[bmuagpjw] prime waltz

consider the odd integers starting from 9.  bass drum hit if composite, snare drum if prime.  written in 6/4 time (or 6/8 time), beats 1 and 4 always have bass drum hits (integers 12n+9 and 12n+15 respectively, always divisible by 3 so always composite).

Pari/gp:

printbeat(x)=if(ispseudoprime(x), print1("*"), print1("."));

for(line=1, 64, for(half=0, 1, n=(line-1)*2+half+1; six=6*n; if(!half, printf("%3d %4d: ", line, six+3)); printbeat(six+3); printbeat(six+5); printbeat(six+7); if(half, print(), print1(" "))); if(0==line%8, print()))

first column is measure number.  second column is the starting integer of that measure, e.g., measure 1 is integers 9 11 13 15 17 19.  following that are 6 beats (3+3) of percussion with bass drum denoted with a period "." and snare drum denoted with an asterisk "*".

  1    9:  .** .**
  2   21:  .*. .**
  3   33:  ..* .**
  4   45:  .*. .*.
  5   57:  .** ..*
  6   69:  .** ..*
  7   81:  .*. .*.
  8   93:  ..* .**

  9  105:  .** .*.
 10  117:  ... ..*
 11  129:  .*. .**
 12  141:  ... .**
 13  153:  ..* ..*
 14  165:  .*. .*.
 15  177:  .** ...
 16  189:  .** .**

 17  201:  ... ..*
 18  213:  ... ..*
 19  225:  .** .*.
 20  237:  .** ...
 21  249:  .*. .*.
 22  261:  .*. .**
 23  273:  ..* .**
 24  285:  ... .*.

 25  297:  ... ..*
 26  309:  .** .*.
 27  321:  ... ..*
 28  333:  ..* ...
 29  345:  .** .*.
 30  357:  .*. ..*
 31  369:  ..* ..*
 32  381:  .*. .*.

 33  393:  ..* .*.
 34  405:  ..* ...
 35  417:  .** ...
 36  429:  .** ..*
 37  441:  .*. .*.
 38  453:  ..* .**
 39  465:  .*. ...
 40  477:  .*. ..*

 41  489:  .*. ..*
 42  501:  .*. .*.
 43  513:  ... .**
 44  525:  ... ...
 45  537:  ..* ..*
 46  549:  ... .*.
 47  561:  .*. .**
 48  573:  ..* ...

 49  585:  .*. .*.
 50  597:  .** ..*
 51  609:  ..* .**
 52  621:  ... ..*
 53  633:  ... .**
 54  645:  .*. .*.
 55  657:  .** ...
 56  669:  ..* .*.

 57  681:  .*. ..*
 58  693:  ... .*.
 59  705:  ..* ...
 60  717:  .*. ..*
 61  729:  ..* ..*
 62  741:  .*. ..*
 63  753:  ..* .*.
 64  765:  ..* .*.

you could have measure 0, a pickup bar, of just 2 or 3 beats (primes 3 5 7).

vaguely similar to the "binary cheer" from MIT.  stomp and clap.

the classic boom-chick-chick boom-chick-chick waltz rhythm -- corresponding to the maximum number of primes possible in a 6/4 measure -- happens in measure 1 (integers 9 11 13 15 17 19), measure 16 (integers 189 191 193 195 197 199), measure 156 (integers 1869...), measure 271 (integers 3249...), measure 471 (integers 5649...), measure 786 (integers 9429...), and is conjectured to continue to happen infinitely often (generalization of the twin prime conjecture).

c=0; for(i=0, +oo, n=12*i; if(isprime(n+11) && isprime(n+13) && isprime(n+17) && isprime(n+19), print(i+1," ",n+9); c+=1; if(c>100, break)))

no primes occur in measure 44 (integers 525 527 529 531 533 535, divisible by 5 17 23 3 13 5 respectively, part of a record prime gap between 523 and 541), measure 95 (integers 1137..., part of a record prime gap between 1129 and 1151), measure 111 (integers 1329...), measure 112 (integers 1341...) (which is two such measures in a row, part of the record prime gap between 1327 and 1361), and it is easy to prove (in contrast to the twin prime conjecture) that such empty measures occur infinitely often.  in such measures, accent beat 1 to distinguish among the otherwise identical bass drum hits.

or, replace bass drum on any beats other than 1 and 4 with a rest.

previously: on rhythm, depicting primes modulo n visually, and as musical pitches.