34ed7: Difference between revisions
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'''[[Ed7|Division of the 7th harmonic]] into 34 equal parts''' (34ED7) is related to [[12edo|12 EDO]], but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents | '''[[Ed7|Division of the 7th harmonic]] into 34 equal parts''' (34ED7) is related to [[12edo|12 EDO]], but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents. | ||
{| class="wikitable" | 34ED7 is consistent to the 11-[[integer-limit]], but not to the 12-integer-limit. In comparison, 12EDO is only consistent up to the 10-[[integer-limit]]. | ||
== Intervals == | |||
{| class="wikitable mw-collapsible" | |||
|+ Intervals of 34ed7 | |||
|- | |- | ||
! | degree | ! | degree | ||
| Line 184: | Line 188: | ||
| | [[7/4|harmonic seventh]] plus two octaves | | | [[7/4|harmonic seventh]] plus two octaves | ||
|} | |} | ||
== Harmonics == | |||
{{Harmonics in equal|34|7|1|intervals=prime}} | |||
{{Harmonics in equal|34|7|1|intervals=prime|collapsed=1|start=12}} | |||
== Regular temperaments == | == Regular temperaments == | ||
| Line 199: | Line 207: | ||
* [[18/17s equal temperament|AS18/17]] - relative [[AS|ambitonal sequence]] | * [[18/17s equal temperament|AS18/17]] - relative [[AS|ambitonal sequence]] | ||
{{todo|expand}} | |||
[[Category:Ed7]] | [[Category:Ed7]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
Revision as of 00:07, 23 December 2024
| ← 33ed7 | 34ed7 | 35ed7 → |
Division of the 7th harmonic into 34 equal parts (34ED7) is related to 12 EDO, but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents.
34ED7 is consistent to the 11-integer-limit, but not to the 12-integer-limit. In comparison, 12EDO is only consistent up to the 10-integer-limit.
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 99.0831 | 18/17 | |
| 2 | 198.1662 | 28/25 | |
| 3 | 297.2493 | 19/16 | |
| 4 | 396.3325 | 49/39, 34/27 | pseudo-5/4 |
| 5 | 495.4156 | 4/3 | |
| 6 | 594.4987 | 24/17 | |
| 7 | 693.5818 | 136/91 | pseudo-3/2 |
| 8 | 792.6649 | 30/19, 128/81 | |
| 9 | 891.7480 | 77/46 | pseudo-5/3 |
| 10 | 990.8311 | 85/48, 39/22 | |
| 11 | 1089.9143 | 15/8 | |
| 12 | 1188.9974 | 143/72, 175/88 | pseudo-octave |
| 13 | 1288.0805 | 21/10, 40/19 | |
| 14 | 1387.1636 | 49/22 | |
| 15 | 1486.2467 | 33/14 | |
| 16 | 1585.3298 | 5/2 | |
| 17 | 1684.4130 | 119/45, 45/17 | pseudo-8/3 |
| 18 | 1783.4961 | 14/5 | |
| 19 | 1882.5792 | 95/32, 98/33 | pseudo-3/1 |
| 20 | 1981.6623 | 22/7 | |
| 21 | 2080.7454 | 133/40, 10/3 | |
| 22 | 2179.8285 | 88/25 | |
| 23 | 2278.9116 | 56/15 | |
| 24 | 2377.9948 | 154/39, 320/81, 336/85 | pseudo-4/1 |
| 25 | 2477.0779 | 46/11 | |
| 26 | 2576.1610 | 133/30 | |
| 27 | 2675.2441 | 169/36 | |
| 28 | 2774.3272 | 119/24 | pseudo-5/1 |
| 29 | 2873.4103 | 21/4 | pseudo-16/3 |
| 30 | 2972.4934 | 39/7 | |
| 31 | 3071.5766 | 112/19 | pseudo-6/1 |
| 32 | 3170.6597 | 25/4 | |
| 33 | 3269.7428 | 119/18 | |
| 34 | 3368.8259 | exact 7/1 | harmonic seventh plus two octaves |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.0 | -19.4 | -12.0 | +0.0 | +10.2 | +18.2 | +49.2 | -44.3 | +21.3 | +16.3 | -0.0 |
| Relative (%) | -11.1 | -19.6 | -12.1 | +0.0 | +10.3 | +18.4 | +49.7 | -44.7 | +21.5 | +16.5 | -0.0 | |
| Steps (reduced) |
12 (12) |
19 (19) |
28 (28) |
34 (0) |
42 (8) |
45 (11) |
50 (16) |
51 (17) |
55 (21) |
59 (25) |
60 (26) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.1 | +11.3 | +28.0 | -26.9 | -36.8 | -24.3 | +17.1 | -46.2 | -47.5 | +3.4 | -34.2 |
| Relative (%) | -9.2 | +11.4 | +28.2 | -27.2 | -37.1 | -24.5 | +17.3 | -46.7 | -48.0 | +3.5 | -34.5 | |
| Steps (reduced) |
63 (29) |
65 (31) |
66 (32) |
67 (33) |
69 (1) |
71 (3) |
72 (4) |
73 (5) |
74 (6) |
75 (7) |
76 (8) | |
Regular temperaments
34ED7 can also be thought of as a generator of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a cluster temperament with 12 clusters of notes in an octave (quintupole temperament). This temperament is supported by 12EDO, 109EDO, and 121EDO among others.