On Rhythm and Ratio
Our perception of time is weird and wonderful.
Indeed, the psychology of perception in general is fascinating and full
of surprises, especially if you come from a background in the physical sciences or engineering.
Here I consider some aspects of our perception of temporal intervals between musical notes.
The time that elapses between the onsets of two consecutive notes is known as an Inter-Onset Interval (IOI).
Consider in turn two consecutive IOIs: there is a certain ratio between them, as measured, and another ratio as perceived…
Why Care?
There is current theory and archaeological evidence that music may
predate natural language, indeed that much of our cognitive evolution, including language and abstract reasoning, may
have developed as an offshoot of musical communication. Typical intervals for morphemes in linguistics
cluster about the preferred interval for accurate IOI perception, with stress on syllables being a function of interval
ratio in many languages (e.g. Latin languages) while others (e.g. Germanic languages) prefer a steady beat.
In research terms, quantization of IOIs comes under the study of musical metre, which has been an active field.
In practical terms, in addition to furthering our understanding, the ability to model IOI perception allows a
beat tracker
to “get it right”, to listen as we do, allowing, e.g. a sequencer to quantize, or a
score follower
to behave, as a human listener would.
Quantization – What's the Problem?
For IOIs measured to an accuracy of about a millisecond, this ratio will in general appear odd, like 423:237 say; in a rhythmic context
however, we only perceive certain small whole round numbers, like 2:1. The cognitive process of how we get from the
physical stimulus (IOIs we can measure) to what we perceive (the beat we feel) is known
as quantization. As an engineer or scientist, one might be inclined to assume that values can simply
be rounded off to the nearest sensible value: this is not the case in general, and even if it were, there is no way of
determining which sensible value just by examining the ratios per se. In most styles of music,
expressive timing is a crucial part of the feel or groove of a performed rhythm. It is also a
crucial means by which musicians phrase a given passage so as to give judicious emphasis to certain structural
moments, rather like prosody in speech or punctuation in writing.
Basically, there is higher level structure in place from the metre or rhythm, which has a kind of momentum of
expectation, within which individual notes can be lengthened or shortened locally. This is rather like the
way in which syntax in natural language permits a degree of poetic licence – local deviations from the established
grammar that are understood in context. Except that with music, there is no fixed grammar or syntax –
each passage creates its own. The exact nature of the process by which this sense of metre arises is a question
that researchers have been scratching their heads over for decades. The mysterious nature of this process is
all the more remarkable considering that rhythm and the notion of a tactus or primary beat is indubitably one of the
most perceptually salient and essential aspects of most styles of music: it's what we tap our foot to. Indeed, in traditional music theory,
this question is almost completely overlooked, assuming that everybody “just gets it”.
Research
It seemed surprising to me that little or no prior mathematical research appeared to have been done on rounding ratios subject to
given constraints. So I went and did some, specifically catering for the constraints of rhythmic ratios such
as the occurrence of low powers of two.
The results are published in a paper of the
Journal of Mathematics and
Music 5(1):21–34.
There is a supplementary appendix available at
the journal website
and
further material for download (implementations of algorithms, test corpus, etc.) here
. A preprint is available
here.
This work was one component of a project to model metrical perception.
Related work, in preparation, includes metre induction and beat tracking.
Acknowledgement
Thanks to Henkjan Honing for first drawing
my attention to the curious nature of quantization in metre perception.