16/15: Difference between revisions

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The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref> – the difference between the major third [[5/4]] and the fourth [[4/3]]~~, and between~~ [[3/2]] and [[8/5]]. It is a [[~~Mersenne comma~~]].

The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref>.

It is the difference between:

* the major second [[9/8]] and the minor third [[6/5]];

* the major third [[5/4]] and the fourth [[4/3]];

* the perfect fifth [[3/2]] and the minor sixth [[8/5]];

* the major sixth [[5/3]] and the minor seventh [[16/9]];

* the major seventh [[15/8]] and the perfect octave [[2/1]].

== Approximation ==

16/15 is very accurately approximated by [[43edo]] (4\43).

== Temperaments ==

When this ratio is taken as a comma to be tempered, it produces [[father]] temperament, ~~where~~ 4/3 and 5/4 are equated~~. In this temperament~~, major thirds and fifths become [[octave complement]]s of each other.

When this ratio is taken as a [[comma]] to be [[tempering out|tempered out]], it produces [[father]] temperament, and lends itself the name '''father comma'''. In this [[exotemperament]], 4/3 and 5/4 are equated, and major thirds and fifths become [[octave complement]]s of each other. It is a [[Mersenne comma]].

The following [[linear temperament]]s are [[generate]]d by a [[~]]16/15:

* [[Vavoom]]

* [[Stockhausenic]]

In addition, this [[fractional-octave temperaments]] are generated by a ~16/15:

* [[Quintosec]]

Some [[11th-octave temperaments]] treat ~16/15 as the period, including [[hendecatonic]].

== See also ==

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* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

* [[16/15 equal-step tuning]] – equal multiplication of this interval

== Notes ==

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[[Category:Second]]

[[Category:Semitone]]

[[Category:Father]]

[[Category:Commas named after their interval size]]

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 }}
 {{Wikipedia|Semitone}}
-The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref> – the difference between the major third [[5/4]] and the fourth [[4/3]], and between [[3/2]] and [[8/5]]. It is a [[Mersenne comma]].
+The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref>.
+It is the difference between:
+* the major second [[9/8]] and the minor third [[6/5]];
+* the major third [[5/4]] and the fourth [[4/3]];
+* the perfect fifth [[3/2]] and the minor sixth [[8/5]];
+* the major sixth [[5/3]] and the minor seventh [[16/9]];
+* the major seventh [[15/8]] and the perfect octave [[2/1]].
+== Approximation ==
+/15 is very accurately approximated by [[43edo]] (4\43).
 == Temperaments ==
-When this ratio is taken as a comma to be tempered, it produces [[father]] temperament, where 4/3 and 5/4 are equated. In this temperament, major thirds and fifths become [[octave complement]]s of each other.
+When this ratio is taken as a [[comma]] to be [[tempering out|tempered out]], it produces [[father]] temperament, and lends itself the name '''father comma'''. In this [[exotemperament]], 4/3 and 5/4 are equated, and major thirds and fifths become [[octave complement]]s of each other. It is a [[Mersenne comma]].
+The following [[linear temperament]]s are [[generate]]d by a [[~]]16/15:
+* [[Vavoom]]
+* [[Stockhausenic]]
+In addition, this [[fractional-octave temperaments]] are generated by a ~16/15:
+* [[Quintosec]]
+{{todo|complete list}}
+Some [[11th-octave temperaments]] treat ~16/15 as the period, including [[hendecatonic]].
 == See also ==
@@ Line 18: / Line 38: @@
 * [[Gallery of just intervals]]
 * [[List of superparticular intervals]]
-* [[16/15 equal-step tuning]] – equal multiplication of this interval
 == Notes ==
@@ Line 25: / Line 44: @@
 [[Category:Second]]
 [[Category:Semitone]]
+[[Category:Father]]
+[[Category:Commas named after their interval size]]