17-limit: Difference between revisions

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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.

== Terminology and notation ==

Conceptualization systems disagree on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]], and as a result the disagreement propagates to all intervals of [[harmonic class|HC17]].

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.

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]], and [[5/4]] is known to be a major third.

In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.

The names tabulated in [[#Intervals]] are common names and do not follow this discussion yet.

== Edo approximations ==

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: '''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.

== ~~17-limit intervals~~ ==

== Intervals ==

Here are all the [[21-odd-limit]] intervals of 17:

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 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.
+== Terminology and notation ==
+Conceptualization systems disagree on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]], and as a result the disagreement propagates to all intervals of [[harmonic class|HC17]].
+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.
+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]], and [[5/4]] is known to be a major third.
+In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.
+The names tabulated in [[#Intervals]] are common names and do not follow this discussion yet.
 == Edo approximations ==
@@ Line 11: / Line 25: @@
 : '''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.
-== 17-limit intervals ==
+== Intervals ==
 Here are all the [[21-odd-limit]] intervals of 17: