A117578 Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.
1, 2, 3, 5, 6, 7, 8, 9, 12, 15, 16, 18, 19, 22, 23, 24, 25, 26, 27, 31, 35, 36, 37, 41, 46, 53, 58, 72, 80, 87, 94, 111, 121, 130, 149, 159, 183, 217, 282, 311, 388, 422, 624, 935, 1171, 1323, 1578, 1600, 2554, 12348, 14842, 17461
Offset: 1
Keywords
Examples
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.
Links
- Tonalsoft Encyclopedia of Microtonal Music Theory, Consistency
Programs
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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 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: condi := proc(n, s) # distinct consistency of edo n with respect to consonance set s local i, d; for i from 1 to nops(s) do if not h(n, s[i])=round(n*l2(s[i])) then RETURN(false) fi od; for i from 1 to nops(s) do d[i] := h(n, s[i]) od; if not nops(convert(d, set))=nops(s) then RETURN(false) fi; RETURN(true) end: condl := proc(n) # highest distinct odd-limit consistency for edo n local c; c := 3; while condi(n, diamond(c)) do c := c+2 od; c-2 end: