16/15

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
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Semitone

The 5-limit superparticular interval 16/15 is the just diatonic semitone, classic(al) diatonic semitone or ptolemaic diatonic semitone.

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:

• 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.

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

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