17-limit: Difference between revisions

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== 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.

* In [[Helmholtz–Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone.

~~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 case for mapping it to either category may include:

* 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.

* Number of steps in the chain of fifths. The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps.

* 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}}.

* Size of the associated formal commas. The formal comma of the chromatic mapping, 2187/2176, is simpler and smaller than that of the diatonic mapping, 4131/4096, though both are generally considered small enough as commas which do not alter the interval category. The chromatic mapping has the advantage of keeping the Pythagorean order of diatonic semitone < chromatic semitone in the intervals of 17.

* Interactions with other primes. On one hand, 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}}. On the other hand, 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 [[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 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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 == 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.
+* In [[Helmholtz–Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone.
-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 case for mapping it to either category may include:
-* 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.
+* Number of steps in the chain of fifths. The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps.
-* 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}}.
+* Size of the associated formal commas. The formal comma of the chromatic mapping, 2187/2176, is simpler and smaller than that of the diatonic mapping, 4131/4096, though both are generally considered small enough as commas which do not alter the interval category. The chromatic mapping has the advantage of keeping the Pythagorean order of diatonic semitone < chromatic semitone in the intervals of 17.
+* Interactions with other primes. On one hand, 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}}. On the other hand, 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 [[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 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.