147edo: Difference between revisions
Factual error. 149 not 147 is the smallest edo uniquely consistent in the 17-limit |
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== Theory == | |||
147edo has a very accurate fifth. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[32805/32768]] in the [[5-limit]], as well as [[225/224]] and [[3125/3087]] in the [[7-limit]], supporting [[garibaldi]]; [[243/242]] in the [[11-limit]]; [[364/363]] in the [[13-limit]]; [[442/441]] and [[595/594]] in the [[17-limit]]. It is the [[optimal patent val]] for 11-limit [[karadeniz]], the 41 & 106 temperament. Another val that can be used is the 147c val, with a sharp mapping of [[5/4]] (from [[49edo]]) instead of a slightly flat one, to go along with the sharp tendency of every other prime up to 17. This val tempers out [[126/125]] and [[1728/1715]] in the 7-limit, as well as [[176/175]], 243/242, [[441/440]], and [[540/539]] in the 11-limit, supporting [[myna]] in the 7- and 11-limits. | |||
One particular subgroup that 147edo serves as a [[microtemperament]] in regard to, with errors of less than half a cent for most basic intervals, is 2.3.13.23, which is commonly associated with [[17edo]]. In fact, 147edo is close to the optimal tuning for the remarkable rank-2 temperament [[shoal]] (17 & 113), which tempers out [[3888/3887]] and [[12168/12167]], is generated by the interval of [[26/23]] (less than 0.01{{c}} off in 147edo), divides [[8/3]] into eight equal parts, and serves as a [[circulating temperament]] of 17edo. Additionally, it equates a stack of three [[256/243|pythagorean limmas]] with [[299/256]] and a stack of four with [[16/13]], tempering out 4294967296/4290323193 and the [[tridecapyth comma]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|147}} | |||
=== Subsets and supersets === | |||
Since 147 = 3 × 7<sup>2</sup>, 147edo has subset edos {{EDOs| 3, 7, 21 and 49 }}. | |||
[[ | [[441edo]], which triples it, provides strong corrections on the 5th and 7th harmonics and is a very notable 7-limit system. | ||
[[ | == Scales == | ||
[[ | * [[Baldy6]] | ||
[[ | * [[Baldy11]] | ||
[[Category: | * [[Baldy17]] | ||
[[Category:Baldy]] | |||
Latest revision as of 19:35, 2 March 2026
| ← 146edo | 147edo | 148edo → |
(semiconvergent)
147 equal divisions of the octave (abbreviated 147edo or 147ed2), also called 147-tone equal temperament (147tet) or 147 equal temperament (147et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 147 equal parts of about 8.16 ¢ each. Each step represents a frequency ratio of 21/147, or the 147th root of 2.
Theory
147edo has a very accurate fifth. Using the patent val, the equal temperament tempers out 32805/32768 in the 5-limit, as well as 225/224 and 3125/3087 in the 7-limit, supporting garibaldi; 243/242 in the 11-limit; 364/363 in the 13-limit; 442/441 and 595/594 in the 17-limit. It is the optimal patent val for 11-limit karadeniz, the 41 & 106 temperament. Another val that can be used is the 147c val, with a sharp mapping of 5/4 (from 49edo) instead of a slightly flat one, to go along with the sharp tendency of every other prime up to 17. This val tempers out 126/125 and 1728/1715 in the 7-limit, as well as 176/175, 243/242, 441/440, and 540/539 in the 11-limit, supporting myna in the 7- and 11-limits.
One particular subgroup that 147edo serves as a microtemperament in regard to, with errors of less than half a cent for most basic intervals, is 2.3.13.23, which is commonly associated with 17edo. In fact, 147edo is close to the optimal tuning for the remarkable rank-2 temperament shoal (17 & 113), which tempers out 3888/3887 and 12168/12167, is generated by the interval of 26/23 (less than 0.01 ¢ off in 147edo), divides 8/3 into eight equal parts, and serves as a circulating temperament of 17edo. Additionally, it equates a stack of three pythagorean limmas with 299/256 and a stack of four with 16/13, tempering out 4294967296/4290323193 and the tridecapyth comma.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.09 | -2.64 | +2.60 | +3.78 | +0.29 | +1.17 | -3.64 | +0.30 | -1.01 | -2.18 |
| Relative (%) | +0.0 | +1.1 | -32.3 | +31.9 | +46.4 | +3.5 | +14.3 | -44.5 | +3.6 | -12.3 | -26.7 | |
| Steps (reduced) |
147 (0) |
233 (86) |
341 (47) |
413 (119) |
509 (68) |
544 (103) |
601 (13) |
624 (36) |
665 (77) |
714 (126) |
728 (140) | |
Subsets and supersets
Since 147 = 3 × 72, 147edo has subset edos 3, 7, 21 and 49.
441edo, which triples it, provides strong corrections on the 5th and 7th harmonics and is a very notable 7-limit system.