# 11edt

 ← 10edt 11edt 12edt →
Prime factorization 11 (prime)
Step size 172.905¢
Octave 7\11edt (1210.34¢)
(semiconvergent)
Consistency limit 5
Distinct consistency limit 4

11 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 11edt or 11ed3), is a nonoctave tuning system that divides the interval of 3/1 into 11 equal parts of about 173 ¢ each. Each step represents a frequency ratio of 31/11, or the 11th root of 3.

## Theory

From a no-two point of view, 11edt has a 5/3 major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic 7/1, which is off by almost half a step (or about a semitone), which causes it to temper out 49/45 in the 7-limit. 11edt is at the extreme end of arcturus temperament, defined by tempering out 15625/15309 in the 3.5.7 subgroup. It gives an equalized interpretation for the sub-arcturus MOS scale.

The 11th harmonic, 11/1, only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out 35/33 and 77/75 in the 11-limit. In the 3.5.11 subgroup, it tempers out 125/121.

### Relation to edos

11edt can be seen as a very stretched version of 7edo, with octaves sharpened by 10.3 cents. The octave stretching makes the 3/2 perfect fifth in better tune, while preserving a just 3/1 tritave.

### Prime harmonics

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced)
7
(7)
11
(0)
14
(3)
16
(5)
18
(7)
19
(8)
21
(10)
22
(0)
23
(1)
24
(2)
25
(3)
Approximation of prime harmonics in 11edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +10.3 +0.0 -19.8 -83.6 -1.6 +55.0 -63.6 -83.3 -68.2 +49.2 -66.3
Relative (%) +6.0 +0.0 -11.5 -48.4 -0.9 +31.8 -36.8 -48.2 -39.5 +28.5 -38.3
Steps
(reduced)
7
(7)
11
(0)
16
(5)
19
(8)
24
(2)
26
(4)
28
(6)
29
(7)
31
(9)
34
(1)
34
(1)

### Interval table

# Cents Hekts Approximate ratios Arcturus nonatonic notation (J = 1/1)
0 1/1 J
1 172.9 118.1 11/10, 10/9 J#, Kb
2 345.8 236.2 11/9 K
3 518.7 354.3 4/3, 27/20 L
4 691.6 472.4 3/2, 40/27 M
5 864.5 590.5 5/3, 28/17, 105/64 N
6 1037.4 708.6 29/16, 20/11, 64/35 N#, Ob
7 1210.3 826.7 2/1 O
8 1383.2 944.8 P
9 1556.1 1062.9 Q
10 1729 1181 R
11 1902 1300 J

## Pieces

Mozart's sonata #11 in A Major K331 in 11 EDT (using a 11 => 12 key mapping so octaves become tritaves)