16/15: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone | |||
| Comma = yes | |||
| Name = diatonic semitone, | |||
| Color name = g2, gu 2nd | | Color name = g2, gu 2nd | ||
| Sound = jid_16_15_pluck_adu_dr220.mp3 | | 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>. | |||
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]]. | |||
When this ratio is taken as a comma to be tempered, it produces [[father]] temperament, | == Approximation == | ||
16/15 is very accurately approximated by [[43edo]] (4\43). | |||
== 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]]. | |||
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 == | == See also == | ||
* [[15/8]] – its [[octave complement]] | * [[15/8]] – its [[octave complement]] | ||
* [[45/32]] – its [[fifth complement]] | * [[45/32]] – its [[fifth complement]] | ||
* [[5/4]] – its [[fourth complement]] | |||
* [[256/243]] – the Pythagorean (3-limit) diatonic semitone | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[List of superparticular intervals]] | |||
== Notes == | |||
<references/> | |||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Semitone]] | [[Category:Semitone]] | ||
[[Category: | [[Category:Father]] | ||
[[Category: | [[Category:Commas named after their interval size]] | ||
Latest revision as of 02:03, 6 August 2025
| Interval information |
classic(al) diatonic semitone,
ptolemaic diatonic semitone
reduced,
reduced subharmonic
S6 × S7 × S8
[sound info]
The 5-limit superparticular interval 16/15 is the just diatonic semitone, classic(al) diatonic semitone or ptolemaic diatonic semitone[1].
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 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 complements of each other. It is a Mersenne comma.
The following linear temperaments are generated by a ~16/15:
In addition, this fractional-octave temperaments are generated by a ~16/15:
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