41ed6: Difference between revisions
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41ed6 is similar to [[16edo]], but has the [[ | == Theory == | ||
41ed6 is similar to [[16edo]], but has the 6th harmonic tuned just instead of the [[2/1|octave]], which stretches the octave by 10.5{{c}}. It can be used as a tuning for [[mavila]] and has an antidiatonic ([[2L 5s]]) scale which approximates [[Pelog]] tunings in Indonesian gamelan music. | |||
=== Harmonics === | |||
{{Harmonics in equal|41|6|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|41|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41ed6 (continued)}} | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
{{Todo|expand}} | |||
[[Category:Pelog]] | [[Category:Pelog]] | ||
[[Category:Mavila]] | [[Category:Mavila]] | ||
Revision as of 07:30, 27 May 2025
| ← 40ed6 | 41ed6 | 42ed6 → |
41 equal divisions of the 6th harmonic (abbreviated 41ed6) is a nonoctave tuning system that divides the interval of 6/1 into 41 equal parts of about 75.7 ¢ each. Each step represents a frequency ratio of 61/41, or the 41st root of 6.
Theory
41ed6 is similar to 16edo, but has the 6th harmonic tuned just instead of the octave, which stretches the octave by 10.5 ¢. It can be used as a tuning for mavila and has an antidiatonic (2L 5s) scale which approximates Pelog tunings in Indonesian gamelan music.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.5 | -10.5 | +21.0 | +13.0 | +0.0 | +35.8 | +31.6 | -21.0 | +23.5 | +9.8 | +10.5 |
| Relative (%) | +13.9 | -13.9 | +27.8 | +17.2 | +0.0 | +47.3 | +41.7 | -27.8 | +31.1 | +13.0 | +13.9 | |
| Steps (reduced) |
16 (16) |
25 (25) |
32 (32) |
37 (37) |
41 (0) |
45 (4) |
48 (7) |
50 (9) |
53 (12) |
55 (14) |
57 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +23.3 | -29.4 | +2.5 | -33.6 | +12.8 | -10.5 | -28.5 | +34.0 | +25.2 | +20.4 | +19.1 | +21.0 |
| Relative (%) | +30.7 | -38.8 | +3.3 | -44.4 | +16.9 | -13.9 | -37.6 | +45.0 | +33.4 | +26.9 | +25.2 | +27.8 | |
| Steps (reduced) |
59 (18) |
60 (19) |
62 (21) |
63 (22) |
65 (24) |
66 (25) |
67 (26) |
69 (28) |
70 (29) |
71 (30) |
72 (31) |
73 (32) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 75.7 | 22/21, 23/22, 24/23, 25/24 |
| 2 | 151.3 | 12/11, 23/21, 25/23 |
| 3 | 227 | 8/7, 25/22 |
| 4 | 302.6 | 25/21 |
| 5 | 378.3 | |
| 6 | 453.9 | 13/10, 30/23 |
| 7 | 529.6 | 23/17 |
| 8 | 605.3 | 17/12, 27/19 |
| 9 | 680.9 | |
| 10 | 756.6 | 17/11 |
| 11 | 832.2 | 21/13, 29/18 |
| 12 | 907.9 | 22/13 |
| 13 | 983.5 | 23/13, 30/17 |
| 14 | 1059.2 | 24/13 |
| 15 | 1134.9 | 25/13, 29/15 |
| 16 | 1210.5 | |
| 17 | 1286.2 | 21/10 |
| 18 | 1361.8 | 11/5 |
| 19 | 1437.5 | 16/7, 23/10 |
| 20 | 1513.1 | 12/5 |
| 21 | 1588.8 | 5/2 |
| 22 | 1664.5 | 21/8 |
| 23 | 1740.1 | 30/11 |
| 24 | 1815.8 | 20/7 |
| 25 | 1891.4 | |
| 26 | 1967.1 | 25/8 |
| 27 | 2042.8 | 13/4 |
| 28 | 2118.4 | 17/5 |
| 29 | 2194.1 | |
| 30 | 2269.7 | 26/7 |
| 31 | 2345.4 | |
| 32 | 2421 | |
| 33 | 2496.7 | |
| 34 | 2572.4 | |
| 35 | 2648 | 23/5 |
| 36 | 2723.7 | 29/6 |
| 37 | 2799.3 | |
| 38 | 2875 | 21/4 |
| 39 | 2950.6 | 11/2 |
| 40 | 3026.3 | 23/4 |
| 41 | 3102 | 6/1 |