247edo: Difference between revisions
Expand on theory; +subsets and supersets |
247cg |
||
| Line 2: | Line 2: | ||
{{EDO intro}} | {{EDO intro}} | ||
[[Prime harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] are all about halfway between 247edo's steps, so 247edo lacks [[consistency]] to the [[5-odd-limit|5]] and higher odd limits. It is the largest numbered edo that the closest approximation to 3/2 is flatter than that of [[12edo]] (700¢, [[Compton family|compton fifth]]). | [[Prime harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] are all about halfway between 247edo's steps, so 247edo lacks [[consistency]] to the [[5-odd-limit|5]] and higher odd limits. It is the largest numbered edo that the closest approximation to 3/2 is flatter than that of [[12edo]] (700¢, [[Compton family|compton fifth]]). 247edo tunes the 2.9.13.15.21 [[subgroup]] very well, as every other step of the monstrous [[494edo]]. | ||
The [[wart_notation|247cg val]] has lower errors: this edo has a [[stretched_and_compressed_tuning|flat tendency]], so its tuning accuracy may be improved by an octave stretch of approximately +0.8{{cent}}. 247cg is a good tuning for [[miracle]], tempering out [[225/224]] and [[1029/1024]] in the [[7-limit]], [[243/242]], [[385/384]], [[441/440]], and [[540/539]] in the [[11-limit]], [[847/845]] in the [[13-limit]], and [[375/374]] and [[561/560]] in the [[17-limit]]. Alternatively, using the [[patent val]], 247edo tempers out [[126/125]], [[243/242]] and [[1029/1024]] in the 11-limit, [[support]]ing the [[hemivalentino]] temperament (31 & 61e). | |||
=== Odd harmonics === | === Odd harmonics === | ||
Revision as of 01:38, 30 October 2024
| ← 246edo | 247edo | 248edo → |
Prime harmonics 3, 5, 7, and 11 are all about halfway between 247edo's steps, so 247edo lacks consistency to the 5 and higher odd limits. It is the largest numbered edo that the closest approximation to 3/2 is flatter than that of 12edo (700¢, compton fifth). 247edo tunes the 2.9.13.15.21 subgroup very well, as every other step of the monstrous 494edo.
The 247cg val has lower errors: this edo has a flat tendency, so its tuning accuracy may be improved by an octave stretch of approximately +0.8 ¢. 247cg is a good tuning for miracle, tempering out 225/224 and 1029/1024 in the 7-limit, 243/242, 385/384, 441/440, and 540/539 in the 11-limit, 847/845 in the 13-limit, and 375/374 and 561/560 in the 17-limit. Alternatively, using the patent val, 247edo tempers out 126/125, 243/242 and 1029/1024 in the 11-limit, supporting the hemivalentino temperament (31 & 61e).
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.36 | +2.35 | -2.02 | +0.14 | -2.33 | -0.04 | -0.01 | +1.93 | -1.16 | +0.47 | -1.55 | -0.16 | -2.22 | +0.38 | +1.52 |
| Relative (%) | -48.6 | +48.4 | -41.7 | +2.9 | -48.0 | -0.9 | -0.2 | +39.7 | -23.8 | +9.8 | -32.0 | -3.2 | -45.7 | +7.9 | +31.4 | |
| Steps (reduced) |
391 (144) |
574 (80) |
693 (199) |
783 (42) |
854 (113) |
914 (173) |
965 (224) |
1010 (22) |
1049 (61) |
1085 (97) |
1117 (129) |
1147 (159) |
1174 (186) |
1200 (212) |
1224 (236) | |
Subsets and supersets
Since 247 factors into 13 × 19, 247edo contains 13edo and 19edo as its subsets. 494edo, which doubles it, provides excellent correction to all the lower prime harmonics.