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XronoMorph: Algorithmic Generation of Perfectly
Balanced and Well-Formed Rhythms
Andrew J. Milne
MARCS Institute, Western
Sydney University
NSW 2751, Australia
a.milne@westernsydney.edu.au
Steffen A. Herff
MARCS Institute, Western
Sydney University
NSW 2751, Australia
s.herff@westernsydney.edu.au
David Bulger
Macquarie University
Sydney
NSW 2109, Australia
david.bulger@mq.edu.au
William A. Sethares
University of
Wisconsin-Madison
WI 53706, USA
sethares@gmail.com
Roger T. Dean
MARCS Institute, Western
Sydney University
NSW 2751, Australia
roger.dean@westernsydney.edu.au
ABSTRACT
We present an application—XronoMorph—for the algorith-
mic generation of rhythms in the context of creative compo-
sition and performance, and of musical analysis and educa-
tion. XronoMorph makes use of visual and geometrical con-
ceptualizations of rhythms, and allows the user to smoothly
morph between rhythms. Sonification of the user generated
geometrical constructs is possible using a built-in sampler,
VST and AU plugins, or standalone synthesizers via MIDI.
The algorithms are based on two underlying mathemati-
cal principles: perfect balance and wel l-formedness, both
of which can be derived from coefficients of the discrete
Fourier transform of the rhythm. The mathematical back-
ground, musical implications, and their implementation in
the software are discussed.
Author Keywords
music, rhythm, scales, balance, evenness, perfect balance,
well-formedness, discrete Fourier transform
ACM Classification
H.5.5 [Information Interfaces and Presentation] Sound and
Music Computing --- Systems
1. INTRODUCTION
The composition and performance of interesting rhythms
is often hindered by the limitations of traditional mu-
sic notation and conceptualization. Simultaneously, many
new music students struggle in generating an intuitive
understanding of poly-rhythmic structures. This paper
presents XronoMorph, an application designed for the visual
and geometrical exploration and construction of interesting
rhythms based on two underlying mathematical principles:
perfect balance (PB ) and well-formedness (WF ).
As shown in Figures 2–4, the temporal structure of pe-
riodic rhythms, meters, riffs and ostinatos can be conve-
niently represented as points on a circle: the clockwise angle
Licensed under a Creative Commons Attribution
4.0 International License (CC BY 4.0). Copyright
remains with the author(s).
NIME’16, July 11-15, 2016, Griffith University, Brisbane, Australia.
.
of each point indicates when it is sounded and the circular-
ity represents the rhythm’s periodicity.
1.1 An Introduction to XronoMorph
XronoMorph has a PB mode and a WF mode for the re-
spective classes of rhythms (PB mode in Fig. 1a, WF mode
in Fig. 1b). The rhythmic patterns are visualized by poly-
gons inscribed in a circle. A small disk rotates clockwise
around the circle and when it hits a polygon vertex a MIDI
event is triggered. The speed at which the disk rotates (the
length of the period) is controlled by the long horizontal
slider at the top. Each rhythm is visualized with its own
configuration of underlying polygons. Each such polygon
can be assigned a MIDI pitch, velocity, duration, and chan-
nel, and directed to up to three, out of a total of twelve,
tracks. Each of these twelve tracks can be thought of as an
“instrumentalist” who plays any polygon being sent to it.
Each track produces sound from a built-in sampler, from
a plugin AU or VST synthesizer, or directs the MIDI to a
port to drive a standalone software or hardware synthesizer.
In this way an “ensemble” of twelve “instrumentalists” can
be formed, and each polygon can be played by up to three
of these “instrumentalists”. This means that the orchestra-
tion/sonification of a given rhythm can be easily changed
during performance and composition.
A large number of user presets can be stored allowing
rhythms to be easily switched between during live perfor-
mance. Rhythms can also be saved as MIDI or audio loops
for later processing in a sequencer or digital audio work-
station; alternatively, they can be saved as Scala scale files,
allowing XronoMorph to be used for designing perfectly bal-
anced and well-formed microtonal scales.
XronoMorph realizes three principal novelties. Firstly,
perfect balance is a recently developed concept [12], which
has not previously been instantiated in a musical applica-
tion. Secondly, although well-formedness (equivalently mo-
ments of symmetry) is a well-established theoretical concept
applied to scales and rhythms [20, 5], it has not been real-
ized in a rhythm application (existing pitch-based applica-
tions include Hex [15] and associated synthesizers [14, 18]);
furthermore, our parameterization of WF and the sounding
of a full hierarchy of WF rhythms is novel [13]. Thirdly, be-
cause the PB and WF rhythms have continuously variable
parameters, differing rhythms can be smoothly morphed
between. We are aware of only one other rhythm app—
Rhythmorpher—designed for rhythmic morphing, and this
uses a very different set of rhythmic parameters [23].
The mathematical principles of evenness and balance,
388
(a) XronoMorph in PB mode: A sum of perfectly balanced rhythms is depicted by polygons inscribed in a circle.
The controls below the circle allow the type of polygon to be chosen and its rotation to be smoothly adjusted.
(b) XronoMorph in WF mode: A hierarchy of well-formed rhythms is depicted by polygons inscribed in a circle.
The large horizontal slider at the bottom (the r-slider) is used to smoothly control beat size ratio.
Figure 1: The user interface of XronoMorph in PB mode (a) and WF mode (b). The rhythm is represented
by polygons 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, and channel specific to
that polygon. The controls below-right the circle allow each polygon’s MIDI parameters to be specified. At
the top is a slider to control the length of the period of rhythmic repetition (the tempo). To the right of the
polygons are tracks, which can play built-in samples, host plugins to sonify the polygons, or send MIDI to
standalone synthesizers. To the right is a large bank of slots where users can store their rhythms as presets.
389
which underlie well-formedness and perfect balance, as well
as their implementation in XronoMorph, are now detailed.
1.2 Mathematical Background
A natural mathematical representation of points on a unit
circle (e.g., rhythmic events) is as a vector of unit-magnitude
complex numbers arranged in circular order. Building on
research by Lewin [10], Quinn [16], and Amiot [2], in re-
cent work [12, 13] we have shown how the magnitudes of
the first two coefficients of the discrete Fourier transform
of this vector identify two musically relevant properties of
the resulting rhythm. Unity minus the magnitude of the ze-
roth coefficient quantifies the rhythm’s balance, whilst the
magnitude of the first coefficient quantifies the rhythm’s
evenness. Balance measures the distance of the rhythm’s
centre of gravity (mean position) from the centre of the cir-
cle; evenness measures the similarity of the rhythm to an
isochronous rhythm with the same number of events (ignor-
ing their relative phases).
More formally, the vector x∈[0,1)Khas Kreal-numbered
time values (for the Krhythmic events) normalized to lie
between between 0 and 1 (the period has a size of 1), and
ordered by size so x0< x1<· · · < xK−1. For example,
for the ˇ“ˇ“ˇ“(ˇ“ˇ“ˇ“ˇ“(“diatonic” rhythm used in Sub-Saharan
African music [17], x=0
12 ,2
12 ,4
12 ,5
12 ,7
12 ,9
12 ,11
12 . The el-
ements of this vector are then mapped to the unit circle in
the complex plane with z[k] = e2πix[k]∈C, so z∈CK.
Each complex element z[k] of zhas unit magnitude, and
its angle represents its time location as a proportion of the
period (whose angle is 2π).
The tth coefficient of the discrete Fourier transform of the
scale vector is given by
Fz[t] = 1
K
K−1
X
k=0
z[k] e−2πitk/K .(1)
As outlined above, the zeroth and first coefficients respec-
tively quantify balance and evenness.
2. BALANCE AND PERFECT BALANCE:
THE ZEROTH COEFFICIENT
Unity minus the magnitude of the zeroth coefficient of the
DFT of xgives the balance of the rhythm:
balance = 1 − |F z[0]| ∈ [0,1] , where
Fz[0] = 1
K
K−1
X
k=0
z[k].(2)
The maximum possible value for balance is 1, and this is
termed perfect balance (PB ). The balance of a rhythm can
be thought of as the distance of the rhythm’s centre of grav-
ity (mean position) from the centre of the circle. If the bal-
ance is 1, the centre of gravity is precisely at the centre of
the circle. This means that if each rhythmic event were a
weight placed onto the rim of a vertical bicycle wheel, the
wheel will have no preferred rotation.
Rhythms with equally-sized steps (isochronous rhythms,
which are also perfectly even) are PB. Isochronous rhythms
can be visualized by a regular K-gon placed within a circle
(e.g. one of the squares in Figure 2a). However, there is
also a complicated manifold of irregular (non-isochronous)
perfectly balanced rhythms. Obtaining perfect balance un-
der additional constraints provides a useful way to narrow
down the manifold of possibilities to a smaller selection of
musically interesting rhythms.
An important consequence of the definition of perfect bal-
ance is that the sum of any two or more PB rhythms is also
(a) PB shuffle rhythm com-
prising two squares (4-
gons). The greatest com-
mon divisor of 4 and 4 is 4,
hence the rhythm has rota-
tional symmetry.
(b) PB rhythm comprising
two digons (2-gons) and an
equilateral triangle (3-gon).
The greatest common divi-
sor of 2, 2, and 3 is 1, hence
this rhythm does not have
rotational symmetry.
Figure 2: Two perfectly balanced rhythms in a 12-
fold isochronous “grid”. The first rhythm is PB over
the whole period, but not over its fundamental pe-
riod of repetition, which is one quarter of the circle.
The second rhythm is one of just two PB rhythms
in 12 that are PB over their fundamental period of
repetition.
PB (as shown in Figs. 2 and 3); as we show later, this en-
ables complex multilayered rhythms to be constructed from
the summation of simpler PB rhythms.
2.1 Perfectly Balanced Sums
In order to constrain perfect balance, we start with a re-
quirement for the rhythm to be a subset of Nisochronous
pulses (i.e., all its events align with an N-fold grid); though
this constraint is later relaxed when more than one PB
rhythm is combined.
For any Nthat is prime, there is only one perfectly bal-
anced pattern, which is simply a regular N-gon (prime N
are, therefore, not of great interest with perfect balance).
However, when Nis not prime, perfectly balanced
rhythms can be formed from the sum (union) of regular
K-gons where K|N(which means Kdivides N).
For example, when N= 12, we can combine 2-gons
(digons), 3-gons (equilateral triangles), 4-gons (squares),
and 6-gons (regular hexagons), each such regular polygon
being placed in one of its N/K distinct rotations. Any sum
of such polygons will produce a rhythm that is, in total, bal-
anced. Of particular interest are those rhythms where the
set of differently sized Kare coprime (no common divisors
greater than 1)—such rhythms do not have rotational sym-
metry and are perfectly balanced over their fundamental
period of repetition. This is illustrated in Figure 2.
For any Nthat is a product of no more than two dis-
tinct primes, all possible perfectly balanced rhythms can be
formed by summing regular K-gons where Kis prime and
K|N; thus the regular K-gons are elemental.
Those Nthat are the product of three or more distinct
primes, however, are particularly interesting. Between 1
and 100, the only such values are 30, 42, 60, 66, 70, 78, 84,
and 90. They contain PB patterns that cannot be created
from a simple sum of regular K-gons; these PB patterns
can only be created from integer combinations of regular K-
gons (i.e., subtracting as well as adding polygons, thereby
allowing vertices to be cancelled out [12]). (Indeed, all PB
patterns in an N-fold grid can be produced from an integer
combination of regular polygons [12, Thm. 5].) Figure 3
shows two such patterns in N= 30.
These two patterns exemplify a useful property, which is
that there is no PB polygon that can be subtracted without
producing a sonically “nonsensical” (hence illegal) negative
390
5/30
6/30
12/3018/30
24/30
25/30
(a) triangle + pentagon −
digon make a 6-element pat-
tern in a 30-fold period.
0/30
6/30
7/30
13/3017/30
23/30
24/30
(b) 2 digons + 3 pentagons
−3 digons −2 triangles in
a 30-fold period [3].
Figure 3: Perfectly balanced integer combina-
tions of intersecting regular polygons in a 30-fold
isochronous grid. When the vertex of one positive-
weighted polygon (solid line) coincides with the ver-
tex of one negative-weighted polygon (dashed line)
they cancel out to zero.
weight. Thus these patterns are also elemental, albeit irreg-
ular. So, for any N, a set of elemental rhythms exists, such
that all possible PB rhythms can be constructed by only
summing elemental rhythms (subtraction no longer being
necessary). If Nhas more than two prime factors, then
some of its elemental rhythms are irregular: in N= 30,
there are 6 irregular elemental PB patterns; in N= 42,
there are 18 such patterns; in N= 66, there are more than
100. Clearly, for larger values of Nwith three or more prime
factors, the number of such patterns explodes.
2.2 Perfect Balance in XronoMorph
In order to accommodate a musically sufficient number of
possibilities, XronoMorph allows the following PB rhythms
to be chosen and summed: all regular K-gons up to 12,
all regular prime-K-gons up to 29, and all six irregular el-
emental polygons in 30. This allows a wide variety of PB
rhythms to be produced (in future versions, we plan to allow
a wider variety of polygons to be specified or generated by
the user and stored as presets). Each such polygon can be
independently rotated—either snapping to a specified N-
grid, or smoothly (thereby allowing PB rhythms that are
not grid-based).
The principal user-parameters for defining PB rhythms
are the choice of polygons (up to 8 may be simultaneously
sounded) and the independent rotation of each of these poly-
gons. The circle in Figure 1a shows a rhythm that consists
of five underlying PB geometrical shapes, each of which has
been independently rotated.
3. EVENNESS AND WELL-FORMEDNESS:
THE FIRST COEFFICIENT
As first shown by Amiot and Noll [2], the magnitude of the
first coefficient of the DFT of xgives the evenness of the
rhythm:
evenness =|Fz[1]| ∈ [0,1] , where
Fz[1] = 1
K
K−1
X
k=0
z[k] e−2πik/K .(3)
The maximum possible value for evenness is 1, and this is
termed perfect evenness. The evenness of a K-event rhythm
can be thought of as a quantification of its similarity to a
K-equal division of the period that has been rotated so
as to maximize this similarity. Following from this, the
only rhythms that are perfectly even are those with equally-
sized steps (isochronous rhythms, or regular K-gons). This
is quite different to perfect balance, where there is a con-
tinuum of possibilities. However, as we show later, when
evenness is maximized under constraints that imply per-
fect evenness is unobtainable, musically interesting results
occur—notably, when we constrain the rhythm to contain
no more than two interonset intervals (IOIs), the resulting
rhythms are well-formed [5].
3.1 Well-Formed Hierarchies
There are two commonly discussed types of periodic rhythm
(or, analogously, scale) that result from maximizing even-
ness under musically sensible constraints. The first are Eu-
clidean rhythms [19] (which can be generated by existing
apps such as SequenceApp, Rhythm Necklace, Euclidean
sequencer, Gibber, and many others), which result when
evenness is maximized under the constraint of Kevents
in an N-fold isochronous grid (interesting when Nand K
are coprime). The second are well-formed (WF ) rhythms
or scales [5], (also known as moments of symmetry [20]),
which result when evenness is maximized under the con-
straint of no more than two sizes of IOIs. This means a
WF rhythm can be described by a word such as `s,s`ss`,
or ```s, etc., where `denotes a large interonset interval, s
denotes a small interonset interval; for any given number
of large and small IOIs, their maximally even arrangement
always forms a well-formed word [6]).
XronoMorph uses well-formed rhythms for the follow-
ing three reasons: a) they are a superset of Euclidean;
b) they can produce rhythms that do not fit into an
isochronous grid (although grid-based rhythms are of ob-
vious utility in music, subtle deviations from the grid are
vitally important, as is the interesting possibility of deeply
non-isochronous rhythms—discussed below—that are max-
imally distant from any possible grid); c) they invite a prin-
cipled approach for producing a hierarchy of interlocking
WF rhythms.
Let the length of `divided by the length of sbe denoted
r(a real number between 1 and infinity). Any given WF
pattern is a subset of a higher-level WF pattern that is
derived by the use of two different morphisms [21, 4] (or,
equivalently in this context, parallel rewrites [11]): when
r < 2, the morphism is `7→ `s and s7→ `; when r≥2,
the morphism is `7→ `s and s7→ s[13]. A consequence of
this is that when the r-value for the lowest level is a rational
number, a higher level (and all levels higher than that level)
will be isochronous (when ris irrational, no higher level is
ever precisely isochronic).
This can be perhaps most simply explained by reference
to musical scales in 12-tone equal temperament. The WF
tetractys D, G, A (`s`) is a subset of the WF pentatonic
scale D, F, G, A, C (`ss`s), which is a subset of the WF
diatonic scale D, E, F, G, A, B, C (`s```s`), which is a subset
of the evenly tempered (“isochronous”) chromatic scale D,
D], E, F, F], G, G], A, A], B, C, C](where `=s). If these
patterns are interpreted rhythmically, and each such level
is considered as a separate rhythmic stream (perhaps each
played with a distinct timbre), then combining them makes
a complex hierarchy of rhythms. This rhythmic hierarchy
is illustrated in Fig. 4a.
This method of generating successive levels results in ev-
ery rhythmic event being duplicated in all higher levels. For
example, all three beats in the lowest level are addition-
ally played by the remaining three higher levels. Naturally,
this gives a strong accent to low-level beats, and amplifies
the inherently hierarchical nature of WF rhythmic struc-
tures. However, XronoMorph allows an interesting alterna-
tive strategy, which is to treat each successive level as the
complement of all lower levels, so it plays only when no lower
391
(a) A WF hierarchy with
universal levels. (b) A WF hierarchy with
complementary levels.
Figure 4: Four levels in the tetractys-pentatonic-
diatonic-chromatic well-formed hierarchy, shown as
universal and complementary forms (as defined in
the main text).
level is also playing (this is done by toggling the “U/C”—
universe/complement—buttons for each level in the epony-
mous column). For example, consider a lower level which,
if expressed as a scale rather than as a rhythm, corresponds
to the white-note diatonic scale, while the next higher level
corresponds to a twelve-pitch chromatic scale. When “C” is
selected for the chromatic level, instead of playing all twelve
events in the latter rhythm, only those events not occurring
in the lower-level pattern are played. Using the scalic anal-
ogy, this means using only the black-note pentatonic scale,
which is the complement of the white-note diatonic in a
chromatic universe.
Interestingly, these complementary well-formed rhythms
are themselves well-formed [1, Prop. 3.2], but they are dis-
placed with respect to each other, so they never coincide.
This non-redundant rhythmic structure is somewhat remi-
niscent of the multiple interlocking parts used by Latin per-
cussion or gamelan percussion ensembles—although each
individual part is relatively simple, in combination, they
produce a complex and interwoven totality. This comple-
mentary hierarchy is illustrated in Figure 4b.
The number of levels that need to be ascended before
isochrony is reached is a function of the ratio of the large
and small IOIs of the lowest level; indeed, as mentioned
above, if this ratio is irrational, isochrony will never be
reached (though it may be closely approximated). Inter-
estingly, there are a number of ratios based on the golden
section that ensure isochrony is never closely approximated
[22, 13].
3.2 Well-Formedness in XronoMorph
The principal user-parameters for defining WF rhythms are:
a) the number of large IOIs of the lowest-level rhythm, b)
the number of small IOIs of the lowest level rhythm, c)
the ratio of the large IOI and the small IOI. This ratio is
controlled by the large horizontal r-slider at the bottom
of the interface, and it traverses the range 1 to ∞(using
the mapping ratio = 1/(1 −t), where t∈[0,1) is the left-
right position of the slider). With these values chosen, a
hierarchy of six WF rhythms is constructed, each of which
can be switched on or off, and between complementary and
universal mode (as described earlier).
As the r-slider is moved, the visualization and sonification
of the rhythmic hierarchy continuously updates. Above the
r-slider are six levels of numbers. When the r-slider lines up
with one of these numbers on a given level (or when one of
these numbers is clicked on), the corresponding level (and
all higher levels) of the resulting rhythm is isochronous with
that number of pulses. As the r-slider is smoothly moved
away from these numbers, each previously isochronous level
smoothly shifts to having two differing IOIs. Certain r-
slider locations are indicated with a φsymbol. These ratios
are related to the golden section and, at such positions, no
rhythmic level approaches isochrony—they are maximally
distant from any isochronous grid (of whatever granular-
ity). For this reason, we term these rhythms deeply non-
isochronous. Perhaps counter-intuitively, we have found
these rhythms to be rather groovy. Figure 1b shows a
WF hierarchy with six levels; this is related to the pre-
viously mentioned tetractys-pentatonic-diatonic-chromatic
hierarchy, but the r-slider has a value of 5/3, which results
in the highest level being a 19-fold (rather than 12-fold)
isochronous pulse. All levels are in complementary mode.
4. CASE STUDIES
4.1 Education
Part of a course in audio engineering at the University of
Wisconsin (ECE401) studies the perception of sound. This
is focused at two levels: on timbre (where both temporal
and spectral influences are important) and on rhythm. One
module considers a taxonomy of rhythm: from isochronous
pulses to polyrhythms, and then “upwards” through the
metrical hierarchy. XronoMorph provides an excellent ex-
perimental platform for the demonstration and investiga-
tion of the various ways of characterizing rhythmic pat-
terns. For example, one homework set considers rhythms
that are (1) isochronous, (2) polyrhythmic, (3) well-formed,
(4) perfectly balanced, (5) Euclidean, and (6) rotationally
symmetric. Students are asked to create examples of each of
these, and then to provide examples of (n) that are not (m),
for instance, well-formed rhythms that are not Euclidean, or
PB rhythms that are not rotationally symmetric. The soft-
ware makes the task feasible, and allows instant feedback
on the perceptible meaning of the various definitions.
The final project in this class is a relatively free assign-
ment where students choose their own subject (within the
audio realm) and prepare a term paper. In the fall semester
2015, one student (Matthew Cortner) conducted a pilot
study with the goal of determining if the properties of “bal-
ance”and“evenness”were perceptually salient. Is it possible
to tell, just by listening, if a rhythm is perfectly balanced or
even (or neither)? Because of the difficulty of describing to
naive listeners what is meant by terms such as balance and
evenness, an experiment was designed to test whether lis-
teners could better distinguish perturbations of isochronous
rhythms and non-isochronous PB rhythms than they could
perturbations of unbalanced and uneven rhythms. Positive
results would show that the specified property (balance or
evenenness) is perceptually salient, at least in the sense that
it matters in discrimination experiments. The results of the
pilot study [7] are encouraging, though the small sample
size precludes any statistically significant results.
4.2 Composition and Performance
XronoMorph facilitates the production of a wide variety of
complex rhythms, many of which would be hard to compose
or perform manually. The ability to smoothly transition be-
tween rhythms as well as abruptly switching between com-
plex but related rhythms also opens up novel compositional
and performative possibilities.
The sonification can be done with unpitched sounds, in
which case purely rhythmic patterns can be created. Some
of the WF rhythms are reflective of rhythms found in non-
Western music; for example, aksak additive rhythms like
2 + 3 + 3 + 2 + 3 are often well-formed [9], as are Sub-
Saharan rhythms such as the previously mentioned “dia-
tonic” rhythm [17]. The PB rhythms include polyrhythms
(e.g., 3 against 2, or 3 against 4) that are also common in
392
Sub-Saharan music; they also include polyrhythms where
the individual streams are phase-shifted so they never co-
incide (e.g., Fig. 3b, where the 3-fold rhythm and two 2-
fold rhythms are respectively displaced). Beyond these dis-
placed polyrhythms, we also find the fascinating rhythmic
structures formed by the irregular elemental PB patterns,
which sonify combinations of positively- and negatively-
weighted (cancelling) polyrhythms.
When the sonification is made with pitched sounds, we
may find that melodies (hockets) emerge from perceptual
streaming of proximal pitches between levels. When the
user changes the rotation or pitch of each polygon, the emer-
gent melody also changes. An interesting feature of such
melodies is that they typically arise without compositional
forethought, but since they arise from such a highly orga-
nized structure, they frequently exhibit æsthetic promise.
Another possibility is to use the raw MIDI output to seed
other algorithmic generation systems. For example, aus-
traLYSIS (http://www.australysis.com) have performed
using WF rhythms to drive Serial Collaborator [8] to pro-
duce rhythmically informed serial transformations of previ-
ously written tone rows.
5. CONCLUSION
We have introduced XronoMorph, an application for the al-
gorithmic generation of perfectly balanced and well-formed
rhythms. The software makes use of visualizations and soni-
fications of a geometrical conceptualization of rhythms to
allow a novel approach for their construction. It is built
around two underlying mathematical principles whose math-
ematical background and implementation in XronoMorph
have been detailed.
The multilevel rhythmic structures generated by
XronoMorph have levels that are PB, or WF, both individu-
ally and in combination. This leads to interwoven structures
evoking a sense of deep organization and self similarity that
is reminiscent of fractals.
Using the algorithmic approach described here, intel-
ligent compositional input is still required—not all well-
formed and perfectly balanced rhythms, or transitions be-
tween them, will sound appropriate. Furthermore, effective
choices for pitches and durations are still required. But we
have found this tool to be both inspiring and surprising in
its musical output. We hope that the visual and geometrical
conceptualization of rhythms demonstrated in XronoMorph
will help in the composition and performance of new and in-
teresting rhythms, and facilitate an intuitive understanding
of complex rhythms found in real-world music.
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