49ed6: Difference between revisions
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== Theory == | |||
49ed6 is very nearly identical to [[19edo]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy. | |||
Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo. | The fifth is ~696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[flattone]] fourth. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo. | ||
Usable prime harmonics include the 3 | Usable prime harmonics include the [[3/1|3]] (about 3 cents flat), the [[5/1|5]] (about a cent flat), the [[7/1|7]] (about 14 cents flat) and the [[13/1|13]] (about 9 cents flat). The 7 and 13 in particular are much improved; with pure octaves they are too far out of tune to be usable for most, but the situation changes with the stretched version. | ||
Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp. | Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are [[44ed5]] and [[93ed30]]. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp. | ||
Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has. | Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has. | ||
=== Harmonics === | |||
{{Harmonics in equal|49|6|1}} | |||
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 49ed6 (continued)}} | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
[[Category:19edo]] | [[Category:19edo]] | ||
[[Category:Godzilla]] | [[Category:Godzilla]] | ||
[[Category:Meantone]] | [[Category:Meantone]] | ||
Revision as of 12:05, 23 January 2025
| ← 48ed6 | 49ed6 | 50ed6 → |
49 equal divisions of the 6th harmonic (abbreviated 49ed6) is a nonoctave tuning system that divides the interval of 6/1 into 49 equal parts of about 63.3 ¢ each. Each step represents a frequency ratio of 61/49, or the 49th root of 6.
Theory
49ed6 is very nearly identical to 19edo, but with the 6/1 rather than the 2/1 being just. It is extremely close to the zeta peak near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.
The fifth is ~696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of 31edo. The fourth is less accurate than in 19edo, and is close in size to a flattone fourth. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.
Usable prime harmonics include the 3 (about 3 cents flat), the 5 (about a cent flat), the 7 (about 14 cents flat) and the 13 (about 9 cents flat). The 7 and 13 in particular are much improved; with pure octaves they are too far out of tune to be usable for most, but the situation changes with the stretched version.
Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.
Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.8 | -2.8 | +5.6 | -0.9 | +0.0 | -13.7 | +8.4 | -5.6 | +1.9 | +26.8 | +2.8 |
| Relative (%) | +4.4 | -4.4 | +8.8 | -1.4 | +0.0 | -21.6 | +13.3 | -8.8 | +3.0 | +42.4 | +4.4 | |
| Steps (reduced) |
19 (19) |
30 (30) |
38 (38) |
44 (44) |
49 (0) |
53 (4) |
57 (8) |
60 (11) |
63 (14) |
66 (17) |
68 (19) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.2 | -10.9 | -3.7 | +11.2 | -30.5 | -2.8 | +30.2 | +4.7 | -16.4 | +29.6 | +16.0 | +5.6 |
| Relative (%) | -14.5 | -17.1 | -5.8 | +17.7 | -48.1 | -4.4 | +47.7 | +7.4 | -26.0 | +46.8 | +25.2 | +8.8 | |
| Steps (reduced) |
70 (21) |
72 (23) |
74 (25) |
76 (27) |
77 (28) |
79 (30) |
81 (32) |
82 (33) |
83 (34) |
85 (36) |
86 (37) |
87 (38) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 63.3 | 26/25, 27/26, 28/27, 29/28, 30/29 |
| 2 | 126.6 | 14/13, 29/27 |
| 3 | 189.9 | 29/26 |
| 4 | 253.2 | 15/13, 22/19, 29/25 |
| 5 | 316.5 | 6/5 |
| 6 | 379.8 | |
| 7 | 443.1 | 31/24 |
| 8 | 506.4 | |
| 9 | 569.7 | 25/18, 32/23 |
| 10 | 633.1 | 13/9, 23/16 |
| 11 | 696.4 | 3/2 |
| 12 | 759.7 | 14/9, 31/20 |
| 13 | 823 | 29/18 |
| 14 | 886.3 | 5/3 |
| 15 | 949.6 | 19/11, 26/15 |
| 16 | 1012.9 | 9/5 |
| 17 | 1076.2 | 13/7, 28/15 |
| 18 | 1139.5 | 27/14, 29/15, 31/16 |
| 19 | 1202.8 | 2/1 |
| 20 | 1266.1 | 25/12, 27/13, 29/14 |
| 21 | 1329.4 | 28/13 |
| 22 | 1392.7 | 29/13 |
| 23 | 1456 | |
| 24 | 1519.3 | 12/5 |
| 25 | 1582.6 | 5/2 |
| 26 | 1645.9 | 31/12 |
| 27 | 1709.2 | |
| 28 | 1772.5 | 25/9 |
| 29 | 1835.9 | 26/9 |
| 30 | 1899.2 | 3/1 |
| 31 | 1962.5 | 28/9, 31/10 |
| 32 | 2025.8 | 29/9 |
| 33 | 2089.1 | 10/3 |
| 34 | 2152.4 | |
| 35 | 2215.7 | 18/5 |
| 36 | 2279 | |
| 37 | 2342.3 | 27/7, 31/8 |
| 38 | 2405.6 | 4/1 |
| 39 | 2468.9 | 25/6 |
| 40 | 2532.2 | |
| 41 | 2595.5 | |
| 42 | 2658.8 | |
| 43 | 2722.1 | 29/6 |
| 44 | 2785.4 | 5/1 |
| 45 | 2848.7 | 26/5, 31/6 |
| 46 | 2912 | |
| 47 | 2975.3 | |
| 48 | 3038.6 | 29/5 |
| 49 | 3102 | 6/1 |