Sound / Patterns / Small experiments
Different rhythms, the same shadow
Two rhythms can share every circular interval count. Listen to a small example, then generate pairs of your own.
A measurement can tell us a great deal about a pattern and still miss what makes it different from another. Two small rhythms make that omission audible.
I had been exploring how an arrangement produces a spectrum. Turning the question round led somewhere more surprising: does knowing the spectrum tell us the arrangement?
Four hits around a circle
These patterns each put four hits into twelve equally spaced positions. Read left to right, then return to the beginning. X is a hit; a dot is a rest.
A XX..X.X..... B XX.X...X....
Starting at the first hit, A has gaps of 1, 3, 2, and 6 positions. B has gaps of 1, 2, 4, and 5. Even rotating or reversing a pattern cannot turn one into the other.
Same clicks, different places.
A: positions 1, 2, 5, and 7.
B: positions 1, 2, 4, and 8. Both clips use identical synthetic clicks, with no extra accent on the downbeat. Each bar lasts two seconds: four quarter-note beats at 120 BPM, three grid positions per beat.
The shared shadow
Choose any two hits and measure the shorter distance between them around the circle. Both patterns have exactly one pair at each distance from 1 to 6. Their pairwise interval inventories are identical. The relation is called homometry, or the Z-relation when excluding rotations and inversions in musical set theory.
That equality also gives matching Fourier magnitudes for the circular hit patterns. Phase information distinguishes them. The classic four-element example and the equivalence between interval content and Fourier magnitudes are discussed in Amiot and Sethares’ An Algebra for Periodic Rhythms and Scales, section 4.2.2 and Example 9 (PDF).
In my generated clips, the FFT magnitudes of complete loops match to floating-point precision. Their waveforms differ. A shorter analysis window need not show matching spectra: it can reveal where the hits occur. These are mathematical and synthesis checks, not a listening study.
Can we generate these at random?
Yes. Randomness chooses the pattern; an exact rule guarantees the relationship. A useful recipe is to fill six of twelve positions, then put the partner’s hits in the six empty positions. Both have the same circular interval counts. Finally, reject any pair that is merely a rotation or reversal.
| Distance | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| A | Enable JavaScript to calculate | |||||
| B | Enable JavaScript to calculate | |||||
The initial six-hit example is listed above. Enable JavaScript to generate and hear new pairs.
Filled positions are hits. The twelve positions form a circle; each number is a position, not a pitch. Starts silent, plays four bars, and stops when this page is hidden. No added downbeat accent. Audio begins only when you press Hear.
This generator chooses equally among fifteen underlying six-hit pair types, then varies their common starting position, direction, and A/B order. Random choices can repeat. It does not sample all musical rhythms, and it does not judge how well a pair will work in a song.
Why filling the gaps works
For any circular shift, count positions containing a hit in both the original pattern and its shifted copy. Call this count C. If a circle has N positions and the pattern has k hits, the corresponding count for its complement is N − 2k + C. With six hits in twelve positions, the extra term vanishes. Every shift count matches.
The complement relation and finite enumerations are treated in Jedrzejewski and Johnson’s The Structure of Z-related Sets, sections 2–3. I independently enumerated the 924 six-hit subsets, removed rotations and reversals, and recovered fifteen distinct pairs. The page constructs that catalogue before drawing from it.
The four-hit example is more constrained: with twelve positions, there is only one nontrivial pair type after rotations and reversals are removed. Randomising that case can vary its orientation, but does not create new underlying interval structures. Using six hits gives the generator more room.
What this could mean in a track
A phrase could alternate between partners while preserving its circular interval inventory. The gaps and placement change, even though that particular statistical description stays fixed. Accents, different sounds on different hits, swing, and a backing beat can all add information beyond the model.
The same sets can describe pitch classes, but the distinction matters: matching a pitch-class interval inventory does not mean that two chords have identical acoustic spectra. Here the demonstrated audio comparison uses equally spaced onsets and identical pulses.
There is also information beyond pairs. B contains a three-hit group at relative positions 0, 1, and 3; A contains no translated copy of it. Counting triples distinguishes what counting pairs leaves unresolved.
I like the possibility of variation with a deliberately preserved constraint. It gives randomness something precise to work inside, while leaving the musical decisions open.
Keep the workings
A question, a correction, or a connection of your own?
I’d be glad to hear it.