35edt: Difference between revisions
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{{Harmonics in equal|35|3|1|intervals=integer|columns=11}} | {{Harmonics in equal|35|3|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}} | {{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}} | ||
=== Subsets and supersets === | |||
Since 35 factors into primes as {{nowrap| 5 × 7 }}, 35edt has subset edts [[5edt]] and [[7edt]]. | |||
== Intervals == | == Intervals == | ||
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|} | |} | ||
<nowiki/>* As a 2.3.5.7.11.17-subgroup temperament | <nowiki/>* As a 2.3.5.7.11.17-subgroup temperament | ||
== See also == | |||
* [[22edo]] – relative edo | |||
* [[57ed6]] – relative ed6 | |||
* [[79ed12]] – relative ed12 | |||
Latest revision as of 09:37, 27 May 2025
| ← 34edt | 35edt | 36edt → |
35 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 35edt or 35ed3), is a nonoctave tuning system that divides the interval of 3/1 into 35 equal parts of about 54.3 ¢ each. Each step represents a frequency ratio of 31/35, or the 35th root of 3.
Theory
35edt is related to 22edo, but with the perfect twelfth rather than the octave being just. The octave is about 4.4854 cents compressed. Like 22edo, 35edt is consistent to the 12-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.5 | +0.0 | -9.0 | -14.9 | -4.5 | +0.4 | -13.5 | +0.0 | -19.4 | -21.4 | -9.0 |
| Relative (%) | -8.3 | +0.0 | -16.5 | -27.4 | -8.3 | +0.6 | -24.8 | +0.0 | -35.7 | -39.3 | -16.5 | |
| Steps (reduced) |
22 (22) |
35 (0) |
44 (9) |
51 (16) |
57 (22) |
62 (27) |
66 (31) |
70 (0) |
73 (3) |
76 (6) |
79 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.5 | -4.1 | -14.9 | -17.9 | -14.2 | -4.5 | +10.6 | -23.9 | +0.4 | -25.8 | +5.9 | -13.5 |
| Relative (%) | +28.5 | -7.6 | -27.4 | -33.0 | -26.2 | -8.3 | +19.5 | -43.9 | +0.6 | -47.6 | +10.8 | -24.8 | |
| Steps (reduced) |
82 (12) |
84 (14) |
86 (16) |
88 (18) |
90 (20) |
92 (22) |
94 (24) |
95 (25) |
97 (27) |
98 (28) |
100 (30) |
101 (31) | |
Subsets and supersets
Since 35 factors into primes as 5 × 7, 35edt has subset edts 5edt and 7edt.
Intervals
| # | Cents | Hekts | Approximate ratios* |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 1/1 |
| 1 | 54.3 | 37.1 | 33/32, 36/35 |
| 2 | 108.7 | 74.3 | 15/14, 16/15, 17/16, 18/17 |
| 3 | 163.0 | 111.4 | 10/9, 11/10, 12/11 |
| 4 | 217.4 | 148.6 | 8/7, 9/8 |
| 5 | 271.7 | 185.7 | 7/6 |
| 6 | 326.0 | 222.9 | 6/5 |
| 7 | 380.4 | 260.0 | 5/4 |
| 8 | 434.7 | 297.1 | 9/7 |
| 9 | 489.1 | 334.3 | 4/3 |
| 10 | 543.4 | 371.4 | 11/8, 15/11, 27/20 |
| 11 | 597.8 | 408.6 | 7/5, 10/7, 17/12, 24/17 |
| 12 | 652.1 | 445.7 | 16/11, 22/15 |
| 13 | 706.4 | 482.9 | 3/2 |
| 14 | 760.8 | 520.0 | 11/7, 14/9 |
| 15 | 815.1 | 557.1 | 8/5 |
| 16 | 869.5 | 594.3 | 5/3, 18/11, 33/20 |
| 17 | 923.8 | 631.4 | 12/7, 17/10 |
| 18 | 978.1 | 668.6 | 7/4, 30/17 |
| 19 | 1032.5 | 705.7 | 9/5, 11/6, 20/11 |
| 20 | 1086.8 | 742.9 | 15/8 |
| 21 | 1141.2 | 780.0 | 21/11, 27/14 |
| 22 | 1195.5 | 817.1 | 2/1 |
| 23 | 1249.9 | 854.3 | 33/16, 45/22 |
| 24 | 1304.2 | 891.4 | 15/7, 17/8, 21/10, 36/17 |
| 25 | 1358.5 | 928.6 | 11/5, 20/9, 24/11 |
| 26 | 1412.9 | 965.7 | 9/4 |
| 27 | 1467.2 | 1002.9 | 7/3 |
| 28 | 1521.6 | 1040.0 | 12/5 |
| 29 | 1575.9 | 1077.1 | 5/2 |
| 30 | 1630.2 | 1114.3 | 18/7 |
| 31 | 1684.6 | 1151.4 | 8/3, 21/8 |
| 32 | 1738.9 | 1188.6 | 11/4, 27/10, 30/11 |
| 33 | 1793.3 | 1225.7 | 14/5, 17/6, 45/16, 48/17 |
| 34 | 1847.6 | 1262.9 | 32/11, 35/12 |
| 35 | 1902.0 | 1300.0 | 3/1 |
* As a 2.3.5.7.11.17-subgroup temperament