

A117577


Equal divisions of the octave with nondecreasing consistency levels.


3



1, 2, 3, 4, 5, 12, 19, 22, 26, 29, 41, 58, 72, 80, 94, 282, 311, 2554, 12348, 14842, 17461
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OFFSET

1,2


COMMENTS

An equal temperament is consistent at level N (odd integer) if all the intervals in the Nlimit 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.


LINKS

Table of n, a(n) for n=1..21.
Tonalsoft Encyclopedia of Microtonal Music Theory, Consistency


EXAMPLE

3EDO 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.


MAPLE

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 ith 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 oddlimit consistency for edo n local c; c := 3; while consis(n, diamond(c)) do c := c+2 od; c2 end:


CROSSREFS

Cf. A116474, A116475, A117578.
Sequence in context: A093713 A057472 A333390 * A109849 A007662 A069469
Adjacent sequences: A117574 A117575 A117576 * A117578 A117579 A117580


KEYWORD

nonn


AUTHOR

Gene Ward Smith, Mar 29 2006


STATUS

approved



