11-limit

Prime limit

The 11-limit consists of all justly tuned intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. The 11-limit is the 5th prime limit and is a superset of the 7-limit and a subset of the 13-limit. Some examples of 11-limit intervals are 14/11, 11/8, 27/22 and 99/98.

These things are contained by the 11-limit, but not the 7-limit:

The 11-odd-limit;
Mode 6 of the harmonic or subharmonic series.

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the 7-limit introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of 12edo, the 11-limit introduces neutral intervals, superfourths and subfifths, which fall in between major, minor and perfect interval categories and thus demand new distinctions. It is thus inescapably xenharmonic.

Edo approximation

Here is a list of edos which represent 11-limit intervals with better accuracy (decreasing TE error): 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 and so on.

Here is a list of edos which tunes the 11-limit well relative to their size (TE relative error < 5%): 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 and so on.

Note: wart notation is used to specify the val chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

Intervals

11-odd-limit intervals

Ratio	Color Name		Harmonic Solfege
12/11	1u2	lu 2nd	fu-sol
11/10	1og2	logu 2nd	mi-fu
10/9	y2	yo 2nd	re-mi
9/8	w2	wa 2nd	do-re
8/7	r2	ru 2nd	ta-do
7/6	z3	zo 3rd	sol-ta
6/5	g3	gu 3rd	mi-sol, ti-re
11/9	1o3	ilo 3rd	re-fu
5/4	y3	yo 3rd	do-mi
14/11	1uz4	luzo 4th	fu-ta
9/7	r3	ru 3rd	ta-re
4/3	w4	wa 4th	do-fa
11/8	1o4	ilo 4th	do-fu
7/5	zg5	zogu 5th	mi-ta
10/7	ry4	ruyo 4th	ta-mi
16/11	1u5	lu 5th	fu-do
3/2	w5	wa 5th	do-sol
14/9	z6	zo 6th	re-ta
11/7	1or5	loru 5th	ta-fu
8/5	g6	gu 6th	mi-do
18/11	1u6	lu 6th	fu-re
5/3	y6	yo 6th	sol-mi
12/7	r6	ru 6th	ta-sol
7/4	z7	zo 7th	do-ta
16/9	w7	wa 7th	re-do
9/5	g7	gu 7th	mi-re
20/11	1uy7	luyo 7th	fu-mi
11/6	1o7	ilo 7th	sol-fu
2/1	w8	wa 8ve	do-do

Selected 15-odd-limit intervals

Here are all the 15-odd-limit intervals of 11:

Ratio	Site (¢)	Color name
12/11	150.637	1u2	lu 2nd
11/10	165.004	1og2	logu 2nd
11/9	347.408	1o3	ilo 3rd
14/11	417.508	1uz4	lu 4th
15/11	536.951	1uy4	luyo 4th
11/8	551.318	1o4	ilo 4th
16/11	648.682	1u5	lu 5th
22/15	663.049	1og5	logu 5th
11/7	782.492	1or5	loru 5th
18/11	852.592	1u6	lu 6th
20/11	1034.996	1uy7	luyo 7th
11/6	1049.363	1o7	ilo 7th