17-limit: Difference between revisions

The '''17-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 17. It is the 7th [[prime limit]] and is thus a superset of the [[13-limit]] and a subset of the [[19-limit]]. It adds to the [[13-limit]] a semitone of about 105¢ – [[17/16]] – and several other intervals between the 17th [[harmonic]] and the lower ones.

The 17-limit is a [[Rank and codimension|rank-7]] system, and 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.

In [[Functional Just System]], 17/16 is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone. The case for it being a diatonic semitone includes:

* The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps, and the associated [[comma]] 4131/4096 is small enough to be considered a comma which does not alter the interval category.  

* It is preferable for the intervals [[17/14]] and [[21/17]] to be thirds since they work well in tertian harmony. If [[7/4]] is known to be a seventh, then 17/16 must be a second.

 

In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:

* It is preferable for otonal intervals to be positive and utonal intervals to be negative in the chain of fifths.  

* It is preferable for the interval [[17/15]] to be a major second since it is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. If [[5/4]] is known to be a major third, then 17/16 must be an augmented unison.  

== Edo approximations ==

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.

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17.