Early processing of pitch and rhythm: integrated or parallel?

I’ll guess you’ll have to read it to know the answer…

Hear Res 2013 Oct 5. pii: S0378-5955(13)00240-2
Thalamocortical mechanisms for integrating musical tone and rhythm

Musacchia G, Large E, Schroeder CE
Communication Sciences and Disorders, Montclair State University, 1515 Broad Street, Bldg B, Bloomfield, NJ 07003, USA; Center for Molecular & Behavioral Neuroscience, Rutgers University, 197 University Avenue, Newark, NJ 07102, USA. musacchiag@mail.montclair.edu

Studies over several decades have identified many of the neuronal substrates of music perception by pursuing pitch and rhythm perception separately. Here, we address the question of how these mechanisms interact, starting with the observation that the peripheral pathways of the so-called “Core” and “Matrix” thalamocortical system provide the anatomical bases for tone and rhythm channels. We then examine the hypothesis that these specialized inputs integrate acoustic content within rhythm context in auditory cortex using classical types of “driving” and “modulatory” mechanisms. This hypothesis provides a framework for deriving testable predictions about the early stages of music processing. Furthermore, because thalamocortical circuits are shared by speech and music processing, such a model provides concrete implications for how music experience contributes to the development of robust speech encoding mechanisms.

Week 8 Assignment

We’re almost there!

I was thinking of the Fall recess, but it is also true of your pilot studies and the Music in Time conference at Harvard…

Our next class meeting will be on October 31. Before then, there are a few outstanding things that need to be done:

1. Group projects: Complete step 3 and send me all the materials as requested on the front page as soon as possible. Before you put the finishing touch, make sure to visit the MARL platform. Make sure to send me all text in a format that is easy to copy and paste (.doc, .txt., .rtf); all sound files should be in .mp3 format and labeled as explicitly as possible. As soon as I have all your materials, I will email them to Mary and her assistant; I will ask them to contact you directly if there are issues/questions.

2. Individual projects: All proposals should be posted on your individual page by Tuesday, October 22, 11:59 PM. All peer-reviews should be posted as soon as possible, but no later than Tuesday, October 29, 11:59 PM. The earlier you post your comments/suggestions, the most helpful it will be to the author!

The assignment for next class is as follows:

1. General discussion: Read/review London (2011) & Martens (2011); you might also want to read Repp’s response to the London piece (I’ve posted it on classes*v2). If you will not be attending class, please post a response to the Forum, by Tuesday, October 29, 11:59 PM, so that everybody attending will have time to read it (and respond if they wish to). Let me know if you would like to try to Skype in; I’ll see how we can arrange it on our end.

2. Student-led discussion: Swing ratio (Chris); check Forum for instructions.

Bibliography additions

(I will add to this post, as I read through new material.)

KVIFTE, T. (2007). Categories and timing: On the perception of meter. Ethnomusicology51(1), 64-84.

Summary: Kvifte’s article is largely theoretical; he argues several central points: first, he considers a reversal of a ‘common fast pulse’ theoretical paradigm, forwarded among certain metric theorists such as London particularly in the context of ‘complex’ or non-isochronous meters. He proposes a ‘common slow pulse’ paradigm, which, in broad strokes, holds that lower metric levels are divisive, and higher levels are additive. Second, Kvifte makes a distinction between ‘models of metric timing’ and ‘models of metric category’. Finally, Kvifte, although agreeing with London on several points, disagrees with the latter’s requiring that a non-isochronous level be upheld by an isochronous lower level.

Use: While Kvifte’s overall discussion is worth thinking about some more, most of it has no direct bearing on my work. Yet, in his discussion of the vexed binary of ‘additive meter’ vs. ‘divisive meter’, Kvifte surfaces some very useful and relevant ideas for my project. First, he quotes London (it seems the quote comes from London’s Grove article on ‘Rhythm’): “It is acknowledged that some melodic pattern may be heard in a number of different metric contexts”. This is exactly at least part of what I am investigating: theoretically, the rhythm I investigate projects a number of equally plausible meters (pulse hierarchies). Kvifte continues, digging up Curt Sachs (Rhythm and Tempo 1953). In response to Sachs distinction, Kvifte writes, “To perceive a rhythm as additive is fundamentally different than perceiving it as divisive” (67). This again is exactly what I investigate. I could say alternatively that I am testing this claim empirically, although not through the binary ‘additive’ vs. ‘divisive’. Later, he re-writes the same idea more summarily, “The point is that it is possible to perceive a given musical sound in both way [additive vs. divisive time], with distinctly different musical experiences” (67). Indeed, my tentative hypothesis is that study participants will not be able to identify the same rhythm when it is played in different metric contexts; that is, I suspect the rhythm will be experienced in ways distinctly different enough that the participant would not be able to discern rhythmic compositional identity.


TOUSSAINT, G., CAMPBELL, M., & BROWN, N. (2011). Computational models of symbolic rhythm similarity: Correlation with human judgments. Analytical Approaches to World Music1(2), 380-430.

This article has proved useful in that it points toward other potentially scholarship specifically relevant to my research subject. The authors write, ‘It is well known that the perception of musical rhythm is dependent on the underlying meter in which the rhythm is embedded’ (382). This seems to be more or less the same idea that I wish to test, although the specifics and the theoretical grounding will likely prove different. The authors attach several sources by which one can expand their summary statement. I will look into the following:

Johnson-Laird, P. N. 1991. “Rhythm And Meter: A Theory At The Computational Level.” Psychomusicology 10.2: 88–106.

Shmulevich, I. & Povel, D.-J. 2000. “Measures Of Temporal Pattern Complexity.” Journal of New Music Research 29.1: 61–69.

Palmer, C. & Krumhansl, C. L. 1990. “Mental Representations For Musical Meter.” Journal of Experimental Psychology – Human Perception and Performance 16:4: 728–41.

Longuet-Higgins, H. C. & Lee, C. S. 1984. “The Rhythmic Interpretation Of Monophonic Music.” Music Perception 1.4: 424–41.

Performance Analysis

I understand that everyone is very busy with midterm projects of various kinds, so I’ve chosen a relatively short article. This will give us a chance to delve into the details of the paper, so please do read it thoroughly. The article I’ve chosen also focuses on a specific musical example from the standard repertoire (Chopin’s Étude in E, Op. 10/3), which makes for a nice change of pace.

REPP, B. (1997). “The Timing Implications of Musical Structures”


This article makes the following claims: 1) musical structure in and of itself gives rise to constraints on expressive timing patterns; 2) these constraints give rise to normative timing profiles for expressive performance; 3) deviation from this norm requires cognitive effort (imagination)

Four pieces of evidence support these claims:
1) Statistical analysis of large samples of expressive performances reveals a remarkable similarity in timing profiles between two groups: a) professional pianists b) advanced students/amateur pianists
2) Inexpressive performances – expressive variations (of a very similar timing profile) were still present when pianists were asked to play without expression
3) When asked to detect variations in timing, subjects had the most trouble at moments where the music “dictates” that a timing variation take place; in other words, listeners expected to hear a specific timing pattern that deviated from absolute strictness
4) When asked to tap in synch with a metronomically timed Chopin excerpt, subjects still tapped in an expressive manner resembling the normative timing profile extracted from list item #1

1) Add to my summary, if you feel like something important is missing. The purpose is to provide a basic outline of the article.
2) What do you make of the following claim on p. 63? “Clearly, all these expressive ritardandos are related to the musical structure, particularly the melodic segmentation, though harmonic progression and metre may also contribute.”
3) In the third experiment, subjects had difficulty identifying variations where expressive lengthening was expected. How do these moments align with the metrical structure of the music? Do the results confirm or contradict what we might expect from the perspective of attentional energy?
4) On p. 63, Repp writes that “Deviation from this bland norm requires cognitive effort and imagination, whereas adherence to the norm merely requires musical competence.” In this statement, Repp takes a step toward defining true artistry in opposition to competent musicianship, paradoxically locating the former in deviations from what the music inherently tells us to do. What do you make of this?

CFP for “Music and Mathematics” RMA Study Day

The last topic listed (“Quantitative studies of musical scores and performances”) is relevant to this course.


RMA Study Day
Music and Mathematics

Saturday 12th April 2014 at the University of Leeds

Keynote speaker: Alan Marsden

From the fundamental mathematics of sound to a wide variety of high-level theoretical and compositional systems, mathematics has frequently shown itself to be a compelling, fruitful, and provocative discipline from which to approach the study of music.  This one-day conference aims to bring together scholars whose work takes inspiration from both fields and present a snapshot of the different ways that the affinities between music and mathematics are being explored today.

Proposals are invited for papers of 20 minutes, lecture recitals of c.30 minutes, and themed sessions/panels/roundtables of up to 90 minutes that address, in any way, the broad theme of music and mathematics.  Possible topics include:

o       Mathematical music theory and analysis
o       Compositional uses of mathematics
o       Composers whose creative processes might have involved mathematics and/or number
o       Present-day and historical mathematical perspectives on tuning
o       Philosophical and historical perspectives on the relationship between music and mathematics
o       Methodological issues in translating between mathematics and music
o       Quantitative studies of musical scores and performances

Abstracts of up to 250 words (750 words for themed sessions), with titles, should be sent to Daniel Holden (D.Holden@leeds.ac.uk) no later than Friday 6th December 2013.  Please also include a cover sheet detailing your name, institutional affiliation (if any), email address, AV requirements, and any other special requests.

Individual Project Proposal



How can the ethnomusicological study of drumming in cultural contexts inform the use of drumming in music therapy?


Music therapy is the use of music as a tool to bring about mental, physical, emotional or social change and healing in the participant.  This is separate from “traditional music and healing” found in various cultures, which is considered an alternative to modern healthcare and is usually the primary healthcare system of a society.

Importance of Question:

Music therapy is a new and important field of therapy that has real and distinct effects within practical world of psychology.  It has been used for multiple types of patients, from pedophiles to soldiers with PTSD to children with autism. Therefore, the furthering of the study of this field is increasingly important in understanding and developing further and better tailored treatments.  At the same time, the study of music to non-Western cultures has uncovered many traditions of music that have social purposes and effects, many of which involve healing.  However, these two fields have rarely intersected.  By investigating what about these musical cultural traditions leads to the social effects, these de facto techniques can be applied to the study of music therapy and advance its reach and effectiveness.


This paper will most likely consist of a literature review, then drawing conclusions and connections that will form an interdisciplinary thesis. (It is possible that an experimental proposal will come out of this endeavor as well, but no promises.) The first half of the paper will consist of a literature review of music therapy techniques, mainly those that involve drumming to create a focal point. This research will first look at the behavioral experiments and findings that underlie the therapeutic decisions in music therapy, and then the music that is used, the format of the therapy and the findings of experiments on these music therapy techniques.

The next section of the paper will consist of case studies of drumming traditions in various cultures.  As of now, there are two main focuses in this section—the use of drumming as a tool to create a cultural community in an immigrant population (Japanese Taiko drumming in Canada and African drumming in Florida) and the use of drumming as a tool for negotiation and unity in a community (American Indian powwow and drumming in Burundi and South Africa.)

Lastly, these two foci will be connected through an application of cultural concepts to music therapy techniques.


Bensimon, M., Amir, D., & Wolf, Y. (2008). Drumming through trauma: Music therapy with post-traumatic soldiers. The Arts in Psychotherapy, 35(1), 34–48. Science Direct

ABSTRACT: Combat stress reaction is common among soldiers and can develop to a post-traumatic stress disorder (PTSD). This distressing condition embraces symptoms such as feelings of loneliness and isolation from society, intrusive memories, outbursts of anger and generalized feelings of helplessness. Drumming has been receiving considerable attention in music therapy. Only few references relate to such activity among those who suffer from PTSD, and even fewer relate to combat induced post-traumatic syndrome, none of them empirical. The current study presents music therapy group work with six soldiers diagnosed as suffering from combat or terror related PTSD. Data were collected from digital cameras which filmed the sessions, open-ended in-depth interviews, and a self-report of the therapist. Some reduction in PTSD symptoms was observed following drumming, especially increased sense of openness, togetherness, belonging, sharing, closeness, connectedness and intimacy, as well as achieving a non-intimidating access to traumatic memories, facilitating an outlet for rage and regaining a sense of self-control.

RELEVANCE: This paper shows the effectiveness of drumming as a music therapy technique in the treatment of patients with PTSD.  It provides references for a number of background research studies that will be helpful in my analysis of music therapy, as well as an example of the drumming technique itself and the thought process behind it.

Mattern, M. (1996). The powwow as a public arena for negotiating unity and diversity in American Indian life. American Indian Culture and Research Journal, 4, 183–201. Google Scholar

ABSTRACT: In this article I will argue that the powwow can best be understood in these dual, paradoxical terms: It plays a unifying role in Indian life while providing a public arena for negotiation of differences and disagreements. The unifying role played by powwows is especially significant in light of the diversity within and among tribes. Although others have argued that the powwow plays a unifying role in this context of diversity, much can nevertheless still be added to our understanding of the specific practices that foster this unifying role. In the first part of this article, I will examine specific powwow practices in light of their unifying role. I will interpret the powwow as a communicative arena in which common experiences help create and sustain a common ground of memory, experience, identity, and commit- ment out of disparate experiences and identities.

RELEVANCE: This paper studies an example of the therapeutic aspects of drumming found naturally in a cultural context.  By looking at the case study of the powwow in American Indian culture, I will be able to understand the practice of joint drumming and develop a theory of the characteristics of drumming as a community that can be applied to music therapy.

Moreno, J. (1995). Ethnomusic therapy: An interdisciplinary approach to music and healing. The Arts in Psychotherapy, 22(4). Science Direct

ABSTRACT: Music has historically been, and continues to be an essential component of the practices of traditional healers in most of the tribal and other indigenous cultures throughout the world that are not primarily oriented toward the Western medical model. This is certainly well supported in the ethnomusicological literature, for example, in Musics of Many Cultures (May, 1993), a wide-ranging survey of 19 world music traditions. In the chapters on the musics and cultures of Indonesia, the Australian Aborigines, several sub-Saharan African cultures, North American Indians, Eskimos, South American Indian cultures and others, there is an overwhelming emphasis on the role of music in healing in the traditions of shamanism and spirit possession rituals. A study of these traditions within their cultural contexts can provide the basis for a better understanding of the role of music as therapy in modem health care settings (Moreno, 1988).

RELEVANCE: Moreno is one of the few music therapists to bridge the gap between ethnomusicology and music therapy.  His research (more general than my focus on drumming) is an example of the endeavor I wish to partake in, and is therefore invaluable to my understanding of the field, the possibilities of an interdisciplinary connection and ways to go about forging that connection.

Week 7 Assignment

1. Group Projects: Complete step 3 (refer to front page for details). You are strongly encouraged to set up a group meeting to finalize your plan for building your protocol. When you meet, go over the Group Task #5 handed-out in class today and “pratice” operationalizing the terms/concepts of your hypothesis. Then, review the steps from question to theory to conjecture to hypothesis for your group project. Finally, you can divide up the work that needs to happen for your protocol to be completed.

2. Individual Projects: Review your colleagues’ proposal and post questions, comments, suggestions. The contents of your posts should be motivated by collegial spirit and knowledge-seeking action. “Thought is the blossom; language is the bud; action the fruit behind it.” (Ralph Waldo Emerson)

3. General readings: London (2011) & Martens (2011) on issues of tempo perception. Forum posting this week is optional.

4. Student-led discussion: Performance analysis (Andrew); check the Forum blog for instructions.

Individual project research proposal: Is it the same rhythm if it’s heard in different meters?

Research Question:

(Note that my question has slightly changed shape.)

Broadly, my research project seeks to answer two interacting questions: first, can trained musicians recognize the same, fixed rhythmic pattern when it is construed across a number of different meters; and second, if so, with what degree of ease or difficulty per permuted configuration (fixed rhythm vs. permutation of meter type and rotation)?

I intend to answer this question through experiment, and so the form of my paper involves two sections. The first part uses theory to develop hypotheses, and the second designs the experiment whereby the hypotheses may be answered. It closes with a discussion of what interpretive and confounding issues I can foresee.

The fixed rhythm in question is a percussed phrase that is nearly ubiquitous across a large and diverse repertoire of Brazilian samba music. It consists of nine attacks in a cycle of sixteen. Theoretically, the rhythm can be conceptualized as a hyperdiatonic rhythm (Clough 1991): it is both maximally even and prime-generated (C=16, D=9, G=7). The unique composition of this pattern suggests a number of equally or near-equally plausible, good- or best- fit meters, both isochronous and non-isochronous. But are all these (near) equally possible meters similarly appreciable by the listener or performer? How well one can conceive of a rhythm as being the same when the enforced metric context changes? Even if the rhythm is compositionally the same, does one really hear it that way, as compositional identity? Or does one rather (coming from the other direction now) have to learn a rhythm anew each time he or she changes the situating metric background. Intuitively, I am inclined to side with the latter speculation. It is in the network of these questions that I intend to situate my experiment proposal.

Theory determines which meters and which rotations of the same I choose to test. All meters, both isochronous and non-isochronous, will be prime-generated and maximally even answering to London’s (2012) proposed augmented set of well-formedness constraints (which seek to include NI meters). With C (the size of the cyclic universe) set to a constant of 16, I use generators (G) 2,3,5, and 7, where 5 may be a special case (co-cyclic definition and lack of evidence in source music) and may be dropped. I do not intend to test every unique rotation of each meter as the number of permutations quickly becomes intractable and experimentally cumbersome. Rather, theory determines which rotations I use. To control the permutation size I employ the concept of ‘sampling’ as it occurs in Pressing (1983) to select which rotations I use. I choose those rotations that have the highest and the second highest number of points (the second only if different in cardinality by only 1 from the first) where the rhythm attacks coincides with metric events. The rotation of this rhythm is fixed: always 0101011010101101, where C=16 and ‘1’ represents an attack and ‘0’ no attack. These conditions select a set of meters and rotations. These constitute the test set stimuli.

From the results I hope to interpret whether and how difficulty is a function of either (1) the absolute number of similar points between rhythm and meter or (2) whether the meter is isochronous or not, or (3) some interaction between the two. I suspect that I will get some kind of an obvious answer as to whether or not participants recognize the same rhythm across all meters. Yet such an answer may be only of a general value (perhaps this has been addressed in some way by gestalt psychology). Mostly, as I expect participants will fail to recognize rhythmic pattern identity, I also anticipate some difficulty in being able to control for the potentially confounding influence of the perceived foreignness (and hence difficulty) of an NI metric state.

Finally, it seems like there needs to be some sort of ontological (?) discussion of sameness or not of a rhythm against different metric backgrounds.

Annotated Bibliography

PRESSING, J. (1983). Cognitive isomorphisms between pitch and rhythm in world musics: West africa, the balkans and western tonality. Studies in Music17, 38-61

Abstract (1st paragraph): “This paper compares some diverse musical phenomena in the light of their underlying structural similarities. Specifically, a number of common cyclic structures in pitch and rhythm are found to be isomorphic, and to be understandable in terms of the principles of mathematical group theory. Because such pitch and rhythm patterns are the products of human musical thinking, I call the relationships between them cognitive isomorphisms. By this phrase I do not mean to suggest that detailed cognitive models of such patterns are being presented—rather, that the observed structural similarities are sufficiently compelling, and their relationship to musical perception and training sufficiently direct, to justify the hypothesis that they may result from general cognitive processes.”

Comments: Pressing (1983) is especially important to my individual research project: for its content in general, which now constitutes a central contribution to rhythm and meter theory for non-isochronous and prime-generated rhythms; and for its specifically elaborating the concept of ‘multistability’ as it pertains to rhythm and meter. Item 6 (p.52) on ‘maximal perceptual stability’ from his closing overview will figure heavily in the opening theoretical exposition of my own work, especially: “This multistability appears to correlate closely with the basic patterns’ property of sampling all existing subgroups as equally as possible, a property which follows from their formation from group generators (…). If L is a prime number (…), then the concept of subgroup sampling is not applicable, and may be replaced by sampling runs compose of units of small size.”

CLOUGH, J., & DOUTHETT, J. (1991). Maximally even sets. Journal of Music Theory35(1), 93-173.

Abstract: (n/a)

Comments: This article provides the foundational theory for my using hyper-diatonic and hyper-pentatonic ME rhythms (cum meter). Other important concepts involve: first order vs. second order ME and ‘dlen’ (diatonic length) and ‘clen’ (chromatic length).

LONDON, J. (2012). Hearing in time: psychological aspects of music meter. (2nd ed.). New York: Oxford University Press.

Comments: This work provides the foundational theory defining my meter selection criteria for various values of G and D.


** I would appreciate a suggestion for relevant experimental articles for their designs/paradigms. **

– S P G

Individual Project – Proposal


QUESTION: What can expressive timing tell us about how performers delineate formal boundaries in passages that exhibit form-functional fusion?

SPECIFIC QUESTION: How do different performers interpret, through expressive timing, the structure of the main theme-transition complex of Mozart’s Piano Sonata in C, K. 545, Allegro?


1) Form-functional fusion: the merging of two formal functions within a single unit
MT-TR complex: a formal unit in which the cadence typically found between MT and TR is eliminated, thus giving rise to the the fusion of main theme and transition functions (the point is that the formal boundaries are blurred, at least from a theoretical perspective)

IMPORTANCE OF QUESTION: The current theoretical perspective on form-functional fusion (Caplin 1997) provides no insight into the point at which MT function becomes TR function in a MT-TR complex. A study of performances of Mozart’s K. 545, which contains such a complex, promises to give us some idea of how performers interpret this transitional moment.


The project I’m proposing is analytical. In other words, I’m interested in applying the results of current research to a specific case study (K. 545, Allegro).


I will first determine how performers group the first 12 measures of K. 545 by extracting timing profiles (IOIs) from a large body of commercial recordings.
I will then compare the grouping structure(s) that emerge from this data to musical characteristics in the score.


The format of the final project will be in two main parts: 1) the experiment and its attendant formalities; 2) discussion, which will branch out into the music-theoretical literature (particularly Caplin’s theory of formal functions)

The final product will analyze the results to see if a normative timing profile emerges, from which we can then extract a more nuanced idea of how MT-TR complexes work from a structural standpoint.


PALMER, C. (1989). “Mapping musical thought to musical performance.” Journal of Experimental Psychology: Human Perception and Performance”, 15 (2), 331-346. (PSYCNET)

ABSTRACT: Expressive timing methods are described that map pianists’ musical thoughts to sounded performance. In Experiment 1, 6 pianists performed the same musical excerpt on a computer-monitored keyboard. Each performance contained 3 expressive timing patterns: chord asynchronies, rubato patterns, and overlaps (staccato and legato). Each pattern was strongest in experienced pianists’ performances and decreased when pianists attempted to play unmusically. In Experiment 2 pianists performed another musical excerpt and notated their musical intentions on an unedited score. The notated interpretations correlated with the presence of the 3 methods: The notated melody preceded other events in chords (chord asynchrony); events notated as phrase boundaries showed greatest tempo changes (rubato); and the notated melody showed most consistent amount of overlap between adjacent events (staccato and legato). These results suggest that the mapping of musical thought to musical action is rule-governed, and the same rules produce different interpretations (Palmer, 1989).

RELEVANCE: Palmer’s work concludes that there is a general consensus among performers on how to express phrase boundaries. More specifically, she finds a correlation between such boundaries and tempo variation, indicating that timing is intimately related with the expression of musical form. And so it gives me reason to look at timing profiles for signs of such expression.

REPP, B. H. (1997). The Timing Implications of Musical Structures. In D. Greer (Ed.), Musicology and Sister Disciplines: Past, Present, Future. Paper presented at The 16th International Congress of the International Musicological Society, London (pp. 60–70). London: Oxford UP. (PART OF MY SPECIAL TOPIC READINGS)

ABSTRACT: For the last decade or so, the author has been engaged in empirical research on the small, unnotated variations in musical sound patterns that convey what is commonly referred to as expression in performance. The work has focused on piano performance, in part for methodological reasons and in part because the author is a pianist himself. Among the several parameters of expression, timing (rubato) has received the greatest attention. This paper summarizes the results of recent studies which have yielded four kids of empirical evidence converging on a single hypothesis: that musical structures have specific timing implications which constrain the considerable variety of expressive timing patterns observed in individual performances.

RELEVANCE: This paper addresses the potential results of my experiment in that it posits the existence of normative timing profiles that are suggested by the music itself. In other words, specific musical features constrain the expressive possibilities available to the performer. Greer’s findings suggest that by analyzing performances we can come to understand features inherent in the musical structure. More specifically, if normative timing profiles emerge, they will tell us something about the internal structure of Mozart’s MT-TR complex.


TODD, N. P. (1985). A Model of Expressive Timing in Tonal Music. Music Perception, 3, 33–58. (JSTOR)

ABSTRACT: During a performance, a pianist has direct control over only two variables, duration and intensity (Seashore, 1938). Other factors such as pitch and timbre are determined largely by the composer and the mechanics of the instrument. Thus expressiveness imparted to a performance lies in the departures from metrical rigidity and constant intensity. In this article, the first of the two variables is considered and it is shown how a duration structure can be generated, corresponding to the rubato in a performance, from the musical structure.The main input to the model is the time-span reduction of Lerdahl and Jackendoff’s theory (1977, 1983). Also shown is an interesting analogy between this model and the algorithms of Grosjean, Grosjean, and Lane (1979). Thus the hypothesis that expression is largely determined by musical structure, and the formal parallel between time-span reduction and prosodic structure are given empirical support.

RELEVANCE: Todd places musical phrase-final lengthening into a larger context, noting parallels in speech and other motor sequences. And so the impetus for using this tendency to slow down at a phrase’s end as a way of gleaning one’s conception of a formal structure is given a more universal significance. Todd’s formalization of phrase-final lengthening is also potentially useful in terms of how it might be represented both mathematically and graphically.

Individual Project Proposal

Research Question:

Can different rhythmic quantizations be heard as identical to the rhythm from which they were derived?

“Musical time can be considered to be the product of two time scales: the discrete time intervals of a metrical structure and the continuous time scales of tempo changes and expressive timing” (Clarke 1987a). Studies of quantization rely on this principle:  In musical performance, there is an idealized metric structure and a performed musical surface.  Typical studies of quantization create algorithms that attempt to derive the idealized rhythms (for instance, the rhythmic notation of Western art music) from the music’s performed surface.  In their most basic form, quantization algorithms use some “metric structure,” of discrete time intervals, to organize the disparate IOIs of the performed musical surface.  What if, rather than quantizing a performed surface onto a metric structure, a mechanically produced rhythm is quantized onto a metric structure distant to its own?  What relationships can be generalized about quantization of rhythms from one metric structure to another?  Using these as guiding questions, I wish to investigate which common rhythms are quantizations of other common rhythms and whether they can be heard interchangeably at given tempi.

Methods and Predicted Outcomes:

My project will be largely speculative using prior research as a means of generating hypotheses about the relationship between a rhythm and its various quantizations.  The project will begin with, and depend upon, a research review that explores approaches to quantization (as outlined in DESAIN & HONING 1992) and practical IOI limits for hearing rhythmic events as the same, different, or at all (as in LONDON 2012).  In the second portion of the paper, I will attempt to import these discussions of relationships between idealized musical rhythms and performed ones into my model of the relationships between different idealized rhythmic surfaces.

In many ways, my project presents fewer problems than traditional inquiries in the quantization of musical surfaces.  Issues of rubato and ictus complicate many computational approaches to quantization, particularly those that rely on IOI data as in Murphy’s model (2011).  To practically avoid this problem, real-time notation software has performers play to a click-track.  Similarly, since my model explores quantized rhythmic realizations of different metric structures that are of set, equal lengths, I do not need to deal with issues of composability, which have also complicated practical models of quantization.  As such, my model for quantization will be a good deal simpler than those presented by Desain& Honing, Murphy etc.  The model will map a rhythmic cycle (as a series of IOIs) onto a metric structure (of discrete timepoints) that is equal to the duration of one cycle (sum of the cycle’s IOIs).  Thus, I can consider each rhythmic cycle to be of some set, unit length, thereby eliminating cycle length as a parameter in the model.

As in Murphy (2011,) I will then inflect my model with perceptual guidelines so that it can ascertain what a listener is capable of hearing within a mechanical reproduction of the rhythms.  For instance, I will specifically discuss how the diatonic and hyper-diatonic timelines are quantized realizations of one another (figure 1), as has been suggested in Stover (2009).   At different tempi, the timelines can be alternately heard as the same or different under my proposed model as the result of different IOIs.

Within this paper I plan on deriving my examples from the well-known non-isochronous rhythms of Sub-Saharan West Africa, the Balkans, and India. figure 1 quantized timelinesfigure 1-Quantized Timelines

Annotated Bibliography:

MURPHY, D. (2011). Quantization revisited: A mathematical and computational model. Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance5(1), 21-34. [found on RILM]

Published Abstract: A nascent theory of near division is presented, from which an efficient quantization algorithm for rhythm intervals is derived. Based on a number theoretic analysis of the uniqueness and convergence of this first algorithm, a generalized algorithm is presented. An empirical study of the algorithm’s performance reveals a readily computable criterion within which the perceived ratio may reliably be produced on real performance data. Distribution properties are shown to be reasonable for computation.

Application: Murphy extends previous models of quantization to create one that is locally composable.  Upon applying perceptual constraints to his model, it successfully analyzed Temperley’s KP corpus via rhythm interval quantization.  I will follow suit by applying perceptual constraints as a means of constraining a otherwise unseemly model.

DESAIN, P., & HONING, H. (1999). Computational models of beat induction: The rule-based approach. Journal of New Music Research28(1), 29-42. [found on Google Scholar]

Published Abstract: This paper is a report of ongoing research on the computational modeling of beat induction which aims at achieving a better understanding of the perceptual processes involved by ordering and reformulating existing models. One family of rule-based beat induction models is described (Longuet-Higgins and Lee, 1982; Lee, 1985; Longuet-Higgins, 1994), along with the presentation of analysis methods that allow an evaluation of the models in terms of their in- and output spaces, abstracting from internal detail. It builds on work described in (Desain and Honing, 1994b). The present paper elaborates these methods and presents the results obtained. It will be shown that they can be used to characterize the differences between these models, a point that was difficult to assess previously. Furthermore, the first results of using the method to improve the existing rule-based models are presented, by describing the most effective version of a specific rule, and the most effective parameter settings.

DESAIN, P., & HONING, H. (1992). The quantization problem: Traditional and connectionist approaches. In M. BALABAN, K. EBCIOGLU & O. LASKE (Eds.), Understanding Music with AI: Perspectives on Music Cognition (pp. 448-463). Cambridge: MIT Press.  [found on RILM]

Published Abstract: Quantization separates continuous time fluctuations from the discrete metrical time in performance of music. Traditional and AI methods for quantization are explained and compared. A connectionist network of interacting cells is proposed, which directs the data of rhythmic performance towards an equilibrium state representing a metrical score. This model seems to lack some of the drawbacks of the older methods. The algorithms of the described methods are included as small Common Lisp programs.

Application: Desain and Honing’s two articles cited here serve to show a general quantization model for musical rhythm as a means of finding the metrical score.  I will be able to adopt a fairly traditional (non-AI) approach because I am not concerned with actual performances, only idealized rhythms and how those precise structures are perceived.

STOVER, C. (2009). A theory of flexible rhythmic spaces for diasporic African music. (University of Washington: Doctoral Dissertation). [found through communication with Stephen]

Published Abstract: This study proposes a model of flexible spans of time to describe some of the ways  in which the actual performed notes of Afro-Cuban musicians locate temporally, as mediated by the improvisational, call-and-response nature of the music as well as the overall teleological motion of the performance. It begins by addressing the ever-evolving discursive terrain around meter, beat hierarchy, and timelines, including various recent and historical perspectives, and as a dialectic begins to emerge between a listenerly perspective and a performerly one, an engagement with a Husserlian phenomenological epistemology unfolds. A detailed analysis begins, then, with a close phenomenological reading of three African and diasporic timelines, or topoi, in order to make some generalizations about how such metro-rhythmic events operate from structural and cognitive frames of reference. As the focus shifts from a metric orientation to a rhythmic one, the malleability of rhythm at a local level is considered: how parallel metro-rhythmic grids affect a performer’s interpretation of the rhythmic details of the music. I demonstrate how two very different metric construals of the rumba topos provide a basic framework from which to conceive of the rhythmic fabric of rumba as events that take place within flexible spans of time rather than between, or through, or alongside of, fixed points in time, and I propose a beat span model that accounts for, and acts as a restraint on, this flexibility. I engage several modern theoretical concepts of time-reckoning and recent micro-rhythmic theory, in light of beat span, and I look at numerous examples that illustrate how the superimposition of metric strata play out in actual musical performance, culminating in a close reading of a performance by Grup Afrocuba de Matanzas. Finally, the important question of whether we can actually entrain to two metric or rhythmic strands at the same time, or can “merely” shift quickly between them, is addressed, and ultimately I advance the response that not only can we do so, but in many cases we must do so in order to address the music in the way that it demands of us.

Application: Stover’s dissertation explores many Afrodiasporic rhythms and contends that understanding some musical rhythms requires multi-entrainment.  He posits the aforementioned relationship between timelines.  I believe my proposed model will specify some of his findings, in particular I will claim that tempo is fundamental to discussions of perceived similarity between quantizations of a rhythm with different metric structures.

Also Cited:

CLARKE, E. (1987). Levels of structure in the organization of musical time. Contemporary Music Review2, 212-238. [cited in (DESAIN & HONING, 1992)]

LONDON, J. (2012). Hearing in time: Psychological aspects of musical meter. (2 ed.). New York: Oxford University Press.