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'''Division of the 5th harmonic into 24 equal parts''' (24ed5) is related to the [[Miracle|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. | {{Infobox ET}} | ||
'''[[Ed5|Division of the 5th harmonic]] into 24 equal parts''' (24ed5) is related to the [[Miracle|miracle temperament]]. The step size is 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 offers a particularly good tuning of the very low-[[badness]] 5.7.11 [[subgroup temperament]] named as [[juggernaut]], tempering out 125/121. This has a CTE generator of exactly [[7/5]] (in 24ed5 approximated as 5 steps) and a period of 1\[[2ed5]] or the square root of five (which is equated to [[11/5]]). 24ed5 shares 31edo's mappings for 5 and 7, but not 11. | |||
== Interval table == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| Line 133: | Line 137: | ||
| | just major third plus two octaves | | | just major third plus two octaves | ||
|} | |} | ||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 24 | |||
| num = 5 | |||
| denom = 1 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 24 | |||
| num = 5 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
==24ed5 as a generator== | ==24ed5 as a generator== | ||
24ed5 | One step of 24ed5 is just flat of 3\31, an acceptable generator for [[miracle]] temperament, so it can be used as a [[retraction]] of miracle. However, miracle maps 5/1 to 23 secors and 3 dieses, not 24 secors. | ||
Latest revision as of 03:27, 26 May 2026
| ← 23ed5 | 24ed5 | 25ed5 → |
Division of the 5th harmonic into 24 equal parts (24ed5) is related to the miracle temperament. The step size is 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 offers a particularly good tuning of the very low-badness 5.7.11 subgroup temperament named as juggernaut, tempering out 125/121. This has a CTE generator of exactly 7/5 (in 24ed5 approximated as 5 steps) and a period of 1\2ed5 or the square root of five (which is equated to 11/5). 24ed5 shares 31edo's mappings for 5 and 7, but not 11.
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 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -39.0 | -44.4 | +38.0 | +0.0 | +32.6 | -2.0 | -1.0 | +27.3 | -39.0 | +28.2 | -6.4 |
| Relative (%) | -33.6 | -38.3 | +32.8 | +0.0 | +28.1 | -1.7 | -0.9 | +23.5 | -33.6 | +24.2 | -5.5 | |
| Steps (reduced) |
10 (10) |
16 (16) |
21 (21) |
24 (0) |
27 (3) |
29 (5) |
31 (7) |
33 (9) |
34 (10) |
36 (12) |
37 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -28.9 | -41.1 | -44.4 | -40.0 | -28.9 | -11.8 | +10.7 | +38.0 | -46.4 | -10.9 | +28.3 |
| Relative (%) | -24.9 | -35.4 | -38.3 | -34.5 | -24.9 | -10.1 | +9.2 | +32.8 | -40.0 | -9.4 | +24.3 | |
| Steps (reduced) |
38 (14) |
39 (15) |
40 (16) |
41 (17) |
42 (18) |
43 (19) |
44 (20) |
45 (21) |
45 (21) |
46 (22) |
47 (23) | |
24ed5 as a generator
One step of 24ed5 is just flat of 3\31, an acceptable generator for miracle temperament, so it can be used as a retraction of miracle. However, miracle maps 5/1 to 23 secors and 3 dieses, not 24 secors.