cp's OEIS Frontend

A117577 Equal divisions of the octave with nondecreasing consistency levels.

%I A117577 #7 Oct 02 2013 16:16:06
%S A117577 1,2,3,4,5,12,19,22,26,29,41,58,72,80,94,282,311,2554,12348,14842,
%T A117577 17461
%N A117577 Equal divisions of the octave with nondecreasing consistency levels.
%C A117577 An equal temperament is consistent at level N (odd integer) if all the intervals in the N-limit tonality diamond (set of ratios with odd factors of numerator and denominator not exceeding N) are approximated consistently, i.e. the composition of the approximations is the closest approximation of the composition.
%H A117577 Tonalsoft Encyclopedia of Microtonal Music Theory, <a href="http://tonalsoft.com/enc/c/consistent.aspx">Consistency</a>
%e A117577 3-EDO is consistent through the 5 limit because 6/5, 5/4 and 4/3 map to 1 step and 3/2, 8/5 and 5/3 map to 2 steps and all the compositions work out, for example 6/5 * 5/4 = 3/2 and 1 step + 1 step = 2 steps. It is not consistent through the 7 limit because 8/7 and 7/6 both map to 1 step, but 8/7 * 7/6 = 4/3 also maps to 1 step.
%p A117577 with(padic, ordp): diamond := proc(n) # tonality diamond for odd integer n local i, j, s; s := {}; for i from 1 by 2 to n do for j from 1 by 2 to n do s := s union {r2d2(i/j)} od od; sort(convert(s, list)) end: r2d2 := proc(q) # octave reduction of rational number q 2^(-floor(evalf(ln(q)/ln(2))))*q end: plim := proc(q) # prime limit of rational number q local r, i, p; r := 1; i := 0; while not (r=q) do i := i+1; p := ithprime(i); r := r*p^ordp(q, p) od; i end: vai := proc(n,i) # mapping of i-th prime by patent val for n round(evalf(n*ln(ithprime(i))/ln(2))) end: via := proc(n,l) # the patent val for n of length l local i,v; for i from 1 to l do v[i] := vai(n,i) od; convert(convert(v,array),list) end: h := proc(n, q) # mapping of interval q by patent val n if q=1 then RETURN(0) fi; dotprod(vec(q), via(n,plim(q))) end: consis := proc(n, s) # consistency of edo n with respect to consonance set s local i; for i from 1 to nops(s) do if not h(n, s[i])=round(n*l2(s[i])) then RETURN(false) fi od; RETURN(true) end: consl := proc(n) # highest odd-limit consistency for edo n local c; c := 3; while consis(n, diamond(c)) do c := c+2 od; c-2 end:
%Y A117577 Cf. A116474, A116475, A117578.
%K A117577 nonn
%O A117577 1,2
%A A117577 _Gene Ward Smith_, Mar 29 2006