Conference PaperPDF Available

Circularity in Rhythmic Representation and Composition

Abstract and Figures

Cycle is a software tool for musical composition and improvisation that represents events along a circular timeline. In doing so, it breaks from the linear representational conventions of European Art music and modern Digital Audio Workstations. A user specifies time points on different layers , each of which corresponds to a particular sound. The layers are superimposed on a single circle, which allows a unique visual perspective on the relationships between musical voices given their geometric locations. Positions in-between quantizations are possible, which encourages experimentation with expressive timing and machine rhythms. User-selected transformations affect groups of notes, layers, and the pattern as a whole. Past and future states are also represented, synthesizing linear and cyclical notions of time. This paper will contemplate philosophical questions raised by circular rhythmic notation and will reflect on the ways in which the representational novelties and editing functions of Cycle have inspired creativity in musical composition.
Content may be subject to copyright.
Circularity in Rhythmic Representation and Composition
Scott Barton
Worcester Polytechnic Institute
100 Institute Road
Worcester, MA 01609
sdbarton@wpi.edu
ABSTRACT
Cycle is a software tool for musical composition and im-
provisation that represents events along a circular timeline.
In doing so, it breaks from the linear representational con-
ventions of European Art music and modern Digital Audio
Workstations. A user specifies time points on different lay-
ers, each of which corresponds to a particular sound. The
layers are superimposed on a single circle, which allows a
unique visual perspective on the relationships between mu-
sical voices given their geometric locations. Positions in-
between quantizations are possible, which encourages ex-
perimentation with expressive timing and machine rhythms.
User-selected transformations affect groups of notes, layers,
and the pattern as a whole. Past and future states are also
represented, synthesizing linear and cyclical notions of time.
This paper will contemplate philosophical questions raised
by circular rhythmic notation and will reflect on the ways
in which the representational novelties and editing functions
of Cycle have inspired creativity in musical composition.
Author Keywords
rhythm, composition, interface design, philosophy
CCS Concepts
Applied computing Sound and music comput-
ing; Performing arts; Human-centered computing
Interactive systems and tools;
1. INTRODUCTION
Time is both linear and cyclical. In one sense we are perpet-
ually moving forward, using knowledge gained in the past
to progress towards our goals. At the same time, we peri-
odically return to familiar states that define that which is
central in our lives. This notion of centrality is psycholog-
ically fundamental; it is an abstract image schema that is
formed from recurring physical experiences that empower
us with a sense of orientation [13]. It gives us a point of
reference relative to which we move away from and back
to. It allows us to distinguish some states or places as more
important than others. It emerges in our relationship to the
world (e.g. sunrise), in our personal lives (e.g. home) and
in our music (e.g. the tonic of a scale, the downbeat of a
metric cycle). The question then becomes to what extent
Licensed under a Creative Commons Attribution
4.0 International License (CC BY 4.0). Copyright
remains with the author(s).
NIME’20, July 21-25, 2020, Royal Birmingham Conservatoire,
Birmingham City University, Birmingham, United Kingdom.
Figure 1: The user interface of Cycle
do our methods of representation in musical practice rep-
resent this core psychological phenomenon? How do these
methods affect the compositional choices that we make?
2. LINEAR AND CYCLICAL REPRESEN-
TATION OF TIME
Linear representation of temporal events, as is found in no-
tation of European Art music and DAW sequencers, adheres
to a conception of time that shows a seamless past, present,
and future in perfect proportions and clarity. It allows us to
see the qualities and rates of change. It allows the develop-
ment of ideas over periods of time that extend beyond the
contents of our short-term memory. It facilitates gestures
that correspond to image schema that involve motion, link-
age, causation, paths and goals [13] by affording formation
of contours within numerous musical aspects (e.g dynamics,
tempo, energy, density).
In other ways, linear representation of time is incongru-
ous with how we experience the present and preserve it in
memory. Unlike the timeline in a typical DAW, the details
of experiences become fuzzy, simplified or absent as time
moves, and our expectations of the future reach out often
only at arm’s length. We are perpetually grounded in the
present; our awareness provides us a center from which we
depart and to which we return, helping us understand the
world and our place within it. It is a kind of motion that is
better aligned with the circle than with the line.
Time is continuous and multi-dimensional. In music, re-
peated patterns that are represented linearly are discontin-
uous. We depart from a location and move increasingly
away from it. When we reach a point of maximal displace-
ment, we teleport back to a disconnected beginning. Such
is incongruous with music whose unyielding repetitions (e.g.
EDM, minimalism) exude that which is flowing and seam-
less. In this kind of music, as in a circle, a beginning and
an end are the same. As we depart from the beginning,
we simultaneously and paradoxically move both away from
and towards it. As we increasingly expect the beginning
of the pattern to return, the distances between points and
the origin decrease, unlike that seen along a single linear
dimension. The latter shows time as flat: a distortion of
its nature. In cyclical representation, multiple dimensions
allow new kinds of geometries to emerge that distinguish
intervals and patterns in new ways [10, 14].
3. MUSICAL PRACTICE
The idea of circular rhythmic representation has inspired
musicians and musicologists since at least the 13th century
[14], and has found particular traction in recent years. Many
of these more recent efforts focus on rhythmic valuation, ex-
ploring ideas of perfect balance [9], well-formedness [10, 8],
and the qualities that make a rhythm“good”[14]. They pur-
port to find characteristic or normative aspects of rhythmic
expression. As a composer I am inspired by such work, but
I am also compelled to look for that which is evades or chal-
lenges convention; that which is weird but cool; that which
reveals a path to new expressive territory. These desires
have led me to the world of machine rhythms.
3.1 Machine Rhythms
As described in [1] machine rhythms are defined by a com-
plexity that is made possible by (electro) mechanical means.
This complexity is characterized by departure from the low-
integer interval proportions that are typically found in mu-
sical notation (e.g. 1:1 or 2:1). It also is created by concate-
nation and division on multiple hierarchical levels (e.g. a
3.5-beat unit that is then divided into 7). While there is re-
search that suggests we perpetually assimilate and contrast
temporal intervals in the direction of small-integer ratios [5,
7], there is also evidence that shows that we deviate from
simple ratios in rhythm production and perception [12, 6,
2]. Indeed, Dan Trueman’s interest in developing the Cy-
clotron was to explore the Norwegian telespringar, whose
rhythms are sometimes performed (by humans) in the pro-
portions 39:33:28 [16]. These complex rhythms (of both
human and machine type) typically have a lopsided charac-
ter that stands in distinction to their symmetrical, balanced
counterparts, yet they evoke a sense of groove. The iden-
tity of a machine rhythm is illuminated when it is looped,
often requiring multiple iterations before it presents itself.
The precision of mechanical performance can evoke rhyth-
mic qualia that differ from those produced by both com-
plex human-played rhythms as well as variations that are
considered expressive timing. Cycle is a tool that allows
composers to explore the possibilities of machine rhythms.
4. CIRCLE-BASED RHYTHM MACHINES
There are many of examples of machines that represent
rhythms on circles from Raymond Scott’s Circle Machine
(1958) [4] to more recent efforts such as Rhythm Necklace
[11], and Sequence [3]. While these examples boast a num-
ber of useful features, they are all limited in that timepoints
are quantized to isochronous divisions of the cycle, and dif-
ferent cycles are not able to be superimposed on each other.
The first limits the kinds of rhythmic configurations that
can be produced (particularly machine rhythms) while the
second limits a composer’s ability to visualize the temporal
relationships between events on different cycles.
XronoMorph is an application that is designed to explore
rhythmic well-formedness and perfect balance through the
use of different geometries [10]. A variety of different poly-
gons can be chosen and oriented (they can be superimposed
on each other) in a circle that is orbited by a playhead that
sends MIDI events when vertices are encountered. These
polygons can be shifted forward or backward either accord-
ing to a grid (or not). A large number of presets can be
saved and recalled, and there are controls for many parame-
ters including sound source, panning, and volume. A unique
feature is the ability to morph between rhythms.
The software also has limitations. The location and ve-
locity of individual timepoints cannot be edited (there are
only global randomize controls for both time and velocity).
Further, individual timepoints cannot be deleted, thus one
is bound by the chosen geometry (in some sense, this is
the point of this particular approach). Rhythm is as much
about space as it is about sound, thus the ability to create
gaps is a fundamental aspect of rhythmic expression.
In all of the previous examples one struggles to realize
gestures formed from smaller elements such as grace notes
and percussion rudiments. Contours affecting dynamics and
rate are similarly unattainable, particularly over short pe-
riods of time within the duration of a particular cycle.
Cyclotron [15, 16] is a circle-based interface that can gen-
erate a wide variety of rhythmic configurations, including
ones that evade isochronous divisions of a cycle. Points
originate at the center of a larger circle and extend outward
to its circumference, creating spokes, the number of which
is specifiable. Spokes may be quantized to isochronous di-
visions of the cycle or may be moved between such points,
creating complex temporal interval ratios. The cap of each
spoke can be adjusted, the size of which can control a spec-
ified parameter (such as gain). A phasor (playhead) moves
around the circle at a specified rate and when it contacts a
spoke, a control message is output (which could be mapped
to pitch, for example). A unique feature of the interface is
the ability to adjust spoke length (either freely or to quan-
tized positions), which affects the probability of that spoke
being played. Another compelling feature of the program is
the ability to warp time, which allows the phasor (playhead)
to move at irregular rates. Patterns may also be reversed,
which moves each spoke symmetrically about a center axis.
As Trueman notes, the ability to warp time allows one to
create rhythmic nuance and expressivity while using simple
subdivisions of a cycle [16]. This feature is paralleled in
the conventions of Western Art Music notation, where time
can be controlled globally or locally (i.e. musical rate can
be changed via subdivisions or tempo, or some combination
of the two). The local and global also interact in our con-
ceptualization of temporal relationships and characteristics
when listening. The rubato encountered in a performance
of Chopin involves both distinguishing the inequality in ad-
jacent temporal intervals and apprehension of an emergent,
organizing wave that ebbs and flows. The ability both to
shift timepoints in minute intervals and to change the shape
of a playhead’s progression help in the effort to discover (or
describe) complex rhythmic relationships.
Cyclotron also has limitations. By changing spoke length,
notes are displaced from the reference circle. The propor-
tions of visual and auditory intervals between timepoints no
longer align, which undermines one of the key features of
circular notation. The newer version of the program (2008)
is restricted to a single instance of a cycle. Regarding ac-
cessibility, the requirement to either compile the program in
Processing or utilize the command line may be intimidating
to less-technical musicians.
None of the aforementioned examples address linear as-
pects of temporal perception, particularly in the form of
past and future states. The design ob jectives of Cycle thus
were to synthesize useful features found in these various
tools as well as develop new ones that would encourage the
exploration of machine rhythms, represent cyclical and lin-
ear aspects of time, and would be accessible to a broad
group of musicians.
5. DESIGN
Cycle was designed and built starting in early 2019 in the
software environment Max. The first iterations of the pro-
gram featured numerous independent cycles that could be
viewed, edited and played simultaneously. The advantages
of this approach included visual clarity of each component
cycle and independent control of cycle time / speed, which
provided an easy way to create polytempic textures. The
disadvantages of this approach were the difficulty in seeing
the relationship between timepoints on different cycles and
the relatively small size of multiple simultaneous timecircles.
The program was re-written to feature all timepoints on a
single timecircle and is described in the following sections.
5.1 Objectives
In order to create a tool capable of exploring a variety of
rhythms (including those of machinic qualities, the follow-
ing design objectives were established, which enable a user
to:
1. locate points manually or automatically around a cir-
cle according to isochronous divisions or their inter-
mediaries.
2. create various layers that are superimposed upon each
other so that global and local relationships can be per-
ceived.
3. transform groups of notes, including selections, layers
and the pattern as a whole.
4. store and recall a library of patterns that includes both
custom and well-known rhythms.
5. view past and future states of the system in order to
better attend to linear aspects of time.
6. interface with conventional music software to provide
accessibility to a wide group of musicians.
5.2 Features
5.2.1 Timepoints
Cycle (figure 1) uses circular representations of musical time
on which one may superimpose rhythmic patterns. Time-
points can be created either by manually clicking on the
timecircle (the large white circle in the background) or by
entering a value in the tuplet box, which will divide the cycle
(specified in msec) into isochronous timepoints.Timepoints
can be selected, moved, and removed either individually or
in groups. When started, a green playhead orbits the time-
circle and when it reaches a timepoint, it outputs a note and
velocity pair in the form of a MIDI message. The velocity
of the note corresponds to the size of the timepoint, which is
adjustable. The pitch is determined by the associated layer.
5.2.2 Layers
One of the primary motivations of Cycle was to develop an
interface for composing rhythms that would make evident
not only the specifics of a particular element but also the
relationships of that element to others in the pattern (and
the pattern as a whole). The approach here uses layers
that are visually superimposed on each other. Timepoints
in each layer output a distinct MIDI pitch thus each layer
corresponds to a particular sound. The focus section allows
one to view individual layers or all layers together. The
write section specifies the layer that edits will apply to. By
separating focus and write controls, one can work on a layer
either in isolation or in the context of all layers together.
Individual layers can be deleted, and a selected note can be
moved to any other layer, which proves to be a powerful
orchestrational tool.
5.2.3 Transformations
Transformations allow the characteristics of groups of notes
to be affected with a single command. The positions of
timepoints within note groups can be quantized to a de-
sired cycle division or can be shifted backward or forward
in time as a layer. Thin deletes the percentage of timepoints
specified (at random) and Randomize Vel changes the ve-
locity of each timepoint in the selected layer by a random
amount.
5.2.4 Playback and Storage
Playback is controlled by a toggle and the speed of the
playhead’s rotation is defined by the user. Patterns can be
stored and recalled. System states are stored, manipulated
and recalled using coll objects and dictionaries.
5.2.5 Past, Present and Future
A novel feature of Cycle is its representation of patterns
that have been played in the recent past and ones that are
to be played in the near future. After a pattern has played,
its preset number will appear immediately to the left of
the timecycle. Each subsequent cycle shifts the sequence of
previously played cycles further to the left. There are a lim-
ited number of slots that loosely correspond to the limits of
short-term memory. Regarding the future, a small number
of patterns can be queued in boxes to the right that are
shifted (either manually or automatically) towards and into
the current timecycle with each cycle repetition. While the
pattern that is being played is visually primary, represent-
ing our grounding in the present, these additions embrace
linear notions of time, which invite the development of ex-
pansive sequences and gestures.
5.2.6 Accessibility
Preset storage and recall can be mapped to external MIDI
controllers (grid interfaces are particularly useful for this
purpose). The now / beg buttons determine whether a pre-
set is recalled immediately or at the beginning of a cycle.
Notes are sent as standard MIDI messages, enabling the
software to interface with a variety of musical devices.
6. MUSICAL APPLICATIONS
I used Cycle in the composition and performance of a num-
ber of musical works including Machine Rhythm Study No.
1, No. 2 and No. 3. The features of the program influenced
my musical choices in a number of meaningful ways. Having
all timepoints superimposed on a single timeline enhanced
my understanding of the relationships between individual
and groups of elements. It gave me a holistic picture of
accent patterns and global aspects such as density. More
complex patterns can result in crowded representations, but
this is a problem which the ability to toggle focus between
a particular layer and the whole ameliorates. In traditional
notation and in common DAWs, layers are represented in
the form of stacked rows. This forces a viewer to scan be-
tween and reconcile events on multiple lines at once. As
a result, notes are separated by a variety of both horizon-
tal and vertical distances. Increasing the vertical distance
between events makes the evaluation of horizontal relation-
ships more difficult, thus creating a distorted picture of the
temporal relationships within a rhythmic pattern (this also
can be an issue with Cyclotron if spoke distances signifi-
cantly vary). The difference is extreme with large numbers
of rows, but noticeable with even smaller numbers. In Cycle,
all events are oriented relative to one circle, which avoids
these vertical distortions.
The lack of quantized gridlines is liberating. The latter
are idealizations of correctness; Platonic Forms to which
we are magnetically and neurotically attracted. Of course,
quantized values are often useful, and Cycle is capable of
such adjustments. The difference between Cycle and lin-
ear systems is that the choice to quantize within Cycle is
motivated by auditory evaluations and not visual ones. In
the piano roll of a conventional DAW, seeing note-blocks
scattered around precise gridlines reminds us of our tempo-
ral imprecision as human performers and engenders doubt
concerning the rhythmic correctness of our phrase. Many
DAW’s allow gridlines to be turned off, but the user knows
they are always available and are readily resorted to in such
moments of incertitude. Turning off the gridlines in a typ-
ical piano roll of a DAW evokes a feeling of disorientation
rather than liberation. Part of the reason for this is that
there is no common reference for note events. The timecircle
in Cycle purports to address that issue.
While Cycle is fundamentally a tool for generating rhythms,
its use inspired me to think about pitch sequences and rela-
tionships in new kinds of ways. In Machine Rhythm Study
No. 3, all events in a cycle were generated by sequences
made in Cycle. Each sequence was first composed with
percussion, and then was mapped to different pitched in-
struments. Thinning and transposing algorithms were then
applied to each of the instruments to create variety in har-
mony, timbre and register. The initial rhythmic sequence
thus was a temporal template whose slots could be filled
by myriad sounds. The ease of mapping layers to different
sounds facilitated this approach. This was a new method
of composing for me, which inspired my creativity and re-
sulted in musical textures that I otherwise wouldn’t have
made.
Cycle is simple and flexible enough to be used in impro-
visational settings. Performance of the aforementioned Ma-
chine Rhythm Studies involves sequencing and alteration of
patterns in real-time. Mapping controls such as preset num-
bers, quantization, isochronous generation, and note thin-
ning to generic MIDI controls allows a performer to gen-
erate, transform and concatenate musical ideas in an easy
and quick way. This mapping (and the use of the interface
in general) does not require sophisticated technical knowl-
edge, therefore the tool is accessible to a wide variety of
musicians.
7. FUTURE DIRECTIONS
There are a number of areas in which Cycle can be im-
proved or enhanced. Developing the program as a Max for
Live device and / or plugin would increase its accessibility,
particularly for those musicians who are most comfortable
in the realm of DAW’s. Currently, there is a single playhead
that rotates around the timecircle. In the future, multiple
playheads could be present so that each layer could run ac-
cording to its own time scale, or a single global time scale
may be shared between all instances, allowing them to be
synchronized. This would enable the polytempic textures
that were attainable in the early versions of the program.
More global transformations such as mirroring a sequence
about an axis could be incorporated. In practice, Dan True-
man writes about the use of Cyclotron as a metronome that
can help guide performers in the articulation of unusual or
complex rhythms [16]. This idea is inspiring. Cycle and
musical works are available at (scottbarton.info).
8. REFERENCES
[1] S. Barton. Creativity in the Generation of Machine
Rhythms. In Proceedings of the 1st Conference on
Computer Simulation of Musical Creativity, 2016.
[2] S. Barton, L. Getz, and M. Kubovy. Systematic
variation in rhythm production as tempo changes.
Music Perception: An Interdisciplinary Journal,
34(3):303–312, 2017.
[3] D. Diakopoulos. Sequence - Euclidean Sequencer for
iPhone. http://sequence.nyquistresearch.com/.
accessed: 2020-01-27.
[4] encyclotronic. Manhattan Research Circle Machine.
https://encyclotronic.com/synthesizers/
manhattan-research/
manhattan-research-circle-machine-r1481/.
accessed:2020-01-31.
[5] P. Fraisse. Les structures rythmiques [The Rhythmic
Structures]. Publications Universitaires de Louvain.,
Louvain, Belgium, 1956.
[6] A. Gabrielsson. Interplay between analysis and
synthesis in studies of music performance and music
experience. Music Perception, 3(1):59–86, 1985.
[7] G. T. Hoopen, T. Sasaki, Y. Nakajima, G. Remijn,
B. Massier, K. S. Rhebergen, and W. Holleman.
Time-Shrinking and Categorical Temporal Ratio
Perception Evidence for a 1:1 Temporal Category.
Music Perception, 24(1):1–22, 2006.
[8] J. London. Hearing in time : psychological aspects of
musical meter. Oxford University Press, New York,
2004.
[9] A. J. Milne, D. Bulger, and S. A. Herff. Exploring the
space of perfectly balanced rhythms and scales.
Journal of Mathematics and Music, 11(2-3):101–133,
2017.
[10] A. J. Milne, S. A. Herff, D. Bulger, W. A. Sethares,
and R. T. Dean. XronoMorph: algorithmic generation
of perfectly balanced and well-formed rhythms. In
Proceedings of the 2016 international Conference on
New Interfaces for Musical Expression, 2016.
[11] M. O’Reilly and S. Tarakajian. Rhythm Necklace.
http://rhythmnecklace.com/. accessed: 2020-01-27.
[12] B. H. Repp, W. Luke Windsor, and P. Desain. Effects
of Tempo on the Timing of Simple Musical Rhythms.
Music Perception, 19(4):565–593, 2002.
[13] B. Snyder. Music and memory: An introduction. MIT
press, 2000.
[14] G. T. Toussaint. The Geometry of Musical Rhythm:
What Makes a “Good” Rhythm Good? Chapman and
Hall/CRC, 2016.
[15] D. Trueman. The Cyclotron: A Tool for Tweaking
Time... (...and some Related Ruminations about
Rhythm). http:
//dtrueman.mycpanel.princeton.edu/Cyclotron/,
2007. accessed: 2020-01-29.
[16] D. Trueman. The Cyclotron: a Tool for Playing with
Time. In In Proceedings of the International
Computer Music Conference (ICMC) 2008., Sonic
Arts Research Centre, Queen’s University Belfast,
Northern Ireland, 2008.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Periodic scales and meters typically embody "organizational principles" – their pitches and onset times are not randomly distributed, but structured by rules or constraints. Identifying such principles is useful for understanding existing music and for generating novel music. In this paper, we identify and discuss a novel organizational principle for scales and rhythms that we feel is of both theoretical interest and of practical utility: perfect balance. When distributed around the circle, perfectly balanced rhythms and scales have their "centre of gravity" at the centre of the circle. The present paper serves as a repository of the theorems and definitions crucial to perfect balance. It also further explores its mathematical ramifications by linking the existing theorems to algebraic number theory and computational optimizations. On the accompanying webpage, http://www.dynamictonality.com/perfect_balance_files/, we provide audio samples of perfectly balanced rhythmic loops and microtonal scales, computational routines, and video demonstrations of some of the concepts.
Article
Full-text available
We investigated the effect of tempo on the production of the syncopated 3-2 son clave rhythm. We recorded eleven experienced percussionists performing the clave pattern at tempi ranging from 70 bpm to 210 bpm. As tempo increased, percussionists shortened the longest intervals and lengthened the shortest interval towards an intermediate interval that is located in the first and second positions in the pattern. This intermediate interval was stable across tempi. Contrary to prior studies, we found that the complexity of interval ratios had little effect on production accuracy or stability and the “short” interval in the pattern was not particularly stable. These results suggest that as tempo is varied, (1) experienced musicians systematically distort rhythmic intervals, (2) rhythmic configuration, and not just the complexity of interval ratios, affects the production of rhythmic intervals, and (3) the distinction between long and short intervals is context-dependent.
Conference Paper
Full-text available
This paper explores musical, psychological and philosophical ideas about how humans and machines interact in creative processes. It argues that creativity is a function of both generator and receiver, and that these roles can be amorphous in the creation and experience of electronic music. It offers an approach to structuring temporal spaces for rhythmic composition, which leads to the idea of machine rhythms, which are proposed as a promising area for creative expression.
Conference Paper
Full-text available
We present an application—XronoMorph—for the algorithmic generation of rhythms in the context of creative composition and performance, and of musical analysis and education. XronoMorph makes use of visual and geometrical conceptualizations 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 mathematical principles: perfect balance and well-formedness, both of which can be derived from coefficients of the discrete Fourier transform of the rhythm. The mathematical background , musical implications, and their implementation in the software are discussed.
Article
Full-text available
IN A PREVIOUS STUDY, we presented psychophysical evidence that time-shrinking (TS), an illusion of time perception that empty durations preceded by shorter ones can be conspicuously underestimated, gives rise to categorical perception on the temporal dimension (Sasaki, Nakajima, & ten Hoopen, 1998). In the present study, we first survey studies of categorical rhythm perception and then describe four experiments that provide further evidence that TS causes categorical perception on the temporal dimension. In the first experiment, participants judged the similarity between pairs of /t1/t2/ patterns (slashes denote short sound markers delimiting the empty time intervals t1 and t2). A cluster analysis and a scaling analysis showed that patterns liable to TS piled up in a 1:1 category. The second and third experiments are improved replications in which the sum of t and t2 in the /t1/t2/ patterns is kept constant at 320 ms. The results showed that the 12 patterns /115/205/, /120/200/,..., /165/155/,/170/150/ formed a 1:1 category. The fourth experiment utilizes a cross-modality matching procedure to establish the subjective temporal ratio of the /t1/t2/ patterns and a 1:1 category was established containing the 11 patterns /120/200/, /125/195/,..., /165/155/, /170/150/. On basis of these converging results we estimate a domain of perceived 1: 1 ratios as a function of total pattern duration (t1 + t2) between 160 and 480 ms. We discuss the implications of this study for rhythm perception and production.
Book
The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explains how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields. For the music community, the book also introduces the distance approach to phylogenetic analysis and illustrates its application to the study of musical rhythm. Accessible to both academics and musicians, the text requires a minimal set of prerequisites. Emphasizing a visual geometric treatment of musical rhythm and its underlying structures, the author-an eminent computer scientist and music theory researcher-presents new symbolic geometric approaches and often compares them to existing methods. He shows how distance geometry and phylogenetic analysis can be used in comparative musicology, ethnomusicology, and evolutionary musicology research. The book also strengthens the bridge between these disciplines and mathematical music theory. Many concepts are illustrated with examples using a group of six distinguished rhythms that feature prominently in world music, including the clave son. Exploring the mathematical properties of good rhythms, this book offers an original computational geometric approach for analyzing musical rhythm and its underlying structures. With numerous figures to complement the explanations, it is suitable for a wide audience, from musicians, composers, and electronic music programmers to music theorists and psychologists to computer scientists and mathematicians. It can also be used in an undergraduate course on music technology, music and computers, or music and mathematics.
Article
This article consists of three parts. In the first part, early empirical research on music performance is reviewed—with special emphasis on the contributions by C. E. Seashore and his co-workers at Iowa University in the 1930s. The second part presents a model for interplay between analysis and synthesis in studies of music performance and its relationship to listeners' experience of the music. The model means that music performance is analyzed with regard to various physical properties, and their relationships to listeners' experience are investigated by means of synthesized sound sequences that are systematically varied in different aspects. In the third part, this idea is illustrated by examples from an extensive research project on musical rhythm. It is shown that performance of musical rhythm is characterized by various systematic variations regarding the durations of the sound events relative to strict mechanical regularity and that these variations may be related to various aspects of the experienced rhythm.
Book
This book develops a theory of musical meter based on psychological research in temporal perception, cognition, and motor behavior. Meter is regarded as a kind of entrainment, a synchronization of attention and actions to the rhythms of the environment. Drawing on research on the ability to make durational discriminations and categorizations at various tempos, as well as evidence from neurobiology, the "speed limits" for meter are given: the inter-onset interval for metric elements must be greater than 100ms (10 per second) and less than 1.5-2.00 seconds. Care is taken to distinguish rhythms or patterns of duration from meters, the listener/performer's complex patterns of expectation and attention. It is thus shown that metric behaviors are highly tempo-dependent. Ambiguities may arise when a rhythmic pattern may be regarded under more than one meter, and conflicts may arise when a pattern of durations contradicts the ongoing meter. The music-theoretical core of the book is its development of a set of metric well-formedness constraints, which limit the temporal range and organization of patterns of metric entrainment. A consideration of the rhythmic practices of various non-western cultures, including some African and Indian music, leads to an additional well-formedness constraint, that of maximal evenness. This allows for meters that involve uneven (i.e., non-isochronous) beats or beat subdivisions. The book concludes with the many meters hypothesis, which proposes that a large number of expressively timed temporal templates are acquired that are readily used when listening in familiar musical contexts.