17-limit: Difference between revisions
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In 17-limit [[just intonation]], all ratios in the system will contain no primes higher than 17. The 17-limit adds to the [[13-limit]] a "minor ninth" of about 105¢ -- [[17/16]] -- and several other intervals between the 17th overtone and the lower ones. | In 17-limit [[just intonation]], all ratios in the system will contain no primes higher than 17. The 17-limit adds to the [[13-limit]] a "minor ninth" of about 105¢ -- [[17/16]] -- and several other intervals between the 17th overtone and the lower ones. | ||
The 17-prime-limit can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is | The 17-prime-limit can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed. | ||
[[ | A list of [[edo]]s with progressively better tunings for 17-limit intervals: {{EDOs| 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 }} and so on. Another list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error|relative error]] < 5.4%): {{EDOs| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on. | ||
== 17-limit intervals == | == 17-limit intervals == | ||