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Computational Creation and Morphing of Multilevel Rhythms by Control of Evenness

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The published version is available at http://www.mitpressjournals.org/doi/abs/10.1162/COMJ_a_00343#.VwZByMcq7X8. The application "MeanTimes" described in this paper has been subsumed by the new application "XronoMorph" available at http://www.dynamictonality.com/xronomorph.html 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.
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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 and
s), repeating subsequences
(e.g., there are two
instances of the sequence
swithin 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, ...,M1. 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 (8.6in 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 ,therearekbeats of size s,where j
and kare coprime, and there are no beats with a size
that is not or s.Foraperiodoftotallengthd,this
implies d=j+ks.Inageneralsetting,and scan
take any possible sizes. For this article, however,
we will apply the additional constraint that s;
that is, that the -sized beats are “long” or “large”
and the s-sized beats are “short” or “small.” (In a
pitch-based context, the use of ,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>,butthesconstraint simplifies some of the
exposition.
We additionally denote the step-size ratio r=
/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 m1consecutivebeats.
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 and 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” (sss). Such patterns minimize,
as much as possible, the clumping together of the
-sized beats and the clumping together of the
s-sized beats—put differently, the -sized beats are
spaced as far apart as possible, and so are the s-sized
beats.
By covarying the values of and s(and hence r),
while simultaneously ensuring that the length of
the period ( j+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:when=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 ,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 (s)isnestedin
the well-formed “tetractys” D–G–A (s), which
is nested in the well-formed five-note pentatonic
scale D–F–G–A–C (sss), which is nested inside
the well-formed seven-note diatonic scale D–E–
F–G–A–B, C (sℓℓℓs), 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, =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 and s,
respectively: Their ratio r=/scan be smoothly
varied with a slider along the entire continuum
that preserves that rhythm’s structure (i.e., given
jand k,therange1r<within which s).
The sizes of and 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 and 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 s, then make
two different pairs of substitutions (each pair
of substitutions is termed a morphism): the first
morphism is '→ sand s'→ ;thesecondmorphism
is '→ sand s'→ s.Applyingthefirstmorphismto
sgives s;applyingthesecondmorphismgives
ss,hencetwonewwordshavebeengenerated.The
two morphisms are then re-applied to the two new
words, resulting in four more new words: sℓℓs,
sss,sℓℓ,andsss. Applying the morphisms again
results in eight new words and, in general, the ith
iteration provides 2inew words, making a total of
2i+11words.Bothmorphismscanbeunderstood
as splitting each long beat into a new long and
anewshortbeat('→ s). In the first morphism,
the size of the new long beat corresponds to the
size of the old short beat (s'→ ); 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 GE;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 and sin
the period. It does not, however, define the values
of and s.Forthat,thelength(temporalduration)
dof the period and the ratio rof the two step sizes
are required. As previously established, d=j+ks,
and by definition, r=/s.Thesetwoequations
imply that =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: (4,1s),(3,2s),
(2,3s),and(1,4s).The
minimal evenness values
occur when r→∞.In
general, for any
well-formed rhythm with
just one long
beat—e.g., (1,4s)—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
1r<.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)=$(ji+k
i,ji), ri2/1,
(ji,ji+k
i), ri2/1.(2)
A. J. Milne and R. T. Dean 45
Second, let us consider the sizes of the long and
short pulses:
(i+1,si+1)=$(si,lisi), ri2/1,
(lisi,si), ri2/1.(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 ('→ sand s'→ ,or'→ s
and s'→ s).
Given that ri+1=(i+1)/(si+1), and substituting
i=risiinto Equation 3 implies
ri+1=
1
ri1,ri2/1,
ri1, ri2/1.
(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)/21.618 is the golden section. When
r0takes such a value, all ris=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 (2,3s)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 (2,3s)
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
(2,3s)“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 (2,3s)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.Thismeansthatallthelatterspulsesare
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
'→ s. We could have chosen the opposite ordering:
'→ s. This implies changing the first morphism
'→ sand s'→ (as defined earlier) to '→ s
and s'→ ;andchangingthesecondmorphism
'→ sand s'→ s(as defined earlier) to '→ sand
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, Fand 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 ,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
0is split into two sizes 1and s1. Furthermore, one
of these takes the size 0s0,andtheothertakes
the size s0. By analogy with musical scales, we take
the new size 0s0as the size of the chromatic
alteration; that is, a Level 0 event is “sharpened”
by delaying it by 0s0,anditis“flattened”by
advancing it by 0s0. 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,thens,thens,thenℓℓ,then
ℓℓ,thens, and finally s(proceeding from the last
element to the first). Hence, the probability mass
function over all possible 2-tuples is p(ℓℓ)=3/7,
p(s)=2/7, p(s)=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,thens,
then sℓℓ,thenℓℓℓ,thenℓℓs,thens, and finally sℓℓ.
Hence, the probability mass function over 3-tuples is
p(ℓℓℓ)=1/7, p(ℓℓs)=2/7, p(s)=2/7, p(sℓℓ)=2/7,
p(ss)=0, p(ss)=0, p(ss)=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.
Acomposersorimprovisorssubjectiveimpres-
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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A. J. Milne and R. T. Dean 53
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