16/15: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = classic/just diatonic semitone, classic/just minor second
| Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone
| Comma = yes
| 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>.


The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''classic''' or '''just diatonic semitone''' – the difference between the major third [[5/4]] and the fourth [[4/3]], and between [[3/2]] and [[8/5]].
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).
 
{{Interval edo approximation|16/15}}


== Temperaments ==
== 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, the following [[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 (temperament)|hendecatonic]].


== See also ==
== See also ==
Line 14: Line 37:
* [[45/32]] – its [[fifth complement]]
* [[45/32]] – its [[fifth complement]]
* [[5/4]] – its [[fourth complement]]
* [[5/4]] – its [[fourth complement]]
* [[256/243]] - the Pythagorean (3-limit) diatonic semitone
* [[256/243]] – the Pythagorean (3-limit) diatonic semitone
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
* [[16/15ths equal temperament|AS16/15]] - its ambitonal sequence
 
== Notes ==
<references/>


[[Category:Second]]
[[Category:Second]]
[[Category:Semitone]]
[[Category:Semitone]]
[[Category:Father]]
[[Category:Commas named for the generator of their temperament]]

Latest revision as of 10:46, 5 April 2026

Interval information
Ratio 16/15
Factorization 24 × 3-1 × 5-1
Monzo [4 -1 -1⟩
Size in cents 111.7313¢
Names just diatonic semitone,
classic(al) diatonic semitone,
ptolemaic diatonic semitone
Color name g2, gu 2nd
FJS name [math]\displaystyle{ \text{m2}_{5} }[/math]
Special properties square superparticular,
reduced,
reduced subharmonic
Tenney norm (log2 nd) 7.90689
Weil norm (log2 max(n, d)) 8
Wilson norm (sopfr(nd)) 16
Comma size large
S-expressions S4,
S6⋅S7⋅S8

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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).


Edo approximations for 16/15 (111.73 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
10 1\10 120.00 +8.27 +6.89
11 1\11 109.09 -2.64 -2.42
21 2\21 114.29 +2.55 +4.47
22 2\22 109.09 -2.64 -4.84
32 3\32 112.50 +0.77 +2.05
33 3\33 109.09 -2.64 -7.26
42 4\42 114.29 +2.55 +8.94
43 4\43 111.63 -0.10 -0.37
44 4\44 109.09 -2.64 -9.68
53 5\53 113.21 +1.48 +6.52
54 5\54 111.11 -0.62 -2.79
64 6\64 112.50 +0.77 +4.10
65 6\65 110.77 -0.96 -5.21
75 7\75 112.00 +0.27 +1.68
76 7\76 110.53 -1.20 -7.63

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, the following fractional-octave temperaments are generated by a ~16/15:

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

See also

Notes

  1. ↑ For reference, see 5-limit.
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