11edt: Difference between revisions
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== Theory == | == Theory == | ||
11edt can be seen as a very [[stretched and compressed tuning|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. | 11edt can be seen as a very [[stretched and compressed tuning|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. This stretching significantly improves the approximation of harmonics 5 and 11 as well. | ||
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 [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]]. | 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 [[9L 2s (3/1-equivalent)|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]]. | The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. On account of this near-just 11th harmonic, it might be a practical simple [[2.3.11 subgroup]] system. 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 === | === Harmonics === | ||
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}} | {{Harmonics in equal|11|3|1|intervals=integer|columns=11}} | ||
Revision as of 05:15, 18 October 2026
| ← 10edt | 11edt | 12edt → |
(semiconvergent)
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. This stretching significantly improves the approximation of harmonics 5 and 11 as well.
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. On account of this near-just 11th harmonic, it might be a practical simple 2.3.11 subgroup system. 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
| 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) | |
| 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
- Piano Sonata No. 11 in A major, K. 331 – using a 11 → 12 key mapping so octaves become tritaves