# 11-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. Some examples of 11-limit intervals are 14/11, 11/8, 27/22 and 99/98. The 11-limit is the 5th prime limit and is thus a superset of the 7-limit and a subset of the 13-limit.

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 approximations

A list of edos which represent 11-limit intervals with better accuracy: 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 and so on.

Another list of edos which tunes the 11-limit well relative to their size (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

## Music

birdshite stalactite
Andrew Heathwaite
Dave Hill
Ben Johnston
• String Quartet No. 6 (1980) – Bandcamp | YouTube – performed by Kepler Quartet
Claudi Meneghin
Chris Vaisvil
Randy Wells