24ed5: Difference between revisions
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== Theory == | == Theory == | ||
From a no-twos-or-threes point of view, 24ed5 tempers out 125/121 in the 11-limit, equating [[11/5]] with the | From a no-twos-or-threes point of view, 24ed5 tempers out 125/121 in the 11-limit, equating [[11/5]] with the 1393-cent major ninth of [[2ed5]]. This implies a period of 1\2ed5 and a generator of approximately [[7/5]] (in this case 5\24ed5). | ||
== Interval table == | == Interval table == | ||
Revision as of 08:31, 11 July 2023
| ← 23ed5 | 24ed5 | 25ed5 → |
Division of the 5th harmonic into 24 equal parts (24ed5) is related to the miracle temperament. The step size about 116.0964 cents. It is similar to every third step of 31edo, but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.
Theory
From a no-twos-or-threes point of view, 24ed5 tempers out 125/121 in the 11-limit, equating 11/5 with the 1393-cent major ninth of 2ed5. This implies a period of 1\2ed5 and a generator of approximately 7/5 (in this case 5\24ed5).
Interval table
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 116.0964 | 16/15, 15/14 | |
| 2 | 232.1928 | 8/7 | |
| 3 | 348.2892 | 11/9 | |
| 4 | 464.3856 | 17/13 | |
| 5 | 580.4820 | 7/5 | |
| 6 | 696.5784 | meantone fifth (pseudo-3/2) | |
| 7 | 812.6748 | 8/5 | |
| 8 | 928.7712 | 65/38 | |
| 9 | 1044.8676 | 11/6 | |
| 10 | 1160.9640 | 45/23 | |
| 11 | 1277.0605 | 23/11 | |
| 12 | 1393.1569 | 38/17, 85/38 | meantone major second plus an octave |
| 13 | 1509.2533 | 55/23 | |
| 14 | 1625.3497 | 23/9 | |
| 15 | 1741.4461 | 30/11 | |
| 16 | 1857.5425 | 38/13 | |
| 17 | 1973.6389 | 25/8 | |
| 18 | 2089.7353 | meantone major sixth plus an octave (pseudo-10/3) | |
| 19 | 2205.8317 | 25/7 | |
| 20 | 2321.9281 | 65/17 | |
| 21 | 2438.0245 | 45/11 | |
| 22 | 2554.1209 | 35/8 | |
| 23 | 2670.2173 | 14/3 | |
| 24 | 2786.3137 | exact 5/1 | just major third plus two octaves |
24ed5 as a generator
24ed5 can also be thought of as a generator of the 2.3.5.7.11.23 subgroup temperament which tempers out 225/224, 243/242, 385/384, and 529/528, which is a cluster temperament with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 46/45 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by 31edo, 82edo, 113edo, and 144edo.