15/14

Interval information

Ratio	15/14
Factorization	2-1 × 3 × 5 × 7-1
Monzo	[-1 1 1 -1⟩
Size in cents	119.4428¢
Names	septimal diatonic semitone, septimal major semitone
Color name	ry1, ruyo unison
FJS name	[math]\displaystyle{ \text{A1}^{5}_{7} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	7.71425
Weil norm (log2 max(n, d))	7.81378
Wilson norm (sopfr(nd))	17
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Septimal diatonic semitone

15/14 is a superparticular ratio with a numerator which is the fifth triangular number. It is traditionally called a diatonic semitone, perhaps for its proximity (and conflation in systems such as septimal meantone and marvel) with the classic diatonic semitone 16/15. However, 15/14 is a chromatic semitone in both Helmholtz–Ellis notation and the Functional Just System, viewed as the apotome 2187/2048 altered by 5120/5103. Marc Sabat has taken to call it the major chromatic semitone in the same material where 21/20 is also named as the minor diatonic semitone.

Because it contains exactly one of each prime up to 7, it appears as the interval between many simple 7-limit ratios. In particular, it is the difference between certain interval qualities of seconds, thirds, sixths, and sevenths: between classical minor and supermajor, and between subminor and classical major. These are the pairs of intervals separated by 15/14:

• 28/27 and 10/9
• 16/15 and 8/7
• 7/6 and 5/4
• 6/5 and 9/7
• 14/9 and 5/3
• 8/5 and 12/7
• 7/4 and 15/8
• 9/5 and 27/14

In addition, it separates the perfect fourth from the larger septimal tritone, and the perfect fifth from the smaller septimal tritone:

• 4/3 and 10/7
• 7/5 and 3/2

It also arises in higher limits as:

• 14/13 and 15/13
• 14/11 and 15/11
• 22/15 and 11/7
• 26/15 and 13/7

Approximation

15/14 is very accurately approximated by 10edo (1\10) and all linus temperaments. The linus comma, 5.6 ¢, is the amount by which a stack of ten 15/14's falls short of the octave.

In combination with 19/17 it forms a good approximation of golden meantone. The untempered combination of five 19/17's and two 15/14's leads to an interval that is sharp to an octave by the mercurial comma: (19/17)⁵ × (15/14)² = 2 / (mercurial comma).

Edo approximations for 15/14 (119.44 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
10	1\10	120.00	+0.56	+0.46
11	1\11	109.09	-10.35	-9.49
20	2\20	120.00	+0.56	+0.93
21	2\21	114.29	-5.16	-9.02
30	3\30	120.00	+0.56	+1.39
31	3\31	116.13	-3.31	-8.56
40	4\40	120.00	+0.56	+1.86
41	4\41	117.07	-2.37	-8.10
50	5\50	120.00	+0.56	+2.32
51	5\51	117.65	-1.80	-7.63
60	6\60	120.00	+0.56	+2.79
61	6\61	118.03	-1.41	-7.17
70	7\70	120.00	+0.56	+3.25
71	7\71	118.31	-1.13	-6.70
80	8\80	120.00	+0.56	+3.71

Temperaments

The following linear temperaments are generated by a ~15/14:

• Septidiasemi
• Subsedia

In addition, this fractional-octave temperament is generated by a ~15/14:

• Tertiosec (1\3)

Several 10th-octave temperaments treat ~15/14 as the period, including decoid and linus.

See also

• 28/15 – its octave complement
• 7/5 – its fifth complement
• List of superparticular intervals
• Gallery of just intervals