17-limit: Difference between revisions

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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:

* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for ~~otonal intervals~~ to be positive and ~~utonal intervals~~ to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.

* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.

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.

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 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:
-* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for otonal intervals to be positive and utonal intervals to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.
+* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.
 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.