17-limit: Difference between revisions

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

* 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.

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