68ed12: Difference between revisions
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== Theory == | == Theory == | ||
68ed12 is very nearly identical to [[19edo]], but with the 12/1 rather than the [[2/1]] being just. This results in octaves being stretched by about 2.02 [[cent]]s. | 68ed12 is very nearly identical to [[19edo]], but with the 12/1 rather than the [[2/1]] being just. This results in octaves being stretched by about 2.02 [[cent]]s. Like 19edo, 68ed12 is [[consistent]] to the [[integer limit|10-integer-limit]]. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|68|12|1|intervals=integer|columns=11}} | {{Harmonics in equal|68|12|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|68|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 68ed12 (continued)}} | {{Harmonics in equal|68|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 68ed12 (continued)}} | ||
=== Subsets and supersets === | |||
Since 68 factors into primes as {{nowrap| 2<sup>2</sup> × 17 }}, 68ed12 has subset ed12's {{EDs|equave=12| 2, 4, 17, and 34 }}. | |||
== Intervals == | == Intervals == | ||
Latest revision as of 13:08, 30 March 2025
| ← 67ed12 | 68ed12 | 69ed12 → |
68 equal divisions of the 12th harmonic (abbreviated 68ed12) is a nonoctave tuning system that divides the interval of 12/1 into 68 equal parts of about 63.3 ¢ each. Each step represents a frequency ratio of 121/68, or the 68th root of 12.
Theory
68ed12 is very nearly identical to 19edo, but with the 12/1 rather than the 2/1 being just. This results in octaves being stretched by about 2.02 cents. Like 19edo, 68ed12 is consistent to the 10-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.0 | -4.0 | +4.0 | -2.7 | -2.0 | -15.8 | +6.1 | -8.1 | -0.7 | +24.1 | +0.0 |
| Relative (%) | +3.2 | -6.4 | +6.4 | -4.3 | -3.2 | -25.0 | +9.6 | -12.8 | -1.1 | +38.1 | +0.0 | |
| Steps (reduced) |
19 (19) |
30 (30) |
38 (38) |
44 (44) |
49 (49) |
53 (53) |
57 (57) |
60 (60) |
63 (63) |
66 (66) |
68 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.0 | -13.8 | -6.7 | +8.1 | +29.6 | -6.1 | +26.9 | +1.3 | -19.9 | +26.1 | +12.4 | +2.0 |
| Relative (%) | -19.0 | -21.8 | -10.6 | +12.8 | +46.9 | -9.6 | +42.5 | +2.1 | -31.4 | +41.3 | +19.7 | +3.2 | |
| Steps (reduced) |
70 (2) |
72 (4) |
74 (6) |
76 (8) |
78 (10) |
79 (11) |
81 (13) |
82 (14) |
83 (15) |
85 (17) |
86 (18) |
87 (19) | |
Subsets and supersets
Since 68 factors into primes as 22 × 17, 68ed12 has subset ed12's 2, 4, 17, and 34.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 63.3 | 27/26, 28/27, 29/28 |
| 2 | 126.5 | 14/13, 29/27 |
| 3 | 189.8 | 19/17, 29/26 |
| 4 | 253.1 | 22/19, 29/25, 37/32 |
| 5 | 316.3 | 6/5 |
| 6 | 379.6 | |
| 7 | 442.8 | 22/17, 31/24 |
| 8 | 506.1 | |
| 9 | 569.4 | 25/18, 32/23 |
| 10 | 632.6 | 36/25 |
| 11 | 695.9 | |
| 12 | 759.2 | 31/20 |
| 13 | 822.4 | 29/18, 37/23 |
| 14 | 885.7 | 5/3 |
| 15 | 949 | 19/11, 26/15 |
| 16 | 1012.2 | |
| 17 | 1075.5 | 13/7 |
| 18 | 1138.8 | 27/14, 29/15 |
| 19 | 1202 | 2/1 |
| 20 | 1265.3 | 27/13 |
| 21 | 1328.5 | 28/13 |
| 22 | 1391.8 | 29/13 |
| 23 | 1455.1 | 37/16 |
| 24 | 1518.3 | 12/5 |
| 25 | 1581.6 | |
| 26 | 1644.9 | 31/12 |
| 27 | 1708.1 | |
| 28 | 1771.4 | 25/9 |
| 29 | 1834.7 | 26/9 |
| 30 | 1897.9 | |
| 31 | 1961.2 | 28/9, 31/10 |
| 32 | 2024.4 | 29/9 |
| 33 | 2087.7 | 10/3 |
| 34 | 2151 | |
| 35 | 2214.2 | 18/5 |
| 36 | 2277.5 | |
| 37 | 2340.8 | 27/7 |
| 38 | 2404 | |
| 39 | 2467.3 | 25/6 |
| 40 | 2530.6 | |
| 41 | 2593.8 | |
| 42 | 2657.1 | |
| 43 | 2720.4 | |
| 44 | 2783.6 | 5/1 |
| 45 | 2846.9 | 31/6 |
| 46 | 2910.1 | |
| 47 | 2973.4 | |
| 48 | 3036.7 | |
| 49 | 3099.9 | 6/1 |
| 50 | 3163.2 | |
| 51 | 3226.5 | |
| 52 | 3289.7 | |
| 53 | 3353 | |
| 54 | 3416.3 | 36/5 |
| 55 | 3479.5 | |
| 56 | 3542.8 | 31/4 |
| 57 | 3606.1 | |
| 58 | 3669.3 | 25/3 |
| 59 | 3732.6 | |
| 60 | 3795.8 | |
| 61 | 3859.1 | |
| 62 | 3922.4 | |
| 63 | 3985.6 | 10/1 |
| 64 | 4048.9 | |
| 65 | 4112.2 | |
| 66 | 4175.4 | |
| 67 | 4238.7 | |
| 68 | 4302 | 12/1 |