16/15: Difference between revisions

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'''16/15'''

{{Infobox Interval

|4 -~~1 -1>~~

| Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone

| Comma = yes

| Color name = g2, gu 2nd

| Sound = jid_16_15_pluck_adu_dr220.mp3

}}

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>.

~~111~~.~~73129 cents~~

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]].

[[~~File:jid_16_15_pluck_adu_dr220.mp3~~]] ~~[[:File:jid_16_15_pluck_adu_dr220~~.~~mp3|sound sample]]~~

== Approximation ==

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

~~The 5-limit just diatonic semitone~~ is ~~the~~ [[~~superparticular~~|~~superparticular~~]] ~~interval 16/15 -- the difference between the major third~~ [[~~5/4|5/4~~]] and the ~~fourth~~ [[4/3|4~~/3]]~~, and ~~between~~ [[~~3/2|3/2~~]] ~~and~~ [[~~8/5|8/5~~]].

== Temperaments ==

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]].

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

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

[[~~Category:5-limit~~]]

* [[Vavoom]]

[[Category:~~interval~~]]

* [[Stockhausenic]]

[[Category:~~just_interval~~]]

[[Category:~~ratio~~]]

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

[[Category:~~semitone~~]]

* [[Quintosec]]

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

== See also ==

* [[15/8]] – its [[octave complement]]

* [[45/32]] – its [[fifth complement]]

* [[5/4]] – its [[fourth complement]]

* [[256/243]] – the Pythagorean (3-limit) diatonic semitone

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

== Notes ==

[[Category:Second]]

[[Category:Semitone]]

[[Category:Father]]

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

@@ Line 1: / Line 1: @@
-'''16/15'''
+{{Infobox Interval
-|4 -1 -1&gt;
+| Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone
+| Comma = yes
+| Color name = g2, gu 2nd
+| Sound = jid_16_15_pluck_adu_dr220.mp3
+}}
+{{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>.
-.73129 cents
+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]].
-[[File:jid_16_15_pluck_adu_dr220.mp3]] [[:File:jid_16_15_pluck_adu_dr220.mp3|sound sample]]
+== Approximation ==
+/15 is very accurately approximated by [[43edo]] (4\43).
-The 5-limit just diatonic semitone is the [[superparticular|superparticular]] interval 16/15 -- the difference between the major third [[5/4|5/4]] and the fourth [[4/3|4/3]], and between [[3/2|3/2]] and [[8/5|8/5]].
+== Temperaments ==
+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]].
-When this ratio is taken as a comma to be tempered, it produces [[Father|father]] temperament, where 4/3 and 5/4 are equated. In this temperament, major thirds and fifths become octave-inverses of each other.
+The following [[linear temperament]]s are [[generate]]d by a [[~]]16/15:
-[[Category:5-limit]]
+* [[Vavoom]]
-[[Category:interval]]
+* [[Stockhausenic]]
-[[Category:just_interval]]
-[[Category:ratio]]
+In addition, this [[fractional-octave temperaments]] are generated by a ~16/15:
-[[Category:semitone]]
+* [[Quintosec]]
+{{todo|complete list}}
+Some [[11th-octave temperaments]] treat ~16/15 as the period, including [[hendecatonic]].
+== See also ==
+* [[15/8]] – its [[octave complement]]
+* [[45/32]] – its [[fifth complement]]
+* [[5/4]] – its [[fourth complement]]
+* [[256/243]] – the Pythagorean (3-limit) diatonic semitone
+* [[Gallery of just intervals]]
+* [[List of superparticular intervals]]
+== Notes ==
+<references/>
+[[Category:Second]]
+[[Category:Semitone]]
+[[Category:Father]]
+[[Category:Commas named after their interval size]]