17/16: Difference between revisions

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In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about ~~105¢~~, it is close to the [[12edo]] semitone of ~~100¢~~, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth" ~~– for~~ instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1|13]] and [[15/1|15]].

In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about 105{{cent}}, it is close to the [[12edo]] semitone of 100{{cent}}, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth"—for instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1|13]] and [[15/1|15]].

17/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about ~~99¢~~. The difference between them is [[289/288]], about 6¢. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.

17/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about 99{{c}}. The difference between them is [[289/288]], about 6{{c}}. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.

17/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8¢. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9¢. The difference between ten 17/16's and [[11/6]] is approximately 0.2¢, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6¢.

17/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8{{c}}. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9{{c}}. The difference between ten 17/16's and [[11/6]] is approximately 0.2{{c}}, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6{{c}}.

== Terminology and notation ==

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For 17/16 specifically:

* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.

* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.

* In [[~~Helmholtz-Ellis~~ notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048|Pythagorean augmented unison (2187/2048)]].

* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048|Pythagorean augmented unison (2187/2048)]].

It could also be reasonable to treat 17/16 as the formal comma for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 2187/2048.

The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

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 }}
 {{Wikipedia|Minor diatonic semitone}}
-In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about 105¢, it is close to the [[12edo]] semitone of 100¢, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth" – for instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1|13]] and [[15/1|15]].
+In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about 105{{cent}}, it is close to the [[12edo]] semitone of 100{{cent}}, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth"—for instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1|13]] and [[15/1|15]].
-/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about 99¢. The difference between them is [[289/288]], about 6¢. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.
+/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about 99{{c}}. The difference between them is [[289/288]], about 6{{c}}. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.
-/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8¢. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9¢. The difference between ten 17/16's and [[11/6]] is approximately 0.2¢, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6¢.
+/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8{{c}}. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9{{c}}. The difference between ten 17/16's and [[11/6]] is approximately 0.2{{c}}, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6{{c}}.
 == Terminology and notation ==
@@ Line 15: / Line 15: @@
 For 17/16 specifically:
-* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.
+* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.
-* In [[Helmholtz-Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048|Pythagorean augmented unison (2187/2048)]].
+* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048|Pythagorean augmented unison (2187/2048)]].
+It could also be reasonable to treat 17/16 as the formal comma for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 2187/2048.
 The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.