[seqfan] "polyrhythmic" sequences

[Bob Selcoe]

Hi Seqfans,

 A recent Seqfan posting suggests there's interest in relating sequences and music; specifically, combining two or more sequences in a "musical" way.  I think the concept of POLYRHYTHM in music lends itself quite nicely to this notion.  

I'll assume some people may not be familiar with musical terms, so here's a brief explanation and  example (skip to *** if you're familiar with polyrhythms).   A polyrhythm combines two or more standard rhythms or "beat" cycles simultaneously.   Think of a beat as the pulse listeners tap their toes to when hearing music - an even length of space between each tap.  Convention holds that we call the first beat "one". 

 

 Generally, music is constructed in "measures", where the first beat in the measure is accented.  A good example is a waltz, which has three beats per measure - or a 3-beat cycle - counted: ONE-two-three, ONE-two-three, ONE-two-three.   A polyrhythm "against" a waltz rhythm would have (at least) one other beat cycle, for instance 2 (ONE-two, ONE-two) played at the same time as the 3-beat cycle, with the accents occurring together.  This is called "2 against 3".

 

If we now treat beats as numbers in the counting sequence, "i against j" means that there is a LCM(i,j)-beat cycle; each individual rhythm has equally spaced "taps", and combined the two rhythms have accents occurring together at beats congruent to 1 mod LCM(i,j).  

 

So with the "2 against 3" example, the 2-rhythm (R[2]) has 2 equally-spaced "taps" in a 6-beat cycle, while the 3-rhythm (R[3]) has 3 equally-spaced taps.  Taps occur at beats:  

 

R[2] = {1,4,7,10,13,16,19.} == 1 mod 3: (two taps per cycle: {1,4}; {7,10}, {13,16}...)

R[3] = {1,3,5,7,9,11,13,15,17.} == 1 mod 2: (three taps per cycle:  {1,3,5}, {7,9,11}, {13,15,17...})

 

Where each cycle repeats at beats congruent to 1 mod 6.

 

*** Let's define a "polyrhythmic sequence" as interleaving two or more sequences of numbers congruent to 1 mod k, a(1)=1, where no modulus is a multiple of another, and all moduli cannot share a common factor.  For example, interleaving just two sequences, R[i] and R[j], generates sequence P[i,j].  So using the above example, interleaving R[2] and R[3]:

 

P[2,3] =  {1,3,4,5,7,9,10,11,13.} = A047251.  Its complement is A047238.

 

In music, this is the simplest polyrhythm.  Music students often are introduced to the idea of polyrhythms by being asked to tap "two" with one hand and "three" with the other.  The result is found in the rhythm of the famous main theme to "Carol of the Bells".  https://www.youtube.com/watch?v=1p3lB43eqpU.  

 

Slightly more complicated is "3 against 4" - a 12-beat cycle.  Music students learn this using the mnemonic , "Pass the G-ddamn butter".

 

R[3] = {1,5,9,13,17,21,25.}

R[4] = {1,4,7,10,13,16,19.}

P[3,4] = {1,4,5,7,9,10,13,16,17,19.}  Neither this nor its complement is in OEIS.  I will add them as time permits.



More "rhythms" (subsequences) add more complexity. For example, P[6,9,10] ("6 against 9 and 10") is a 90 beat cycle with R[6] == 1 mod 15, R[9] == 1 mod 10, and R[10] == 1 mod 9.  Generally, P[g,h,i..] is a "polyrhythmic" sequence interleaving subsequences R[z] == 1 mod LCM(P[*]/z). 

 

It might be interesting to see which polyrhythmic sequences currently are in OEIS.  Perhaps we could generate some interesting new ones.    



Cheers,

Bob Selcoe