Pythagorean Polyrhythmic Piano

April 30

This machine is a prototype for a concept ‘rhythm piano’ I started designing during my MA which allows a performer to play beat equivalents of tonal ratios. It forces you to compose using rhythm rather than pitch which is a very exploratory process. Feeling your way around the counter-rhythms allows you to discover unusual structures in swing and syncopation, whilst ensuring everything is easily reproducible.

It works by one motor driving 7 chains, and each chain loop has one or more special wide links on it that can lift and drop a hammer. This in turn activates an electronic drum. The spacing between the wider chain links determines the tempo of beat that that key plays when pressed. Although it’s a tempo I prefer to think of it as a pitch because it’s effectively a very very low frequency tone and the ratios between tones in this instrument based on those in western tonal music.

Thinking of a tempo as low tone is pretty much where the idea for this instrument came from. During the MA I became fascinated by the tipping points between our perception of tones, beats, and silence. A simple way of experiencing this is to slow a square wave so much you can eventually hear each ‘tick’ of it’s wave shape, for most listeners this happens around 7-10Hz [ref]. This in turn led to me asking if how we perceive tonal intervals relates to how we perceive tempo intervals, i.e. is the same processing (or parts of the same processing) activated in the brain when listening to polyrhythmic equivalents of polytonal music? For example, does the cognition of an octave against a root note (2:1) relate to the cognition of snare and bass drum in a 4/4 rock beat (2:1)?

This is where the ‘Pythagorean’ part of this instrument’s name comes from. The ratios of the tempos between keys is based on the just intonation ratios of tones for a perfect 4th (4:3), perfect 5th (3:2) and octave (2:1). Musical instruments are usually tuned to equal temperament, but for this piece the ratios of just intonation relate nicely to to Pythagoras’s mathematical ideal and the ratios 4:3, 3:2 and 2:1 are particularly interesting because along with being ‘nice’ tonic interval ratios they are also the most common counter-rhythm ratios. 3:2 is a hemolia jazz beat, and 4:3 is an even more swingy traditional rhythm. 2:1 is a straightforward rock beat.

The chain lengths for the keys from bottom to top are 6, 8, 9, 12, 16, 18 and 24. These are the smallest units which fit the ratios 1:1, 4:3, 3:2, 2:1, 8:3, 6:2, 4:1 respectively. If you regarded the root note as a C4, this makes this piano’s keys beat-equivalents of C4, F4, G4, C5, F5, G5, C6. The total number of chain links that pass before the cycle repeats is 144, so in fact you could generate the same patterns without using chains but but putting this pattern on a large drum and rotating it over hammers or electronic sensors:

Each chain’s hammer drops onto a small piezo transducer which activates a drum if the performer is holding down that key. Each drum is synthesised by a little digital circuit which allows for a range of sounds that the performer can explore. Using electronic drums is for convenience whilst getting a feel for this prototype, seeing how it feels to play with rhythm, but purely mechanical equivalents are in the pipeline. Note, there are problems with the drum circuit, it’s extremely noisy and eats batteries like no tomorrow. I rushed this build for BEAM day and didn’t design the circuit very well, consequently most of the power is eaten up in making several components very hot. This is why there are a lot of AVR posts on my site as I’ve been redesigning the circuit for future builds.

I’m actively developing other ways of generating polyrhythms mechanically as this is core to my PhD. This extends software-based rhythm improvisation tools exploring more extreme rhythmic effects such as phasing. In the meantime, you can still develop very interesting rhythms with this machine alone, there are strange ratios between certain keys, e.g. 9:8 between the ‘F’ and ‘G’ of this keyboard.

One final note I’ll leave as a thought experiment is that for this keyboard the ratio of rhythms is preserved if you reverse the order of keys, putting the fast tempo on the ‘low’ notes and the slow tempo on the ‘high’ notes. Instead of making tempo proportional to frequency, this makes it proportional to wavelength. One of the nice crossovers of dealing with perfect fractions is you get lots of interesting numerical artefacts like this.



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