11edt

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← 10edt 11edt 12edt →
Prime factorization 11 (prime)
Step size 172.905 ¢ 
Octave 7\11edt (1210.34 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 5

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

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.

From a no-2 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.

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 harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced)
26
(4)
26
(4)
27
(5)
28
(6)
28
(6)
29
(7)
29
(7)
30
(8)
30
(8)
31
(9)
31
(9)
32
(10)

Subsets and supersets

11edt is the fifth prime edt, following 7edt and before 13edt, so it does not contain any nontrivial subset edts.

Intervals

# Cents Hekts Approximate ratios* Arcturus
enneatonic notation (J = 1/1)
0 0.0 0.0 1/1 J
1 172.9 118.1 10/9, 11/10 J#, Kb
2 345.8 236.2 5/4, 6/5, 11/9, 27/22 K
3 518.7 354.3 4/3, 15/11 L
4 691.6 472.4 3/2 M
5 864.5 590.5 5/3, 18/11, 33/20 N
6 1037.4 708.6 9/5, 11/6, 20/11 N#, Ob
7 1210.3 826.7 2/1 O
8 1383.2 944.8 9/4, 11/5 P
9 1556.1 1062.9 5/2, 12/5, 22/9, 27/11 Q
10 1729.0 1181.0 8/3, 11/4 R
11 1902.0 1300.0 3/1 J

* As a 2.3.5.11-subgroup temperament

Music

Modern renderings

Wolfgang Amadeus Mozart

21st century

Chris Vaisvil
  • Frozen Time Occupies Wall Street (2011) – blog | play
  • Molly's Playground (2011) – blog | play

See also