Appendix: Euclidean Rhythm Reference

This appendix catalogues the Euclidean distributions most relevant to Poly’s rhythmic vocabulary, organised by step count. For each pattern, the table lists the onset distribution, traditional associations, and the chapter where it is discussed in detail. Use this as a lookup reference when designing patches — if you know the tradition you are drawing from, find its Euclidean signature here and dial it in directly.

Phase Convention

Every row in this appendix is derived from bjorklund(n, k) in site/src/audio/bjorklund.ts and then, if a rotation is specified, rotated using Poly’s rotate-right convention (a rotation of r moves each onset r steps later in the cycle, wrapping at the end). Rows shown without an explicit rotation are printed at rotation 0 — the canonical Bjorklund output, with the first onset always at step 0. The Pattern column reads left-to-right from step 0, so x at position i means an onset at step i, and the Grouping column is the wrap-around gap sequence between consecutive onsets (each numeral is the distance to the next onset, and the numerals sum to n).

The same E(k,n) distribution can therefore appear in the wild with any of its n rotational spellings — they are all the same Euclidean rhythm, just phased differently against step 0. For example, E(3,16) at rotation 0 puts onsets at steps 0, 5, 10 (gaps 5+5+6, printed as x . . . . x . . . . x . . . . . in the 16-step table below); the same distribution rotated by 6 puts onsets at steps 0, 6, 11 (gaps 6+5+5), and rotated by 11 puts them at steps 0, 5, 11 (gaps 5+6+5). If you derive a row from Bjorklund by hand and get a different Pattern than the table shows, check the rotation before concluding the row is wrong: the two are the same distribution iff their gap sequences are cyclic rotations of each other.

4 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(1,4)	x . . .	4	Whole-note pulse	1
E(2,4)	x . x .	2+2	Backbeat, half-time snare	11
E(3,4)	x . x x	2+1+1	Afrobeat support	4
E(4,4)	x x x x	1+1+1+1	Four-on-the-floor kick	9

E(4,4)

E(4,4) is the most ubiquitous pattern in Western popular music — every hit occupied, every beat marked. It is the foundation of house, techno, disco, and most rock drumming.

5 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(2,5)	x . x . .	2+3	Khafif-e-ramal (Persian)	1
E(3,5)	x . x . x	2+2+1	Gamelan pelog accent	5

E(2,5)

E(2,5) produces a 2+3 grouping found in classical Persian rhythm. In Poly, a 5-step lane at 1/8 subdivision creates a cycle that drifts against 4-based patterns, generating polymetric interaction every 20 steps.

7 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(2,7)	x . . x . . .	3+4	Bendir accent (North Africa)	1
E(3,7)	x . x . x . .	2+2+3	Rachenitsa (Bulgaria, 7/8)	7
E(4,7)	x . x x . x .	2+1+2+2	Longa variant	7
E(5,7)	x . x x . x x	2+1+2+1+1	Dense aksak fill	7

E(3,7)

E(3,7) is the Balkan rachenitsa — three accents across seven pulses, grouping as 2+2+3. Rotation selects the variant: under Poly’s rotate-right convention, rotation 0 gives 2+2+3, rotation 3 gives 3+2+2, and rotation 5 gives 2+3+2.

8 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(3,8)	x . . x . . x .	3+3+2	Tresillo (Cuba, West Africa)	3
E(4,8)	x . x . x . x .	2+2+2+2	Straight eighth-note pulse	9
E(5,8)	x . x x . x x .	2+1+2+1+2	Cinquillo (Cuba)	3
E(7,8)	x . x x x x x x	2+1+1+1+1+1+1	Near-saturated fill	1

E(3,8)

The tresillo E(3,8) is arguably the most widely distributed rhythmic cell on earth. Its 3+3+2 grouping appears in Cuban son, New Orleans second line, reggaeton, Afrobeat, and electronic music. It is the irreducible atom of Afro-Atlantic rhythm.

E(5,8)

The cinquillo E(5,8) is the tresillo’s dense complement. Together, they cover nearly every pulse position in an 8-step cycle.

9 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(2,9)	x . . . x . . . .	4+5	Sparse aksak pulse	7
E(4,9)	x . x . x . x . .	2+2+2+3	Daichovo horo (Bulgaria, 9/8)	7
E(5,9)	x . x x . x . x .	2+1+2+2+2	Dense 9/8 variant	7

E(4,9)

E(4,9) produces the 2+2+2+3 grouping characteristic of the daichovo horo. Four accents across nine pulses create a lopsided cycle that resolves on the fourth beat’s extended duration.

11 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(4,11)	x . . x . . x . . x .	3+3+3+2	Kopanitsa (Bulgaria, 11/8)	7
E(5,11)	x . x . x . x . x . .	2+2+2+2+3	Dense kopanitsa variant	7

E(4,11)

E(4,11) is the kopanitsa pattern — four accents in eleven pulses. At the fast tempos typical of Bulgarian wedding music, this pattern becomes a blur of asymmetric drive. Rotation shifts through the various kopanitsa variants, each with a subtly different character.

12 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(3,12)	x . . . x . . . x . . .	4+4+4	Structural pulse (even ternary)	2
E(4,12)	x . . x . . x . . x . .	3+3+3+3	Triplet feel, shuffle accent	12
E(5,12)	x . . x . x . . x . x .	3+2+3+2+2	Reich phasing seed	8
E(7,12)	x . x . x x . x . x x .	2+2+1+2+2+1+2	Ewe gankogui bell (West Africa)	2
E(7,12)	x . x x . x . x x . x .	2+1+2+2+1+2+2	Aka bòndò dì.kétɔ̀ (Central Africa)	2b
E(9,12)	x . x x x . x x x . x x	2+1+1+2+1+1+2+1+1	Dense 12/8 fill	4

E(7,12) r2

E(7,12) is the timeline pattern Toussaint describes as the most commonly used in sub-Saharan African music. Its seven onsets across twelve pulses create the asymmetric grouping 2+2+1+2+2+1+2 that drives Ewe, Yoruba, and Afrobeat ensemble music.

E(5,12)

E(5,12) distributes five onsets sparsely across twelve positions. It is the seed pattern for Steve Reich-style phasing — two lanes running the same E(5,12) with one gradually drifting in rotation create an evolving canon.

The same seven onsets at rotation 0, x . x x . x . x x . x ., are the dì.kétɔ̀ of the Aka bòndò dance. There Arom does not read them as a timeline: they split two of the groups of three in the oddity model the drums play. See Chapter 2b.

16 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(3,16)	x . . . . x . . . . x . . . . .	5+5+6	Syncopated breakbeat kick	13
E(4,16)	x . . . x . . . x . . . x . . .	4+4+4+4	Straight quarter-note pulse	9
E(5,16)	x . . x . . x . . . x . . x . .	3+3+4+3+3	Bossa nova bass	10
E(7,16)	x . . x . x . x . . x . x . x .	3+2+2+3+2+2+2	Polymetric ghost layer	1
E(9,16)	x . x x . x . x . x x . x . x .	2+1+2+2+2+1+2+2+2	Dense hi-hat fill	9

E(5,16) r10

E(5,16) is the bossa nova bass pattern — five onsets distributed as evenly as possible across sixteen pulses, producing the gap sequence 3-3-4-3-3.¹ This is a genuine Euclidean rhythm. The Cuban son clave sits close to E(5,16) (differing only in the fifth hit’s position: 12 vs 13) but is not itself Euclidean; its gap sequence 3-3-4-2-4 contains a 2 that E(k,n) cannot produce. See Chapter 3 (Afro-Cuban) for the son-vs-rumba clave analysis and Chapter 10 (Brazilian) for the bossa nova reading.

24 Steps

E(k,n)	Pattern	Grouping	Traditional Association	Chapter
E(13,24)	x . x . x . x x . x . x . x . x . x x . x . x .	2+2+2+1+2+2+2+2+2+1+2+2+2	Aka dì.kétɔ̀ mould: ngbɔ̀lù	2b
E(13,24)	x x . x . x . x . x . x x . x . x . x . x . x .	1+2+2+2+2+2+1+2+2+2+2+2+2	Aka dì.kétɔ̀ mould: mò.nzòlì	2b
E(13,24)	x . x . x . x . x x . x . x . x . x . x . x x .	2+2+2+2+1+2+2+2+2+2+2+1+2	Aka dì.kétɔ̀ mould: yómbè, zɔ̀bɔ̀kɔ̀	2b
E(13,24)	x . x x . x . x . x . x . x . x x . x . x . x .	2+1+2+2+2+2+2+2+1+2+2+2+2	Aka dì.kétɔ̀ mould: mò.kóndí	2b
E(13,24)	x . x . x . x . x . x . x x . x . x . x . x . x	2+2+2+2+2+2+1+2+2+2+2+2+1	Aka dì.kétɔ̀ mould: mò.mbénzélé	2b

E(13,24) r4

E(13,24) is the figure of the Aka dì.kétɔ̀, two iron strips struck together. The figure never changes inside; each formula places it at a different offset from the pulsation, so it “works like a mould” (Arom 1991, p. 298). Each row above is one of those placements, and in Poly each is one Rotation value of one lane. The mould in Chapter 2b gives the offsets and has you hear them by turning one knob.

Further Reading

Every table on this page is an application of Toussaint’s result. The Geometry of Musical Rhythm is the book-length treatment, and Gotham’s review is a short critical assessment of it.

Using This Reference

When designing a Poly patch, start by identifying the tradition or groove type you are targeting. Find its Euclidean signature in the tables above. Set your anchor lane’s Steps and Hits accordingly. Then choose complementary patterns for additional lanes — patterns whose step counts create interesting polymetric ratios (prime or coprime values produce longer combined cycles) and whose hit counts provide contrasting density levels.

Remember that Rotation selects the variant within each E(k,n) family. The same steps and hits at different rotations produce patterns with different metric feels. Experiment with all available rotations before settling on a groove — the right rotation can transform a pattern from awkward to compelling.