**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 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 Performance*, *5*(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 Research*, *28*(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 Review*, *2*, 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.