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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 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.

ratio
harmonic solfege
12/11
fu-sol
11/10
mi-fu
10/9
re-mi
9/8
do-re
8/7
ta-do
7/6
sol-ta
6/5
mi-sol, ti-re
11/9
re-fu
5/4
do-mi
14/11
fu-ta
9/7
ta-re
4/3
do-fa
11/8
do-fu
7/5
mi-ta
10/7
ta-mi
16/11
fu-do
3/2
do-sol
14/9
re-ta
11/7
ta-fu
8/5
mi-do
18/11
fu-re
5/3
sol-mi
12/7
ta-sol
7/4
do-ta
16/9
re-do
9/5
mi-re
20/11
fu-mi
11/6
sol-fu
2/1
do-do

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.

Relative to their size, edos which do (relatively) well in supporting 11-limit intervals are: 1edo, 2edo, 3edo, 4edo, 5edo, 6edo, 7edo, 9edo, 10edo, 12edo, 15edo, 22edo, 26edo, 31edo, 41edo, 63edo, 72edo, 87edo, 109edo, 161edo.

11-limit_compare.png

Intervals

Some of the simplest intervals of 11 include:

Interval
Cents Value
12/11
150.637
11/10
165.004
11/9
347.408
14/11
417.508
15/11
536.951
11/8
551.318
16/11
648.682
22/15
663.049
11/7
782.492
18/11
852.592
20/11
1034.996
11/6
1049.363
See: Gallery of Just Intervals

Music

Study #3 play by Dave Hill
Brief 11-ratio composition play by Dave Hill
11 Limit Piano by Chris Vaisvil
11-limit singtervals by Andrew Heathwaite

See also

Harmonic Limit