15-odd-limit

Odd limits:

---

• 1-odd-limit
• 3-odd-limit
• 5-odd-limit
• 7-odd-limit
• 9-odd-limit
• 11-odd-limit
• 13-odd-limit
• 15-odd-limit
• 17-odd-limit
• 19-odd-limit
• 21-odd-limit
• 23-odd-limit
• 25-odd-limit
• 27-odd-limit
• 29-odd-limit
• 31-odd-limit

The 15-odd-limit is the set of all rational intervals for which neither the numerator nor the denominator of the frequency ratio exceeds 15, once all powers of 2 are removed. To the 13-odd-limit, it adds 4 interval pairs involving 15.

Below is a list of all octave-reduced intervals in the 15-odd-limit. This collection of intervals has proven to be useful for illustrating certain characteristics of medium-sized EDOs (~15 to 41 steps).

• 1/1
• 16/15, 15/8
• 15/14, 28/15
• 14/13, 13/7
• 13/12, 24/13
• 12/11, 11/6
• 11/10, 20/11
• 10/9, 9/5
• 9/8, 16/9
• 8/7, 7/4
• 15/13, 26/15
• 7/6, 12/7
• 13/11, 22/13
• 6/5, 5/3
• 11/9, 18/11
• 16/13, 13/8
• 5/4, 8/5
• 14/11, 11/7
• 9/7, 14/9
• 13/10, 20/13
• 4/3, 3/2
• 15/11, 22/15
• 11/8, 16/11
• 18/13, 13/9
• 7/5, 10/7

Ratio	Size (¢)	Color name		Name(s)
16/15	111.731	g2	gu 2nd	classic diatonic semitone
15/14	119.443	ry1	ruyo unison	septimal diatonic semitone
15/13	247.741	3uy2	thuyo 2nd	tridecimal supermajor second / tridecimal second-third
15/11	536.951	1uy4	luyo 4th	undecimal acute fourth
22/15	663.049	1og5	logu 5th	undecimal grave fifth
26/15	952.259	3og7	thogu 7th	tridecimal subminor seventh / tridecimal sixth-seventh
28/15	1080.557	zg8	zogu octave	small septimal major seventh
15/8	1088.269	y7	yo 7th	just major seventh

The smallest EDO which is consistent in the 15-odd-limit is 29edo; that which is distinctly consistent in the same is 111edo.

See also

• Arto and Tendo Theory
• Diamond15 – as a scale