17-limit: Difference between revisions

Line 9:

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~~.

* If [[7/4]] is known to be a seventh, assigning 17/16 to a second will make intervals [[17/14]] and [[21/17]] thirds. This is favorable because 17/14 and 21/17 are important building blocks of {{w|tertian harmony}}.

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.~~

* 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.

* 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.

@@ Line 9: / Line 9: @@
 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.
+* If [[7/4]] is known to be a seventh, assigning 17/16 to a second will make intervals [[17/14]] and [[21/17]] thirds. This is favorable because 17/14 and 21/17 are important building blocks of {{w|tertian harmony}}.
 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.
+* 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.
-* 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.