59ed6: Difference between revisions
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== Theory == | == Theory == | ||
59ed6 corresponds to 22.8243…edo. It can be viewed as a stretched version of [[23edo]]. | 59ed6 corresponds to 22.8243…edo. It can be viewed as a [[stretched and compressed tuning|stretched]] version of [[23edo]] or a compressed version of [[36edt]]. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|59|6|1|intervals=integer|columns=11}} | {{Harmonics in equal|59|6|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}} | {{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}} | ||
=== Subsets and supersets === | |||
59ed6 is the 17th [[prime equal division|prime ed6]], so it does not contain any nontrivial subset ed6's. | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
== See also == | |||
* [[36edt]] – relative edt | |||
Revision as of 09:43, 27 May 2025
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| ← 58ed6 | 59ed6 | 60ed6 → |
59 equal divisions of the 6th harmonic (abbreviated 59ed6) is a nonoctave tuning system that divides the interval of 6/1 into 59 equal parts of about 52.6 ¢ each. Each step represents a frequency ratio of 61/59, or the 59th root of 6.
Theory
59ed6 corresponds to 22.8243…edo. It can be viewed as a stretched version of 23edo or a compressed version of 36edt.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.2 | -9.2 | +18.5 | +0.2 | +0.0 | -4.0 | -24.9 | -18.5 | +9.4 | +2.1 | +9.2 |
| Relative (%) | +17.6 | -17.6 | +35.1 | +0.4 | +0.0 | -7.6 | -47.3 | -35.1 | +17.9 | +4.1 | +17.6 | |
| Steps (reduced) |
23 (23) |
36 (36) |
46 (46) |
53 (53) |
59 (0) |
64 (5) |
68 (9) |
72 (13) |
76 (17) |
79 (20) |
82 (23) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -24.2 | +5.2 | -9.0 | -15.6 | -15.4 | -9.2 | +2.3 | +18.7 | -13.2 | +11.4 | -13.0 | +18.5 |
| Relative (%) | -46.0 | +10.0 | -17.2 | -29.7 | -29.4 | -17.6 | +4.4 | +35.5 | -25.2 | +21.7 | -24.7 | +35.1 | |
| Steps (reduced) |
84 (25) |
87 (28) |
89 (30) |
91 (32) |
93 (34) |
95 (36) |
97 (38) |
99 (40) |
100 (41) |
102 (43) |
103 (44) |
105 (46) | |
Subsets and supersets
59ed6 is the 17th prime ed6, so it does not contain any nontrivial subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 52.6 | 31/30, 34/33 |
| 2 | 105.2 | 33/31 |
| 3 | 157.7 | 23/21, 34/31 |
| 4 | 210.3 | 26/23 |
| 5 | 262.9 | 7/6 |
| 6 | 315.5 | 6/5 |
| 7 | 368 | 21/17, 26/21, 31/25 |
| 8 | 420.6 | 14/11, 23/18 |
| 9 | 473.2 | 25/19 |
| 10 | 525.8 | 19/14, 23/17 |
| 11 | 578.3 | 7/5 |
| 12 | 630.9 | |
| 13 | 683.5 | |
| 14 | 736.1 | 23/15, 26/17, 29/19 |
| 15 | 788.6 | 30/19 |
| 16 | 841.2 | |
| 17 | 893.8 | |
| 18 | 946.4 | 19/11 |
| 19 | 998.9 | |
| 20 | 1051.5 | 11/6 |
| 21 | 1104.1 | 17/9 |
| 22 | 1156.7 | |
| 23 | 1209.2 | |
| 24 | 1261.8 | 27/13, 29/14 |
| 25 | 1314.4 | |
| 26 | 1367 | 11/5 |
| 27 | 1419.5 | 25/11, 34/15 |
| 28 | 1472.1 | |
| 29 | 1524.7 | 29/12 |
| 30 | 1577.3 | |
| 31 | 1629.8 | |
| 32 | 1682.4 | 29/11 |
| 33 | 1735 | 30/11 |
| 34 | 1787.6 | |
| 35 | 1840.1 | 26/9, 29/10 |
| 36 | 1892.7 | |
| 37 | 1945.3 | |
| 38 | 1997.9 | 19/6 |
| 39 | 2050.4 | |
| 40 | 2103 | |
| 41 | 2155.6 | |
| 42 | 2208.2 | 25/7 |
| 43 | 2260.7 | |
| 44 | 2313.3 | 19/5 |
| 45 | 2365.9 | |
| 46 | 2418.5 | |
| 47 | 2471 | 25/6 |
| 48 | 2523.6 | 30/7 |
| 49 | 2576.2 | 31/7 |
| 50 | 2628.8 | |
| 51 | 2681.4 | 33/7 |
| 52 | 2733.9 | 34/7 |
| 53 | 2786.5 | 5/1 |
| 54 | 2839.1 | 31/6 |
| 55 | 2891.7 | |
| 56 | 2944.2 | |
| 57 | 2996.8 | |
| 58 | 3049.4 | |
| 59 | 3102 | 6/1 |
See also
- 36edt – relative edt