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, aberschismic chromatic semitone, major chromatic 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 an interval in 7-limit just intonation measuring about 119.4 cents, traditionally known as the septimal diatonic semitone for its proximity (and conflation in marvel tunings such as septimal meantone) with the classical diatonic semitone 16/15. However, it functions as a chromatic semitone, as is supported by Sagittal notation, Helmholtz–Ellis notation and the Functional Just System, viewed as the Pythagorean apotome altered by an aberschisma. This gives rise to the more precise name aberschismic chromatic semitone, or as Marc Sabat has taken to call it, the major chromatic 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 the difference between:

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

Finally, since it is a superparticular ratio with a numerator which is the fifth triangular number, it is a triangle-particular ratio with factorization (25/24)⋅(36/35).

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