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.

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

embodiment and (e)motion – assignment and leading questions

Hey folks,

For our discussion on “embodiment and (e)motion”, we’ll do one close reading, and one general. I’d like you to prepare the following:

1. (CLOSE READING) Read Phillips-Silver & Trainor (2007). Some guidance: closely read their definition of meter; decide whether they are trying to interpret ‘autonomous’ systems; see if you can pin-down what ‘auditory encoding’ is; debate whether their results (e.g., PDF p.8) suggest correlation or causality; ask whether you agree with their pithy last line; finally consider whether the study design(s) potentially involves some analysis/learning/conceptual mediation which would confound the putative direct connection between ‘movement’ and ‘listening’ – that is, might there not be some other process in between.

2. (GENERAL) Get a sense of the general tenor across the five articles (i.e., review the abstracts). Then read through Iyer (2002). What’s the point of this article? And please reflect on how important you think kinesthetic/physiological aspects are to your own theoretical work and thinking. We’ll try to talk through some perspectives.

In sum:

1. Read Phillips-Silver & Trainor (2007) closely.

2. Read Iyer (2002).

3. Read the five abstracts.

4. Post after the guidance per item 1.

– S P G

Research question/background

There is one rhythmic 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 (cf. Clough 1991): it is both maximally even and prime-generated (c=16, d=9, g=7). If two attacks are removed from where the rhythm ‘clumps’, that is, if the ‘semi-tones’ of the hyperdiatonic rhythm are undone such that every interval is either 2 or 3 sixteenths, then a hyperpentatonic rhythm obtains: the complement of the original rhythm and also maximally even and prime-generated. This second rhythm can also be conceptualized as describing a non-isochronous meter, where the faster pulse stream consists of straight sixteenth notes, and the slow consists of a mixture of 5 eighths and 2 dotted eighths. The metric relationship between these two pulse layers can be described, for example, in one rotation, as 2232223. Music theorists are interested in these scales or rhythms—or even metric descriptions—because they feature unique properties. One of these is that they are just as stable (or just as unstable) in any of the other domains produced by even division of the aggregate (c=16). The ‘problem’ with c=16 is that there is only one domain where even division results: multiples of two, or 2-generation, or g=2. However, 3-generation will yield a maximally even rhythm, which like the hyperpentatonic, can also be interpreted as a meter between two pulses, one isochronous and the other non-isochronous.


The question: if theory suggests that a hyperdiatonic rhythm such as the one described will be multi-stabile (or unstable) across several meters due to its special properties, does this empirically appear to be the case? If, for example, the rhythm is played in various metric contexts (just pulses, for simplicity, and for starters), is the rhythm always just as clear, or more/less clear? Does the listener assume one to be more ‘natural’ or ‘simple’ or ‘obvious’ than the other? And so on.

-S P G

Aka Pygmies, Zoboko

This music is at once monometric and polymetric, and continuously polyrhythmic, and this polyvalence presents rich implications for the cultural experience of musical time. Unpacking in reverse order: the music is polyrhythmic simply because each of the four parts adheres to its own rhythmic phrase. Out of focus, these layer into a rich web of harmonious tone conversing; and in focus, each project a meter or even several meters through which to comprehend any other three parts. The music is polymetric because any give rhythm projects one or more equally plausible meters and there are four rhythms. The music is also monometric because any given meter projected by one instrument may present a plausible way in which to conceive of the other three rhythms. In other words, a transcription by an outsider could use one of a number of possible meters to interpret the rhythmic phrases for the purpose of prescriptive reproduction. But the polyvalence is likely more than a mere exercise in gestalt switching. Holistically, the rhythmic texture of this music presents a sort of conversation in measured time—as opposed to parlando, for example—, where reciprocal elucidation is a potentially emphatic goal of performing. The polyvalence gives voice to both the importance of identity and solidarity; or alternatively, one asserts his or her own existence, but respects his or her relationship to others. The individual derives meaning, knowledge, and self-awareness from the group, and vice versa. In the case of monometric polyrhythm, where, hypothetically, to the insider there may only be one meter, the four distinct rhythms become as individual personalities upon a common existential plane.

I present a brief overview of the process of the music and some highlights from my diachronic and synchronic analyses, which I engage(d) both to make sense of the excerpt to my senses and to suggest something interesting to the reader. These will obviously belie some of my theoretical commitments.

Texture shapes this piece, and it implies at least one approach to comprehension. There are four instrument groups. The first two are quite similar, if not the same: some sort of percussed wooden body. The tambour of the third is quite salient in its difference being some sort of metal idiophone, a ‘bell’. The fourth is a mere, but significant, clap. The entrances of the four instruments (or, choirs, as the case may especially be for the wooden instrument) stagger: wood, wood, bell, and finally the clap. Intuitively, it seems that the duration between entrances is perhaps at least in part determined by the how long it takes to complete at least one cycle. It is as if the performance group has agreed to give the listeners enough time to make sense of, to ‘sympathize’ with one voice, before adding another and so on, so that each additional voice will have the opportunity to present its voice in high definition against the backdrop of the group. Or, perhaps more practically, each next performer needs a moment to figure his or her different rhythm against the one or several in play. After listening several times, I conclude that the duration of one cycle understates the magnitude of the inter-entrance interval.

I suspect that this music is processually cyclic. The two wooden parts challenge my ability to discern a regular rhythmic pattern, which I nevertheless hypothesize to exist. My assumption may be mistaken here, but I can at least confirm regularity with the other two parts. After an introductory phrase, the bell part cycles through a pattern of 24 sixteenth notes. One way to hear the bell part in terms of durations as integers is as 222122|2222122. My own testing-entrainment model distinguished between the ‘3 side’ (three 2s before the ‘12’) and the ‘4 side’ (four 2s before the ‘12’). This accounts for the dividing bar which partitions the asymmetrical ‘half-measures’ into 11+13 sixteenths. This phrase by itself projects numerous meters, both isochronous and non-isochronous. The clapping pattern is perhaps the easiest to ‘figure out’, but may also be the most interesting in terms of the new metric interpretations it proposes. The shortest possible clap cycle is only 3 sixteenths. In other words, if the density referent is arbitrarily assigned as a sixteenth note, the clap rhythm is a straight isochronous run: dotted eighths until the cows come home. Yet, with this simplicity, a rich new complexity: if before we were locked into the bell part—our Western ears holding fast for metric traction—the simple dotted eighth provokes wholesale reconsideration, that is, at least experientially. The first clap comes in on the ‘1’ of the ‘12’ of the second half of the two asymmetrical half-measures of the bell phrase. For my ear, the resultant polyrhythm between the bell and the clap begins to recall the special kind of Ewe 6/8 swing constituted with hemiola, that is, when I allow the claps to project the meter.

In a preliminary summary, this music presents a potpourri of metric possibilities, which I hypothesize to be equally plausible: there is no one metric state that attaches to or accounts for more attack points than any other. The relationships have been very carefully conceived. I found this performance excerpt to be fascinating also for its suggestions of trancelike polyvalence of poly- and monometric multi-stability, which I assume to be available to participant and observer alike, if not in equal portion.

-S P G

Journal of Mathematics and Music

By my count, the Journal of Mathematics and Music [subtitled: Mathematical and Computational Approaches to Music Theory, Analysis, Composition, and Performance] from its inception in 2007 contains about five [+/- a few] articles specifically devoted to some theoretical consideration of rhythm and or meter. The to-some [metric theory enthusiasts] well known Toussaint co-authors two of these (2009). In general, the journal contributors seem to hover around popular mathematic watering holes (or water coolers): scale theory, transformation theory, set theory, which places pitch prima inter pares among musical elements. The smallish seeming number is perhaps glibly suggestive: the journal is young, as is the faction it supports; and there are also roughly about three substantial articles per issue thrice annually.

-S P G