17/12: Difference between revisions

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In [[17-limit]] [[just intonation]], '''17/12''' is the '''~~larger~~ septendecimal tritone''', measuring very nearly 603¢. Its inversion is the smaller septendecimal tritone, [[24/17]], and the interval that separates them is the small [[comma]] [[289/288]], about 6¢. This difference is usually negligible, and tempering out this comma allows the 600¢ half-octave to function as both septendecimal tritones. Thus, every even-numbered [[edo]] system contains a close approximation to these intervals.

In [[17-limit]] [[just intonation]], '''17/12''' is the '''large septendecimal tritone''', measuring very nearly 603¢. Its inversion is the smaller septendecimal tritone, [[24/17]], and the interval that separates them is the small [[comma]] [[289/288]], about 6¢. This difference is usually negligible, and tempering out this comma allows the 600¢ half-octave to function as both septendecimal tritones. Thus, every even-numbered [[edo]] system contains a close approximation to these intervals.

17/12 is the [[mediant]] between the two septimal tritones [[7/5]] and [[10/7]].

== Terminology and notation ==

~~There exists a disagreement in different conceptualization~~ systems on whether 17/12 should be ~~an augmented fourth or~~ a ~~diminished fifth. In~~ [[~~Functional Just System~~]]~~, it is~~ a ~~diminished fifth, separated by~~ [[~~4131/4096]] from [[1024/729~~]], the ~~Pythagorean diminished fifth In~~ [[~~Helmholtz-Ellis notation~~]]~~, it is an augmented fourth, separated by~~ [[~~2187/2176~~]] ~~from [[729/512]], the Pythagorean augmented fourth. The term "large septendecimal tritone" omits the distinction and only describes its melodic property i.e. the size~~.

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]]. See [[17-limit]] for a detailed discussion.

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

For 17/12 specifically:

* In [[Functional Just System]], it is a diminished fifth, separated by [[4131/4096]] from the [[1024/729|Pythagorean diminished fifth (1024/729)]].

* In [[Helmholtz-Ellis notation]], it is an augmented fourth, separated by [[2187/2176]] from the [[729/512|Pythagorean augmented fourth (729/512)]].

The term ''large septendecimal tritone'' omits the distinction and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal tritone of 24/17.

== See also ==

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-In [[17-limit]] [[just intonation]], '''17/12''' is the '''larger septendecimal tritone''', measuring very nearly 603¢. Its inversion is the smaller septendecimal tritone, [[24/17]], and the interval that separates them is the small [[comma]] [[289/288]], about 6¢. This difference is usually negligible, and tempering out this comma allows the 600¢ half-octave to function as both septendecimal tritones. Thus, every even-numbered [[edo]] system contains a close approximation to these intervals.
+In [[17-limit]] [[just intonation]], '''17/12''' is the '''large septendecimal tritone''', measuring very nearly 603¢. Its inversion is the smaller septendecimal tritone, [[24/17]], and the interval that separates them is the small [[comma]] [[289/288]], about 6¢. This difference is usually negligible, and tempering out this comma allows the 600¢ half-octave to function as both septendecimal tritones. Thus, every even-numbered [[edo]] system contains a close approximation to these intervals.
 /12 is the [[mediant]] between the two septimal tritones [[7/5]] and [[10/7]].
 == Terminology and notation ==
-There exists a disagreement in different conceptualization systems on whether 17/12 should be an augmented fourth or a diminished fifth. In [[Functional Just System]], it is a diminished fifth, separated by [[4131/4096]] from [[1024/729]], the Pythagorean diminished fifth In [[Helmholtz-Ellis notation]], it is an augmented fourth, separated by [[2187/2176]] from [[729/512]], the Pythagorean augmented fourth. The term "large septendecimal tritone" omits the distinction and only describes its melodic property i.e. the size.
+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]]. See [[17-limit]] for a detailed discussion.
-In practice, the interval category may, arguably, vary by context. One solution for the JI user who uses expanded [[circle-of-fifths notation]] is to prepare a [[Pythagorean comma]] accidental so that the interval can be notated in either category.
+For 17/12 specifically:
+* In [[Functional Just System]], it is a diminished fifth, separated by [[4131/4096]] from the [[1024/729|Pythagorean diminished fifth (1024/729)]].
+* In [[Helmholtz-Ellis notation]], it is an augmented fourth, separated by [[2187/2176]] from the [[729/512|Pythagorean augmented fourth (729/512)]].
+The term ''large septendecimal tritone'' omits the distinction and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal tritone of 24/17.
 == See also ==