1,2

Consistency in Tonalsoft Encyclopedia of Microtonal Music Theory

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.

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:

Cf. A116474, A116475, A117578.

Sequence in context: A103651 A093713 A057472 this_sequence A069469 A109849 A007662

Adjacent sequences: A117574 A117575 A117576 this_sequence A117578 A117579 A117580

nonn

Gene Ward Smith (genewardsmith(AT)gmail.com), Mar 29 2006