of rhythm and meter that we

will use, and place them in context. We characterize

arhythm as a sequence of sonic events arranged

in time, and thus primarily characterized by their

inter-onset intervals. Often a rhythm comprises a

number of different streams or levels. Each different

stream may be played by a different instrument or

by a different drum in a drum kit—for example, one

stream may be played by the kick drum, another

by the snare, another by the hi-hat. The decision as

to which events are assigned to which stream or,

indeed, whether all events are considered to belong

to a single stream, depends on how much detail

the analysis requires. We use the term level when

we wish to emphasize the hierarchical nature of the

streams whereby higher levels,whichtypicallyhave

higher pitches or brighter timbres, move faster than

lower levels,whichtypicallyhavelowerpitches

or darker timbres. When a higher-level (i.e., faster)

rhythmic stream nests (or is a superset of) a lower-

level (i.e., slower) stream, we refer to events in the

36 Computer Music Journal

Figure 2. Some different

types of event patterning

that may lead to

metricality: periodic (a), as

is inherent in the circular

representation, but there is

no obvious pattern within

the period; a ﬁve-beat

rhythm (b) with only two

event sizes (shown as ℓand

s), repeating subsequences

(e.g., there are two

instances of the sequence

sℓwithin the period), and

reﬂectional symmetry

about the vertical axis; the

third (c) has properties

similar to (b) except it is

also nested by seven

isochronous pulses (the

open circles represent the

unsounded pulses).

former as pulses, and events in the latter as beats.

Under this definition, every beat must coincide

with a pulse, but not every pulse must coincide

with a beat (because the beat rhythm is a subset

of the pulse rhythm). By extension, it is possible

to have multiple rhythmic levels where each level

nests all lower levels. In such a situation, we

refer to events in the lowest rhythmic level under

consideration as beats, and events in all higher levels

as pulses. This hierarchical approach is related to

that taken by Lerdahl and Jackendoff (1983), but

our terminology differs because they term all levels

“beats.” Of course, there are slower metrical levels

below beats, such as measures, or phrases that span

several measures, or even longer-term structures,

but these are not our concern here. Henceforth, we

use the terms event size or event length to specify

the inter-onset interval between that event and the

next event on the same level. An event’s duration

is used to specify how long that event is actually

played for (which may be less, the same as, or more

than its size).

In this article, our focus will be on rhythms that

have a nontrivial degree of metricality.Wetake

arelativelynonrestrictivefunctionalapproachto

the definition of metricality: We use it to mean

the extent to which a rhythm has a repeating

or repeatable pattern and is perceived to have a

predictable structure. Indeed, we consider a meter

to be a mental template or dynamic process that

is sufficiently stable to assign probabilities to

events’ onset times. In other words, a meter is

amentalrepresentation,notaphysical,sounded

manifestation (which is a rhythm). This implies

that a rhythm with high metricality or, indeed a

meter itself, must have some level of organization,

patterning, or regularity across time. Our focus here

will be on temporal patterns of the onsets of discrete

events, and we will mostly ignore variations in their

intensity, spectrum, or individual duration. Similar

arguments will apply across all these domains,

however, and they may interact to determine the

resulting meter (e.g., through the influence of

dynamic accents).

For events to be patterned we have only a

general requirement that they are arranged with

some regularity or symmetry that distinguishes

them from a purely random arrangement. We will,

however, make some concrete suggestions about

possibly relevant regularities. Perhaps the most

basic form of regularity is repetition over a period.

Indeed, periodicity may be reasonably considered

a prerequisite for metricality, although in much

music the temporal period of a given meter may

vary detectably across a piece. As illustrated in

Figure 2, additional patterning may result from

irregular repetitions of event sizes, or irregular

repetitions of subsequences of event sizes. There

may be reflective symmetries where all, or a part

of, a rhythm is repeated but in retrograde form.

The rhythm may exhibit some of the organizational

properties identified by Toussaint (2013), or other

principles such as balance (Milne et al. 2015).

These organizational principles may hold at any

single rhythmic level or any mixture of levels. They

may also hold for unplayed but mentally induced

metrical pulses. For example, suppose a played beat

rhythm that is somewhat unpatterned is a subset of

A. J. Milne and R. T. Dean 37

an unplayed pulse rhythm that is highly patterned. It

is feasible that this beat might then induce metrical

pulses analogous to this unplayed, but highly

organized, pulse rhythm. For example, a beat whose

events all fall at some small integer subdivision of

aperiodmayinduceanalogousisochronouspulses,

whereas a beat whose events do not fall at such

locations may not. This is concretely illustrated

in Figure 2. Note how Figure 2c is a subset of a

low-cardinality pulse (seven per period), while the

similarly structured beat in Figure 2b is not a subset

of a low-cardinality pulse (indeed, for the precise

rhythm depicted, the smallest isochronous superset

has 37 pulses per period, which is likely to be too

fast to be readily induced as a metrical level). For

this reason, we might conjecture that Figure 2c may

be heard as more metrical than Figure 2b, even if the

pulse level is not physically played.

Evenness (Clough and Douthett 1991; Amiot

2007) is a notable organizational principle and one

that is highly pertinent to MeanTimes. The evenness

of a rhythmic stream is the extent to which its

events’ sizes are equal (or, equivalently, the extent

to which that rhythm is isochronous). When a

rhythmic stream is perfectly even (i.e., isochronous),

it has translational symmetry at the most granular

level possible. It also has reflectional symmetry

and is perfectly balanced (the mean position of

the circularly arranged rhythmic events is at the

center of the circle; see Milne et al. 2015), hence it

has a powerful claim for high metricality. Indeed,

the most common meters in Western music (e.g.,

4

4,3

4,and6

8) exemplify perfect evenness at all

levels. In some jazz, progressive rock, sub-Saharan

African music (Rahn 1986), and Eastern European

aksak (Fracile 2003), rhythms and meters typically

comprise nonisochronous beats, underneath a faster

isochronous pulse (which may be played or implied).

There is a class of patterns—called well-formed

(Carey and Clampitt 1989) or moments of symmetry

(Wilson 1975)—that can have varying degrees

of evenness including perfect evenness. These

patterns also have relatively numerous instances of

transpositional symmetry of subsequences, and have

reflectional symmetry (Figures 2b and 2c are both

well-formed rhythms). Furthermore, they group into

anaturalhierarchywhereeverywell-formedpattern

is nested by another. Well-formed patterns form the

basis of the rhythms produced by MeanTimes. We

discuss well-formedness in the section “Maximizing

Evenness under Well-Formedness Constraints,” but

prior to that we will provide a formal definition of

evenness.

Evenness

There are a number of different models of a pattern’s

evenness (Amiot 2007). We use the calculation

introduced by Amiot (2009), for the following

reasons: It makes no assumption of an underlying

isochronous pulse, it has a clear interpretation in

terms of conventional circular statistics (Fisher

1993), and it can be relatively easily understood.

We provide a formal definition in the following

but, for those wishing to skip the mathematics, a

simple summary is that evenness is the similarity

of a rhythm with Mevents to an isochronous (hence

perfectly even) rhythm also with Mevents.

First, the Monset times are placed, in time order,

into a vector. These times are then normalized by

subtracting the lowest time value from all of them

and then dividing them by the length of the whole

period so the resulting values start from 0 and are

all less than 1. This vector is denoted xand its M

elements are indexed by m=0, 1, 2, ...,M−1. The

elements of xare then mapped to the unit circle in

the complex domain by z[m]=e2πix[m]. This results

in a vector of Mcomplex numbers, where the phase

(angle or argument) of the mth element represents

the mth event’s normalized onset time, and all

events have a magnitude of 1.

Each such complex number can itself be thought

of as a vector extending from the center of the circle

to its perimeter; hence zcan be thought of as a

vector of vectors. For example, let us consider two

rhythms all of whose beats fall at a twelfth equal

division of the period. The first is analogous to an or-

dered diatonic pitch-class set and can be represented

by x=(0/12, 2/12, 4/12, 5/12, 7/12, 9/12, 11/12);

the second comprises two “clusters” of beats

x=(0/12, 1/12, 4/12, 5/12, 6/12, 7/12, 11/12). The

mapping of each of these two vectors to the complex

unit circle is illustrated by the solid vectors (arrows)

38 Computer Music Journal

Figure 3. A graphical

representation of the

evenness calculation

described in the main text.

The solid-line vectors

show the “diatonic”

rhythm (a) and the

“cluster” rhythm (b), both

having seven events. The

dashed-line vectors show

the seven isochronous

locations to which the

rhythms are compared.

The numbers around the

circle are the angles

between the mth solid-line

rhythm vector and the mth

dotted-line isochronous

vector—the squiggly line is

the sum of seven vectors

having these respective

angles, each with a

normalized length of 1/7.

in Figure 3 (in this visualization, the imaginary unit

iis at 3 o’clock, the real unit 1 is at 12 o’clock, and

angles are measured clockwise from the latter—this

is a diagonal reflection of the conventional math-

ematical visualization, but results in the familiar

clockface type representation usually depicted in

rhythm-related research). The first element of x,

which is 0 in both of these rhythms, is mapped to 1

in the complex plane and so goes to the 12 o’clock

position, while successive elements proceed clock-

wise around the circle (the angles written around

the circle will be explained forthwith).

The next step is to take the discrete Fourier

transform of this complex vector, and the magnitude

of the first coefficient gives the evenness of the

pattern:

evenness =!!Fx[1]!!=!!!!!

1

M

M

"

m=1

z[m]e

−2πim/M!!!!!

=!!!!!

1

M

M

"

m=1

e2πi(x[m]−m/M)!!!!!.(1)

The summand in the second line, e2πi(x[m]−m/M),

makes it clear that what is being calculated, for each

value of m,isaunit-lengthvectorwhoseangleis

equivalent to the directional (angular) displacement

between the mth event of the beat structure under

investigation and the mth event of an isochronous

meter (both containing Mevents). In Figure 3, the

seven isochronous locations are shown with dashed

vectors, and the clockwise displacement from the

mth isochronous location to the mth beat is shown

by the respective number outside the circle.

The magnitude of the first coefficient, |Fx[1]|,is

the length of the resultant vector, which is the sum

of the Mpreviously mentioned unit-length vectors

whose angles correspond to the respective displace-

ments. Clearly, the smaller the variance of these

displacements, the longer the resultant vector will

be. When the events are maximally even—which

implies all their displacements are identical, so all M

summands are parallel vectors—the normalization

by 1/Mensures that the length of the resultant

(i.e., the evenness value) is unity. In Figure 3, this

summation of the angular displacement vectors

is depicted by the seven small vectors extending

from the center of the circle outwards. Each vector

has a length of 1/7 because of the normalization.

The innermost vector has an angle of 0◦(vertical)

because the displacement between the first beat

and the first isochronous location is zero; the next

vector outwards has an angle corresponding to the

displacement between the second beat and the sec-

ond isochronous location (8.6◦in Figure 3a, −21.4◦

in Figure 3b); and so on. Note how the vector re-

sulting from the sum of these displacement vectors

is slightly longer in Figure 3a (its length is 0.988)

than it is in Figure 3b (its length is 0.924); hence

A. J. Milne and R. T. Dean 39

the diatonic rhythm is more even than the cluster

rhythm.

Using the standard definition of circular variance

(Mardia 1972; Fisher 1993), metrical evenness is

equivalent to unity minus the circular variance of

the displacements. Importantly, the calculation can

be applied equally to any single rhythmic stream

(e.g., beat or pulse) or any combination of such

streams. In all cases, the Mrhythmic events are

compared with Misochronous events occupying the

same period.

Maximizing Evenness under Metrical Constraints

Clearly, this definition implies that a maximally

even rhythm of Mbeats has isochronic beats (a

simple result: this is perfect evenness). As soon

as certain common constraints are applied to the

metrical structure, however, isochronous beats

that also coincide with isochronous pulses become

impossible to achieve, and the resulting patterns

are more interesting. The most obvious example is

when there are Mbeats in the same period as N

isochronous pulses, and Mand Nare coprime (this

means that their greatest common divisor is unity,

which can be concisely written as gcd(M,N)=1).

In this case, there is no way for the beats to be

isochronous and also to coincide with pulses (the

latter being a requirement under our previously

given definitions of beat and pulse).

Consider a twelve-pulse, five-beat rhythm: the

two numbers are coprime so, as just indicated, there

is no way for the beats to fall on pulses and also be

isochronous (perfectly even) under the constraint

that all beats must coincide with a pulse. It is

possible to maximize the evenness of the beats

under these same constraints, however: We have

to choose the most even arrangement of five beats

(i.e., the beats are not isochronous), whose total

length equals twelve isochronous pulses. Clearly,

there are numerous such patterns, but one of them

will be maximally even. To be more concrete, let

there be two beats of size (dotted quarter-note) and

three beats of size (quarter note). Ignoring rotation

(i.e., the position of the period commencement or

the “bar line” relative to the events), there are just

two ways these beats can be arranged (

or ). Of the two possibilities, the latter

pattern is closer to the isochronous beat pattern (the

displacements have less variance) and is, hence, the

pattern of greatest evenness under these constraints.

Indeed, this latter pattern is maximally even with

respect to all possible patterns of five beats in

atwelve-isochronous-pulseperiod(i.e.,including

those with more than two beat sizes). Generally,

for any gcd(M,N)=1, the maximally even pattern

will have no more than two step sizes. This method

for patterning events corresponds precisely to Justin

London’s (2004) well-formed constraint WFC 6

(version 1).

Maximizing Evenness under Well-Formedness

Constraints

Setting gcd(M,N)=1isnot,however,theonlyuse-

ful constraint under which to maximize evenness.

There is an alternative constraint that is particularly

attractive because it makes no assumption about

asupersetofisochronouspulses.Asnotedearlier,

the resulting patterns are known as “well-formed”

(Carey and Clampitt 1989) or, in a tuning-theory

context, “moments of symmetry” (Wilson 1975).

(As shown by these citations, well-formedness was

first applied to pitch-based scales. It is natural to

extend this property from scales, which are events

distributed in pitch with repetition at the octave, to

rhythms, which are events distributed in time with

repetition at the period.) Well-formed patterns are a

superset of the previously described gcd(M,N)=1

patterns, and they are a method for generalizing the

latter into contexts with a nonisochronous pulse or

no underlying pulse.

The well-formed constraint is that there are j

beats of size ℓ,therearekbeats of size s,where j

and kare coprime, and there are no beats with a size

that is not ℓor s.Foraperiodoftotallengthd,this

implies d=jℓ+ks.Inageneralsetting,ℓand scan

take any possible sizes. For this article, however,

we will apply the additional constraint that s≤ℓ;

that is, that the ℓ-sized beats are “long” or “large”

and the s-sized beats are “short” or “small.” (In a

pitch-based context, the use of ℓ,orL,andsfollows

40 Computer Music Journal

the notational convention introduced by Erv Wilson,

and one that is followed by much of the microtonal

community; e.g., xenharmonicwiki.com.) Most of

the properties we outline subsequently also hold for

s>ℓ,butthes≤ℓconstraint simplifies some of the

exposition.

We additionally denote the step-size ratio r=

ℓ/s∈[1, ∞), after Blackwood’s (1985) use of R

to describe the ratio between the sizes of major

and minor seconds in different tunings of the

diatonic scale. This ratio is one of the principal

parameters used in MeanTimes to control the

rhythmic output (see Figure 1). It is useful because,

when ris a rational number whose reduced fraction

is a/b(i.e., a fraction where the numerator and

denominator are both divided by their greatest

common divisor), every beat falls at an Nth division

of the period, where N=ja +kb,where jis the

number of long beats, and kis the number of short

beats. Therefore, if Nis sufficiently small, the

meter has a perceptibly isochronous pulse rhythm.

In MeanTimes, these faster pulse rhythms can

be sounded, if required, along with the main beat.

Conversely, if Nis sufficiently large, or ris irrational

and sufficiently distant from the nearest low-integer

ratio, it is more reasonable to consider the pulse to

be nonisochronous. Furthermore, when r<2, the

scale is proper (Rothenberg 1978) or, equivalently,

coherent (Balzano 1982), which means the total

length of any mconsecutive beats is always larger

than the total length of any m−1consecutivebeats.

Justin London (2004), for example, has suggested

that incoherences (where the above property does

not hold) may make the resulting meters less

stable or, when the clumping of beat sizes is

extreme, they may be heard as alternating meters of

different lengths. For example, the non-well-formed

3+3+2+2+2 (a common compositional pattern,

used in Bernstein’s “America” and often found in

flamenco) may be heard as a repetitive alternation

of 3 +3and2+2+2. In this rhythm, the first two

consecutive beats—each of size 3—have a total

length of 6, and the last three consecutive beats—

each of size 2—also have a total length of 6; hence

there is an incoherence.

Interestingly, the maximally even arrangement

of the two beat sizes ℓand sis invariant across

all possible ratios 1 ≤r≤∞.Forexample,4+3+

4+3+3(so r=4/3), 3 +2+3+2+2(sor=3/2),

and 3 +1+3+1+1(sor=3/1) are all maximally

even arrangements of the beat sizes they comprise

because they all follow the same abstract “well-

formed pattern” (ℓsℓss). Such patterns minimize,

as much as possible, the clumping together of the

ℓ-sized beats and the clumping together of the

s-sized beats—put differently, the ℓ-sized beats are

spaced as far apart as possible, and so are the s-sized

beats.

By covarying the values of ℓand s(and hence r),

while simultaneously ensuring that the length of

the period ( jℓ+ks)isconstant,itispossibletomove

through a continuum of metrical structures, passing

through and connecting meters with differing

numbers of isochronous pulses. It is worth noting

that there are two instances along this continuum

that are degenerate:whenℓ=s(so r=1), and s=0

(so r→∞). At these values, the resulting beats are

perfectly isochronous with, respectively, j+kand

jbeats occurring in each period. This means that

smoothly changing the rvalue over 1 to ∞smoothly

interpolates between these two isochronous beats,

while keeping the period’s length constant: the j+k

events are reduced down to jby having pairs of

events occurring simultaneously.

Other than maximizing evenness (given the above

constraints), well-formed patterns (both meters and

scales) have several other consequent mathematical

properties, which we will elaborate elsewhere (these

properties are also discussed in more depth in

Milne et al. 2011). It is worth noting at this stage,

however, that well-formed patterns have reflectional

symmetry: When the total number of beats is odd,

the axis of reflection falls on one of the beats; when

the number of beats is even, the axis of reflection

falls mid-way between two beats (and may fall

between pulses too, depending on the value of r).

In both cases their ℓ,salphabetic representation is,

therefore, a circular palindrome, as illustrated in

Figures 2b and 2c.

Because well-formed rhythms comprise no more

than two different beat lengths, it follows that

they will have a lower beat-length entropy in

comparison with the huge number of possible

rhythms containing three or more beat lengths. As

A. J. Milne and R. T. Dean 41

discussed later, there are other non-well-formed

patterns that also contain just two beat sizes having

similarly low entropy.

Another highly pertinent property of a well-

formed pattern is that it is always nested inside a

larger well-formed pattern. By nested,wemeanthat

every event in one pattern (the subset) coincides

with an event in the other (the superset). This can

be most easily exemplified by musical scales, where

scale-step sizes are treated as analogous to beat

sizes (early elaborations of the analogy between

a meter and a pitch scale were offered by Rahn

1975; Mazzola 1990; Monahan 1993; Cowell 1996

[1930]). For example, in a pitch-based context, the

well-formed perfect fifth D–A (ℓs)isnestedin

the well-formed “tetractys” D–G–A (ℓsℓ), which

is nested in the well-formed five-note pentatonic

scale D–F–G–A–C (ℓssℓs), which is nested inside

the well-formed seven-note diatonic scale D–E–

F–G–A–B, C (ℓsℓℓℓsℓ), which is nested inside the

well-formed twelve-note chromatic scale D–D♯–E–

F–F♯–G–G♯–A–A♯–B–C–C♯(where, typically, ℓ=s).

But this is just one example of such a hierarchy.

This means that the hierarchical levels, which can

play concurrently, provide a natural analogue for

the hierarchical structure of beats and pulses in

the common conception of meter. This also raises

another possibility, which is that any beat may be

displaced to a neighbouring pulse. This provides a

means to modify well-formed patterns into closely

related (hence still highly structured) non-well-

formed patterns. Both these aspects are explored

in more detail in the section “MeanTimes: The

Application and its Design.”

Creating Real-World Meters and Rhythms

Well-formed rhythms are a common strand within

theories of rhythm and meter (e.g., London 2004;

Toussaint 2013). And, interestingly, it has been

observed that well-formed rhythms are common

in sub-Saharan music (Rahn 1986), and irregular

groupings of strong beats in aksak music often fall

into such well-formed patterns, too (Fracile 2003).

Western music also contains numerous examples

of well-formed rhythms, such as those that formed

the basis for the rules and WFCs discussed by

London.

Despite the appealing conceptual properties of

well-formed patterns, however, there are examples

of important rhythms (and scales) that are not

well-formed. For example, the two beat lengths

may not be distributed with maximal evenness

(e.g., the 3 +3+2+2+2 given earlier) and, in this

case, the pattern may perceptually resolve into an

alternation of two (well-formed) meters 3 +3and

2+2+2. Alternatively, the rhythmic pattern may

contain more than two differing beat lengths; for

example, the five-beat, 16-pulse son clave pattern

(3 +3+4+2+4). So this raises a question of

whether there is a principled way to extend the

framework of well-formed rhythms in such a way

as to include a useful subset of non-well-formed

rhythms.

London relaxes his constraints in the second

version of WFC 6 precisely to enable such meters.

As we show later, our conception of meter as

hierarchies of well-formed rhythms allows such

non-well-formed meters to occur naturally by

substituting a beat on one level with a neighboring

pulse from the next higher level. This is analogous

to chromatic alteration in a diatonic-chromatic

context, whereby one might flatten or sharpen a

diatonic scale degree to obtain a different—yet still

highly structured—scale. For example, flattening the

third degree of the major scale (C–D–E–F–G–A–B)

produces the ascending melodic minor scale (C–D–

E♭–F–G–A–B). This latter scale comprises two step

sizes, but they are now not arranged with maximal

evenness, though the scale is still reasonably even

(its evenness value is not far below 1). Taking a

common metrical example, the son clave pattern

3+3+4+2+4 is not well-formed (after all, it has

three beat lengths). But it can be simply constructed

as a well-formed rhythm whose fifth event has been

“flattened”; that is, it is an altered version of the

well-formed 3 +3+4+3+3. Such displacement

of the beat might also occur because of repeated

accentuation of the specified higher-level pulse

note.

We do not mean to suggest here that well-

formedness and evenness are necessary principles of

rhythm production. Indeed, there are many different

42 Computer Music Journal

Figure 4. Three well-formed

rhythms, each comprising

six rhythmic levels

visualized in MeanTimes

as polygons inscribed in a

circle. Each polygon vertex

represents a beat or pulse.

The small disk between 3

and 4 o’clock rotates

clockwise around the

circle and represents a

“playhead.” When the

playhead passes over a

vertex, the corresponding

MIDI note is played. In (a),

Level 0 is the upwards

pointing isosceles triangle

(its vertices correspond to

the twelve-tone equal

temperment [12TET] pitch

classes C, F, and G), Level

1 is the irregular pentagon

(pentatonic), Level 2 is the

irregular heptagon

(diatonic), Level 3 is the

regular dodecagon

(chromatic). In (b), the

diatonic Level 2 has been

set to counterclockwise

mode (discussed

subsequently). In (c),

Levels 1–3 from (a) have

been set to complementary

mode (discussed

subsequently) so Levels 1

and 2 become digons

(straight lines), and Level 3

becomes an irregular

pentagon (pentatonic).

strategies that may be taken—for example, using

balance rather than evenness (Milne et al. 2015).

We suggest, however, that they are useful principles

relevant to both rhythm production and probably

to perception. As such, they provide a useful means

to reduce the space of rhythms into one that can

be smoothly parameterized and that also contains a

wide variety.

MeanTimes: The Application and its Design

Developing from the analogy between (rhythmic)

meter and (pitch) scale introduced earlier, and the

consideration of evenness and well-formedness,

MeanTimes is constructed as a MIDI rhythm

generator providing a rhythmic parallel to the pitch-

based manipulations used by the freeware Dynamic

Tonality synthesizers and sequencers detailed by

Milne, Sethares, and Plamondon (2008), Sethares and

coworkers (2009), and Prechtl and colleagues (2012).

MeanTimes is built in Cycling ’74’s Max, and can be

downloaded from www.dynamictonality.com as a

standalone application for Windows and Mac OS X.

As detailed subsequently, MeanTimes provides

anumberofparameterstoshapesixseparate

rhythmic levels of a well-formed hierarchy. The

user can set the length, in milliseconds, of the

period of repetition (denoted d), and the numbers of

long beats and short beats into which this period

is divided for the lowest-level rhythm (in this

article, these numbers are denoted jand k,buthave

different designations in MeanTimes’s interface).

The lengths of these beats are denoted ℓand s,

respectively: Their ratio r=ℓ/scan be smoothly

varied with a slider along the entire continuum

that preserves that rhythm’s structure (i.e., given

jand k,therange1≤r<∞within which s≤ℓ).

The sizes of ℓand sare automatically calculated

to ensure the period’s length is invariant across all

values of r.Itisalsopossibletoselectivelysounda

number of “higher-level” well-formed rhythms that

nest the “Level 0” beat rhythm, and that smoothly

change as a function of rover its entire range. As

discussed earlier, at certain values of r,someof

these higher-level rhythms will comprise perfectly

isochronous pulses.

As shown in close-up in Figure 4, each rhythmic

level is visualized as a polygon inscribed in a circle. A

“playhead,” depicted as a small disk, rotates around

the circle and, whenever it “hits” a polygon vertex,

a MIDI note is sent out with a pitch, duration,

channel, and output port that has been designated

for that polygon by the user.

Our approach to parameterizing and explaining

well-formedness is novel because of our focus on the

size of r,ratherthanthesizeofageneratinginterval,

which, with the notable exception of Blackwood

(1985), is the more usual way to parameterize well-

formed pitch-based scales (Wilson 1975; Erlich 2006;

A. J. Milne and R. T. Dean 43

Milne et al. 2011). Indeed, our approach can be seen

as a generalization of Blackwood’s in that it can

additionally encompass any possible well-formed

pattern and the well-formed patterns that embed it

(not just diatonic and chromatic).

In a rhythmic context, parameterizing by ris

sensible because of the importance of isochronies

(and deviations therefrom), which occur when ris

alowintegerratio,whereasinapitchcontext,the

acoustic properties of intervals between nonadjacent

scale members become important. For example, the

interval used to generate a well-formed scale is the

most prevalent in that scale, which may explain

the importance of the pentatonic, diatonic, and

chromatic scales because they are traditionally

generated by the most acoustically simple and

consonant pitch-class interval other than the octave,

the perfect fifth/perfect fourth (Milne, Laney, and

Sharp 2016).

Generating and Parameterizing

Well-Formed Rhythms

The use of ras our primary parameter of inter-

est leads to the following mathematical descrip-

tion. To provide the necessary background, we

first outline the word-theory characterization of

well-formedness recently developed by Clampitt,

Dom´

ınguez, and Noll (2009), and related earlier

work by Wilson (1991). We then explain our novel

extensions of this theory: extensions that allow us

to create parameterized multileveled well-formed

rhythms.

Well-Formed Words

In order to create a well-formed rhythm, it is

necessary to know the pattern in which its long

and short beats are arranged; in other words, it is

necessary to know its well-formed word over the

alphabet of the letters ℓand s(Clampitt, Dom´

ınguez,

and Noll 2009). Up to rotation, each rhythm is

uniquely defined by the numbers ( jand k)oflong

and short beats it contains and, as described earlier,

the two letters (beat sizes) are arranged so they are

clumped as little as possible. The well-formed word

is equivalent to the Christoffel word of slope j/k,or

any conjugate thereof (Berstel et al. 2008; Clampitt,

Dom´

ınguez, and Noll 2009). In this context, a

conjugate is some rotation (also commonly called a

“circular shift”) of the word, where a sequence of

letters are removed from the end of the word and

added to the start.

As demonstrated in principle (the details and

context differ) by Erv Wilson (1991), well-formed

words can be generated by the following simple

algorithm. Start with the word ℓs, then make

two different pairs of substitutions (each pair

of substitutions is termed a morphism): the first

morphism is ℓ'→ ℓsand s'→ ℓ;thesecondmorphism

is ℓ'→ ℓsand s'→ s.Applyingthefirstmorphismto

ℓsgives ℓsℓ;applyingthesecondmorphismgives

ℓss,hencetwonewwordshavebeengenerated.The

two morphisms are then re-applied to the two new

words, resulting in four more new words: ℓsℓℓs,

ℓssℓs,ℓsℓℓ,andℓsss. Applying the morphisms again

results in eight new words and, in general, the ith

iteration provides 2inew words, making a total of

2i+1−1words.Bothmorphismscanbeunderstood

as splitting each long beat into a new long and

anewshortbeat(ℓ'→ ℓs). In the first morphism,

the size of the new long beat corresponds to the

size of the old short beat (s'→ ℓ); in the second

morphism, the new short beat is the same as the old

short beat (s'→ s). These two morphisms produce

only Christoffel words or their conjugates because

they are compositions of the standard Christoffel

morphisms: the first morphism is G◦E;thesecond

is #

D(Berstel et al. 2008).

Calculating Beat Sizes

As described earlier, ( j,k)definesthewell-formed

word—the ordering of the beats of sizes ℓand sin

the period. It does not, however, define the values

of ℓand s.Forthat,thelength(temporalduration)

dof the period and the ratio rof the two step sizes

are required. As previously established, d=jℓ+ks,

and by definition, r=ℓ/s.Thesetwoequations

imply that ℓ=dr /(jr +k)ands=d/(jr +k), which,

in conjunction with the well-formed word, allow

us to calculate the precise timings of all beats as

afunctionoftheperiodlengthd, the numbers of

44 Computer Music Journal

Figure 5. The evenness of

all four ﬁve-beat

well-formed rhythms as a

function of the ratio of

long and short beat sizes.

The patterns are, from

highest evenness to lowest

evenness: (4ℓ,1s),(3ℓ,2s),

(2ℓ,3s),and(1ℓ,4s).The

minimal evenness values

occur when r→∞.In

general, for any

well-formed rhythm with

just one long

beat—e.g., (1ℓ,4s)—this

minimal evenness value

approaches zero, for the

other well-formed rhythms

it approaches values

greater than zero.

long and short beats ( j,k), and the beat-size ratio

r(all of which can be controlled in MeanTimes’s

user interface). It is also trivial to generate different

“modes” (starting points) for the rhythms by rotating

the well-formed word.

Because rhas an infinite range, we scale it

by adjusting a parameter t∈[0, 1], such that r=

1/(1 −t). Hence t=0 implies r=1, and t=1

implies r=∞.Thisensuresthatwhenther-slider

is in its halfway position (i.e., t=0.5), the resulting

rvalue is 2/1. This is a particularly important

beat-size ratio because—as shown in the next

section—it always implies that the next higher-level

well-formed rhythm (which may be thought of as

comprising pulses, some of which coincide with the

lower-level beat) is isochronous.

As previously discussed, as the value of ris

moved throughout its range, it affects the evenness

of the resulting rhythm. When r=1, the long and

short beats are of identical size, so the rhythm is

maximally even (isochronous). As ris increased, the

evenness reduces. For any given rvalue, however,

evenness also varies across different well-formed

rhythms. Usually, the evenness of well-formed

rhythms, all of the same number of events j+k,

is ordered by the size of j.Thatis,thegreaterthe

number of long beats, the greater the evenness of

that well-formed rhythm at any given rvalue. This

is illustrated in Figure 5, which shows the evenness

values for all five-beat well-formed rhythms over

1≤r<∞.Becausetheminimalevennessvalue

differs across rhythms, and also to keep the interface

simple, we do not allow the user direct control of the

evenness value. The changes in evenness (for each

level) are displayed in the interface of MeanTimes,

and they arise from the user-controlled changes in r,

or the other parameters.

ANestedHierarchyofRhythmicLevels

As previously discussed, every well-formed rhythm

is nested inside another higher-level well-formed

rhythm. This allows a hierarchy of such rhythms

to be played simultaneously or successively: Mean-

Times’s interface (Figure 1) includes a set of six

checkboxes to toggle playback of the user-defined

beat rhythm, and the next five higher levels of

pulses.

Let each successive rhythmic level be denoted

i,wherei=0isthebeatrhythm(whose jand k

values are entered by the user), i=1istheLevel

1 well-formed pulse rhythm that nests it, i=2is

the Level 2 well-formed pulse rhythm that nests

the Level 1 pulse rhythm, i=3 is the Level 3 well-

formed pulse rhythm that nests the Level 2 pulse

rhythm, and so forth. This means that any meter

is nested inside a hierarchy of higher-level pulse

rhythms, with each higher-level pulse rhythm level

being a superset of the previous.

We now make some more general statements

about the relationships of the well-formed rhythms

of successive levels. First, let us consider the

numbers of long and short pulses in level i+1asa

function of level i:

(ji+1,k

i+1)=$(ji+k

i,ji), ri≤2/1,

(ji,ji+k

i), ri≥2/1.(2)

A. J. Milne and R. T. Dean 45

Second, let us consider the sizes of the long and

short pulses:

(ℓi+1,si+1)=$(si,li−si), ri≤2/1,

(li−si,si), ri≥2/1.(3)

Equations 2 and 3 are a direct consequence of the

(i+1)th level being derived from the ith level by the

application of one of the morphisms described at

the start of this section (ℓ'→ ℓsand s'→ ℓ,orℓ'→ ℓs

and s'→ s).

Given that ri+1=(ℓi+1)/(si+1), and substituting

ℓi=risiinto Equation 3 implies

ri+1=⎧

⎪

⎨

⎪

⎩

1

ri−1,ri≤2/1,

ri−1, ri≥2/1.

(4)

Some pertinent corollaries flow from these

equations. Usually, once r0is set, r1,r2,r3,andr4

will each take different values. We will point out

some interesting cases. First, Equation 4 shows that

r0=2/1 implies r1=1/1. Furthermore, Equation 2

shows that when r0=2/1, there are two possible

solutions for ( j1,k1), which are ( j0+k0,j0)and

(j0,j0+k0). Together, these imply that both Level

1 rhythms are isochronous with 2 j0+k0pulses, so

the transition between them, which occurs when

r0=2/1, is seamless (no discontinuity).

Second, again when r0=2/1, the rhythm two

levels higher has r2→∞.Furthermore,Equation2

shows that this rhythm has two possible forms,

which are (2 j0+k0,j0+k0)and(2j0+k0,j0). To-

gether, these imply that both are isochronous

rhythms with 2 j0+k0pulses to the period because

the small-sized steps have shrunk to zero size.

This means the transition across this boundary

is also seamless. By iterating this procedure, it is

trivial to show that the same isochronous 2 j0+k0

pulse rhythm is the boundary between all pairs of

higher-level well-formed rhythms above r0=2/1.

These two corollaries mean that any given pulse–

pattern level has no discontinuities across the full

range of legal r0values. Hence it makes sense to

allow users to enable or disable the playing of

higher-level well-formed pulse-patterns for each

level irather than, for example, allowing them to

choose from among a variety of higher-level pulse-

patterns as and when they become available across

the r0continuum. In this way, multiple rhythmic

levels can play simultaneously, and the transitions

across all well-formed boundaries as the r0-slider’s

value is adjusted will always be seamless.

A final corollary is that whenever r0is irrational,

there will never be a higher rhythmic level that

is perfectly even (there is no rithat equals 1). For

many irrational numbers, however, the rifor certain

levels may closely approximate 1. Interestingly,

there is a class of irrational r0values where rivalues

never come close to 1, and produce what we call

deeply nonisochronous rhythms. These are r0values

of the form (aφ+c)/(bφ+d), where the fractions

a/band c/dare adjacent members from level sand

level s+1oftheStern-Brocottree,a/b<c/d,and

φ=(1 +√5)/2≈1.618 is the golden section. When

r0takes such a value, all ri≥s=φ(Wilson 1997).

As can be seen by substituting φinto Equation 4,

whenever a level has r=φ,allhigherlevelsget

“stuck” with this value and so can never come close

to 1 or ∞(the only two values that imply isochrony).

The only other value that gets stuck in this way is

∞(which implies isochrony). Indeed, all rational r0

values have higher levels that ultimately converge

on ∞,whilealmostallirrationalr0values have

higher levels that either “wander” through different

rvalues (in all likelihood some of which will be

close to 1 or ∞and hence close to isochrony) or

converge to φ.Wedenotether0values that reach φ

on the sth level φs.Thesevaluesareshownabove

the r-slider in MeanTimes. Other values of r0may

also produce rhythms with no perceptible isochrony

because the (approximately) isochronous level is too

fast, but the r0=φsvalues minimize convergence to

isochrony in the mathematical limit.

An Example of a Nested Hierarchy

The structure of well-formed rhythms across differ-

ent levels and r0values is therefore complex, yet

highly organized and patterned. We now give a more

concrete illustration of this structure, and of the

mathematics introduced in the previous subsection,

by focusing on the (2ℓ,3s)well-formedbeatrhythm

and the hierarchy of pulse rhythms that nest it over

46 Computer Music Journal

Figure 6. The hierarchy of

well-formed rhythms

nesting a ﬁve-beat (2ℓ,3s)

rhythm across the latter’s

full range of r0values. The

bottom half-diamond

represents the beat

rhythm, the diamonds

above represent the

nesting pulse rhythms. The

horizontal extent of each

rhythm’s diamond shows

the r0values over which it

exists. The numbers down

the side show the number

of events in the rhythms,

while the numbers inside

each half-diamond (e.g.,

5:2) show their numbers of

long and short pulses.

Some rivalues for each

well-formed rhythm are

shown directly above its

diamond. The dotted

vertical lines show the

locations where isochronic

pulses occur, and the

numbers of these pulses

are shown in the top row.

different r0values. This is illustrated in Figure 6,

which is explained forthwith.

Let us start by considering the beat rhythm,

which is shown by the bottom half-diamond. The

user has control of the beat rhythm’s r0value, and

the precise locations of some r0values are indicated

by the numbers directly above this diamond. As

the value of r0is changed, visualize a vertical line

passing over the entire height of the figure; any

horizontal diamond the line passes through is a

higher-level well-formed rhythm that nests all

lower-level rhythms (including the beat rhythm) at

that r0value. The horizontal extent of every such

well-formed rhythm indicates the range of r0values

over which its own rivalues are between 1 and ∞

(some rivalues for each well-formed rhythm are

shown directly above its diamond). The thickness of

each diamond gives a very approximate indication

of the rhythm’s evenness, which is maximal when

ri=1 and minimal when ri→∞(as was illustrated

in Figure 5). The numbers of long and short events

in each well-formed rhythm are indicated by the

figures inside each diamond. For example, the

label 2:3 means there are two long and three short

events. Note how each well-formed pattern swaps

its numbers of long and short events as it passes

through ri=1/1, as implied in Equation 2. Also

note that when ri→∞,thewell-formedrhythmis

degenerate and its total number of events reduces

because its short events vanish.

The level iof any well-formed rhythm is given by

the number of diamonds the r0line passes through

to get to it. Hence the diamond labeled 5:2 and 2:5

is Level 1; the diamond labeled 5:7 and 7:5 and the

diamond labeled 7:2 and 2:7 are both Level 2; the

diamond labeled 7:9 and 9:7 and the diamond labeled

9:2 and 2:9 are Level 3; and so forth (each successive

level is successively colored with alternating black

and white). At any given level, there will always be

awell-formedrhythmavailable(theapparentgaps

at levels 3, 4, and 5 are actually filled by higher-level

well-formed rhythms “above” the current figure).

Let us demonstrate how this works by considering

the nested rhythms that are available when r0=

3/2. At this r0value, the Level 0 beat rhythm—

represented by the bottom diamond—is a rhythmic

analogue of the standard pentatonic scale (which

may help the visualization and auralization). It is

analogous because the pentatonic scale has two

large and three small steps and, assuming a standard

twelve-tone equal temperament (12TET) tuning,

a step-size ratio of 3:2 (its large steps are three

semitones and its small steps are two semitones).

The Level 1 rhythm in the next row up has seven

A. J. Milne and R. T. Dean 47

Figure 7. The evenness,

over all r0values, of the

(2ℓ,3s)“pentatonic”

rhythm and its next three

higher “chromatic” levels.

Note that the r0values on

the horizontal axis refer to

the step-size ratio for the

lowest (2ℓ,3s)level, not

that of the higher-level

rhythmic streams.

nonisochronous pulses and nests the five-beat

pattern. It comprises five long and two short pulses,

and it has a step-size ratio (rvalue) of 2:1 (and so is

arhythmicanalogofthe12TETdiatonicscalewith

five two-semitone steps and two single-semitone

steps). There is also a Level 2 rhythm that nests both

the previous two levels (shown by the next black

diamond up), which contains twelve pulses. When

r0=3/2fortheLevel0rhythm,r2=1fortheLevel

2rhythm.Thismeansthatallthelatter’spulsesare

isochronous, hence this is the rhythmic analog of

the equal-tempered chromatic scale. All higher-level

rhythms have twelve isochronous pulses.

As the value of the beat rhythm’s r0-slider is

changed, MeanTimes automatically selects the

appropriate well-formed rhythm for each higher

level, and finds the appropriate event timings by

recursion of Equations 2–4.

Evenness of the Well-Formed Hierarchy

We previously showed, in Figure 5, how the evenness

of each possible five-beat well-formed rhythm

changes over all r0values. Clearly, the same process

also occurs for the higher-level well-formed rhythms

too. In general, each will have an analogous curve

(maximal at ri=1, minimal at ri→∞)butovera

smaller r0range as levels are ascended. An example,

for the well-formed rhythms, up to Level 3, that nest

the two-long-, three-short-beat rhythm is illustrated

in Figure 7.

In MeanTimes’s interface, the evenness values

for the beat-level and each of the three higher-level

pulses are displayed as bar graphs, which are updated

in real time as the rslider is moved. The color of

the bar also indicates whether each rhythm level

is coherent (its rvalue is less than two), and each

level’s rvalue is also displayed.

Extensions and Modiﬁcations of

Well-Formed Rhythms

In the earlier section “Well-Formed Words,” we

formalized the splitting of each long beat into

anewlongandshortbeatwiththemapping

ℓ'→ ℓs. We could have chosen the opposite ordering:

ℓ'→ sℓ. This implies changing the first morphism

ℓ'→ ℓsand s'→ ℓ(as defined earlier) to ℓ'→ sℓ

and s'→ ℓ;andchangingthesecondmorphism

ℓ'→ ℓsand s'→ s(as defined earlier) to ℓ'→ sℓand

s'→ s.Forconvenience,wewillrefertolevels

resulting from the original morphisms as clockwise,

and those originating from the just introduced

alternatives as counterclockwise. Interestingly,

clockwise and counterclockwise levels are identical

up to rotation (or reflection). This means that by

independently changing the directionality of the

levels in a hierarchy, their relative rotations change.

48 Computer Music Journal

This results in differing beats being duplicated

across the levels. Using a pitch-based analogy,

consider the pentatonic scale A–C–D–E–G. The

clockwise morphism gives A–B–C–D–E–F♯–G; the

counterclockwise morphism gives A–B♭–C–D–E–F–

G. Both resulting scales are diatonic, but the scale

degrees of the notes duplicated in the lower-level

pentatonic scale differ. Numbering from the first

degree of the major scale (G in the first diatonic scale,

and F in the second), the first scale is duplicated

at ˆ

1–ˆ

2–ˆ

4–ˆ

5–ˆ

6, while the second scale is duplicated

at ˆ

2–ˆ

3–ˆ

5–ˆ

6–ˆ

7. These two versions of the diatonic

hierarchy are illustrated, respectively, in Figures 4a

and 4b. Each rhythmic level in MeanTimes has a

directional toggle allowing a wide variety of different

patterns of duplications and rotational relationships

between levels.

Another interesting modification that can be

applied in MeanTimes is to use what we call the

complement of any given rhythmic level. This

can be used to eliminate beat duplication across

levels. As just described, the next level above the

pentatonic scale C–D–E–G–A is the diatonic C–

D–E–F♯–G–A–B (assuming a clockwise morphism).

Instead of using all the notes in the higher level,

it is possible to use just those notes that are not

found in the lower level; in this case, F♯and B. In

this way, each higher level only splits long beats

of the lower level (it fills in the gaps), it no longer

duplicates the lower level. This can be used to

produce multiple interlocking patterns that, in

combination, fully delineate a pulse. Interestingly

(as can be shown from Amiot 2007, Proposition 3.2),

these complementary patterns are also well-formed.

This is illustrated in Figure 4, where the patterns

emanating from a rhythm analogous to the pitch-

class set C–F–G in 12TET (Level 0 has two large

and one small beat, and r0=5/2) are shown when

complementary mode is not engaged for any level

(Figures 4a and 4b), and when it is engaged for all

levels (Figure 4c).

With the hierarchy of well-formed rhythms in

place, it is easy to introduce the possibility of

“chromatic” alteration, or modification, of any

well-formed rhythm. We use the term chromatic to

re-emphasize the relation between meter and scale,

because a chromatic modification can create a new

beat size, distinct from sor ℓ,ratherasachromatic

modification of a scale member (even if it is to an

equal-tempered alternative pitch) may create new

interval step sizes in the result. To do this, we

simply displace any given event by an amount that

is equivalent to the new pulse size that is introduced

by the next higher-level rhythm.

As shown in Equation 3, when a Level 0 well-

formed pattern is advanced to Level 1, the long beat

ℓ0is split into two sizes ℓ1and s1. Furthermore, one

of these takes the size ℓ0−s0,andtheothertakes

the size s0. By analogy with musical scales, we take

the new size ℓ0−s0as the size of the chromatic

alteration; that is, a Level 0 event is “sharpened”

by delaying it by ℓ0−s0,anditis“flattened”by

advancing it by ℓ0−s0. In MeanTimes, a specific

alteration is achieved by dragging a polygon vertex,

and it will snap to a pulse position from the next

higher level. The corresponding beat at all lower

levels will also move in tandem. Making any single

such modification results in a number of different

possibilities for the form of the resulting rhythm:

The new rhythm may simply be a transposition

(mode change) of the original; it may create a rhythm

with the same two beat lengths but arranged in a

non–maximally even form (not making a Christoffel

word, or rotation thereof); it may create a rhythm

with three beat lengths; or it may create a rhythm

with four beat lengths.

We now give examples of such possibilities by

looking at analogous chromatic alterations of the C

major scale. If just the F is sharpened, the resulting

scale is simply a transposition of the original (it has

moved from C major to G major or, equivalently,

from Ionian to Lydian mode). If just the E is flattened,

the resulting scale is the ascending melodic minor,

which contains the original two step sizes (major

and minor seconds) but now arranged in a non-

well-formed pattern. If just the G is sharpened, the

resulting scale is the non-well-formed harmonic

minor scale, which now contains three step sizes

(there is an additional augmented second between F

and G♯). Similar results are obtained if just the A is

flattened, which gives the harmonic major scale.

As shown in Figure 8, all but the first of these

modifications will reduce the analogous rhythm’s

evenness. This is of obvious compositional utility as

A. J. Milne and R. T. Dean 49

Figure 8. The evenness,

over all rvalues, of ﬁve

different seven-beat

patterns. From most to

least even: diatonic (e.g.,

C–D–E–F–G–A–B),

melodic minor (e.g.,

C–D–E♭–F–G–A–B),

harmonic major or minor

(e.g., C–D–E–F–G–A♭–B or

C–D–E♭–F–G–A♭–B), and

double harmonic (e.g.,

C–D♭–E–F–G–A♭–B). The

harmonic major and

harmonic minor are

inversionally equivalent

and so have identical

evenness.

Figure 8

Figure 9. The entropies of

different seven-beat

patterns. Entropies are

calculated for sequences

(n-tuples) of consecutive

event sizes.

Figure 9

a motivic transformation device, but also of interest

in terms of the resultant change in metricality and

rhythmic transparency.

The rvalue has no effect on the entropy of event

sizes, but the actual form of the scale does. This is

shown in Figure 9, which shows the effect on entropy

of the given modifications of well-formedness.

The entropies are calculated from probability

distributions over n-tuples of consecutive events,

with ntaking values from one to five We will

show how these are calculated by considering the

diatonic pattern of ℓℓsℓℓℓs. Stepping through the

diatonic pattern from left to right, the following

2-tuples occur: ℓℓ,thenℓs,thensℓ,thenℓℓ,then

ℓℓ,thenℓs, and finally sℓ(proceeding from the last

element to the first). Hence, the probability mass

function over all possible 2-tuples is p(ℓℓ)=3/7,

p(ℓs)=2/7, p(sℓ)=2/7, and p(ss)=0. The entropy

(in bits) of a probability mass function is calculated

by H(p)=−)ipilog2pi; hence the entropy for this

example is 1.56.

Stepping through the diatonic pattern from left

to right, the following 3-tuples occur: ℓℓs,thenℓsℓ,

then sℓℓ,thenℓℓℓ,thenℓℓs,thenℓsℓ, and finally sℓℓ.

Hence, the probability mass function over 3-tuples is

p(ℓℓℓ)=1/7, p(ℓℓs)=2/7, p(ℓsℓ)=2/7, p(sℓℓ)=2/7,

p(ℓss)=0, p(sℓs)=0, p(ssℓ)=0, and p(sss)=0. This

has an entropy of 1.95. And so forth.

50 Computer Music Journal

Compositional, Analytical, and

Perceptual Implications

MeanTimes permits systematic generation and

perturbation of well-formedness of a meter, and con-

tinuous variation of long/short beat ratio r,which

produces continuous variation in the evenness of the

rhythms generated. Furthermore, different combina-

tions of the levels of the metrical hierarchy (where

increasing level corresponds to shortening event

lengths) may produce different pulse and apparent

beat positions, in addition to those of the lowest

beat level, such that new meters are created without

change in period length. Compositionally, all these

features are of interest for systematic control, and

many of the rhythmic instantiations would not

be readily accessible by manual composition nor

manually performable by instrumentalists. They are

thus quintessentially computer-dependent rhyth-

mic compositional devices, permitting rhythmic

morphing that has as yet been little explored.

To make this conclusion clear, we can consider

some of the compositional rhythmic complexities

that have been used in instrumental works and

then, in contrast, some mechanical or computerized

alternatives. Serialization of rhythm or the use of

irrational rhythms results in rhythmic and metrical

irregularity (even if, particularly in the case of

serialization, rhythmic patterns may repeat often).

But these techniques do not readily allow continuous

variation of event-size ratios, especially in the case

of serialized rhythm. With irrational rhythms or

spatial (i.e., proportional) notation any sequence

of event sizes and durations can be notated, but

the approach does not in itself imply a systematic

way of controlling such sequences, nor would the

resultant score be likely to receive a performance

that completely fulfilled its specifications.

Elliott Carter’s metric modulation (Hobert 2010),

or its improvised relatives in some work of Miles

Davis (1964–1966) or subsequent groups such as

Chick Corea’s ensemble Circle (Dean 1992), offer

devices for abrupt metrical change, with or without

change in period length. For example, Carter often

notates transitions in which a unit of length (be it

pulse or bar) is retained but subdivided by a different

integer. Related devices occur in Davis’s work, but

improvisation does allow more continuous variation

in such changes, as heard in recordings such as those

by Circle and others.

It is perhaps in the mechanically performed and

rhythmically complex music of Conlon Nancarrow

that we find an antecedent to MeanTimes, although

(as far as we are aware) there is little in his writing

or in analyses of his work to suggest a systematic

use of the ideas developed here. Nancarrow was

most interested in coexistence of multiple tempi

and meters, whose barlines rarely coincide, and in

trying to “synthesize the two opposing conceptions

[divisive, additive] of rhythm” (Gann 2006, p. 7).

There is no doubt that a player-piano performance

system such as he used could, like a present-day

computer, realize the rhythmic patterns MeanTimes

can produce.

The combination of perturbing well-formedness

and morphing rhythmic ratios and periods means

that MeanTimes provides a novel compositional

and improvisational tool, and further extensions

to the algorithm can be readily envisaged. Using

the algorithmic approach described here, intel-

ligent compositional input is still required: not

all well-formed rhythms, or transitions between

them, will sound appropriate. Furthermore, effec-

tive choices for the timbres or pitches used for the

multiple levels are still required. Our composing

and performing experience to date with MeanTimes

suggests, however, that we can indeed make inter-

esting music with it, including morphing in and

out of a feeling of groove, and finding some deeply

nonisochronous grooves in the process. We hope to

use it more extensively in composition and impro-

visation in the future. An audio recording of the

first performance with it, given by the two authors

at the Sydney Conservatorium of Music in 2014,

is provided as an online supplement to this arti-

cle (www.mitpressjournals.org/doi/suppl/10.1162

/COMJ a00343). We also include a variety of

demonstrations.

Composition and improvisation are both creative

and analytic processes, and MeanTimes has the

potential to contribute also to the analysis of

rhythm and its performance. At the broadest level,

the theories of well-formedness and evenness offer

new descriptive and comparative approaches to

A. J. Milne and R. T. Dean 51

rhythmic and metrical structure. MeanTimes may

also offer an enhanced approach to the analysis

of rhythmic variation in performance. It is not

most commonly the case that metrically conceived

Western or Asian classical music is performed

with a precision approaching isochrony or precise

repetition of beat sizes. Rather, variation in such beat

sizes is used as an expressive vehicle. By considering

the extrema of beat-size ratios exhibited in timing

variations, in relation to the evenness constraints

of MeanTimes, further insight may be obtained

as to how they may operate and be optimized for

expression.

Acomposer’sorimprovisor’ssubjectiveimpres-

sions of the interest or degree of groove of rhythmic

patterns are of course of central importance to that

person, even though they may be capable of in-depth

rhythmic analysis of the kind just indicated. But

what of the average (musically untrained) listener?

Clearly subjective impressions (i.e., perception) are

necessarily dominant, as they normally do not have

any analytic framework to use as an alternative. The

study of perception of metricality, particularly by

nonmusicians, can be based on at least two levels of

response. First, does a person reproducibly identify

acyclelengthasthemeterisenunciated?This

can be tested in tapping experiments, by asking a

participant to tap along with the repeating metrical

pattern such as to represent any repeating features

the listener notices. The responses can be quantified

in terms of the proportion of times the cycle length

is correctly tapped. In addition, if a respondent taps

in several different positions within the pattern,

these data begin to address the second level of

analysis of perception of metricality: the question

of whether listeners can also identify the repeating

subcycles, or beats, of the meter.

In the absence of strong perceptual data, it is not

justified to speculate extensively about predictors

of metricality. But, overall, we predict that well-

formed meters will be more readily perceptible

than others and that the more even a meter, or its

rhythmic instantiation, the more well perceived

that meter will be. These predictions are based

both on the overall expectation that the higher

the entropy of the pattern, the less likely it is

that the period will be detected and on our own

experience in performing with and evaluating the

rhythmic outputs of MeanTimes.

Conclusion

Our mathematically driven approach to meter treats

events on pulse and beat positions with equity in

relation to evenness, such that well-formed patterns

include not just the simple regular isochronous beats

of conventional 3

4,4

4,6

8,andsoforth,butalsopatterns

comprising pulses organized in two beat sizes—such

as those found in Balkan music. Furthermore, well-

formedness can in this way be defined for patterns

with no underlying isochronous pulse (the beat or

pulse lengths have no common divisor). This means

their long and short beat sizes can be covaried

across a continuum, thereby creating rhythms and

meters that narrowly miss perfect isochronies, or

more radical patterns where obvious isochronies are

absent. Such variation also encompasses a variety

of common non-well-formed meters (again with no

requirement for isochrony at the pulse level). In

these ways, well-formed patterns can be used as

the starting point for creating systematic families

of rhythms that may have musical and perceptual

interest.

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