17-limit: Difference between revisions

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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''' 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 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.

These things are contained by the 17-limit, but not the 13-limit:

* The [[17-odd-limit]];

* Mode 9 of the harmonic or subharmonic series.

== Terminology and notation ==

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 {{Prime limit navigation|17}}
-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''' 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 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.
+These things are contained by the 17-limit, but not the 13-limit:
+* The [[17-odd-limit]];
+* Mode 9 of the harmonic or subharmonic series.
 == Terminology and notation ==