A116474
Equal divisions of the octave with progressively increasing consistency levels.
Original entry on oeis.org
1, 3, 4, 5, 22, 26, 29, 58, 80, 94, 282, 311, 17461
Offset: 3
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.
A116475
Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.
Original entry on oeis.org
1, 3, 9, 27, 41, 58, 87, 111, 149, 217, 282, 388, 1323, 1600, 2554, 17461
Offset: 1
9-EDO is consistent and distinct through the 5 limit because 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3 map to 2, 3, 4, 5, 6 and 7 steps respectively and all the compositions of those intervals are consistent.
A117577
Equal divisions of the octave with nondecreasing consistency levels.
Original entry on oeis.org
1, 2, 3, 4, 5, 12, 19, 22, 26, 29, 41, 58, 72, 80, 94, 282, 311, 2554, 12348, 14842, 17461
Offset: 1
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.
- Tonalsoft Encyclopedia of Microtonal Music Theory, Consistency
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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:
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