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'''[[Ed5|Division of the 5th harmonic]] into 22 equal parts''' ( | {{Infobox ET}} | ||
'''[[Ed5|Division of the 5th harmonic]] into 22 equal parts''' (22ED5) is a good [[hyperpyth]] tuning. The step size about 126.6506 cents. It is compared to [[15edt|15EDT]] and every second step of [[19edo|19EDO]], but with the 5/1 rather than 2/1 or 3/1 being just. | |||
== Intervals == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| Line 60: | Line 62: | ||
| | 10 | | | 10 | ||
| | 1266.5062 | | | 1266.5062 | ||
| | | | | [[26/25|52/25]], 160/77 | ||
| | | | | | ||
|- | |- | ||
| Line 70: | Line 72: | ||
| | 12 | | | 12 | ||
| | 1519.8075 | | | 1519.8075 | ||
| | | | | [[77/64|77/32]], 125/52 | ||
| | | | | | ||
|- | |- | ||
| Line 116: | Line 118: | ||
| | 2659.6631 | | | 2659.6631 | ||
| | 65/14 | | | 65/14 | ||
| | | | | +17.7 cents from [[23/20|23/5]] | ||
|- | |- | ||
| | 22 | | | 22 | ||
| Line 124: | Line 126: | ||
|} | |} | ||
== | == Harmonics == | ||
{{Harmonics in equal | |||
| steps = 22 | |||
| num = 5 | |||
| denom = 1 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 22 | |||
| num = 5 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
== 22ED5 as a generator == | |||
22ED5 can also be thought of as a generator of the 19-limit [[Hemimean clan #Mowglic|mowglic temperament]], which tempers out 351/350, 476/475, 495/494, 513/512, 540/539, and 1701/1690, which is an extension of the [[Syntonic–enneadecal equivalence continuum|mowgli temperament]]. This temperament is supported by [[19edo|19EDO]], [[161edo|161EDO]], and [[180edo|180EDO]] among others. | |||
[[Category: | [[Category:Hyperpyth]] | ||
{{todo|expand}} | |||
Latest revision as of 19:20, 1 August 2025
| ← 21ed5 | 22ed5 | 23ed5 → |
(convergent)
Division of the 5th harmonic into 22 equal parts (22ED5) is a good hyperpyth tuning. The step size about 126.6506 cents. It is compared to 15EDT and every second step of 19EDO, but with the 5/1 rather than 2/1 or 3/1 being just.
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 126.6506 | 14/13 | |
| 2 | 253.3012 | 22/19 | |
| 3 | 379.9519 | 56/45 | |
| 4 | 506.6025 | 75/56 | |
| 5 | 633.2531 | 75/52 | |
| 6 | 759.9037 | ||
| 7 | 886.5544 | 5/3 | |
| 8 | 1013.2050 | 70/39 | -4.4 cents from 9/5 |
| 9 | 1139.8556 | 85/44 | |
| 10 | 1266.5062 | 52/25, 160/77 | |
| 11 | 1393.1569 | 38/17, 85/38 | |
| 12 | 1519.8075 | 77/32, 125/52 | |
| 13 | 1646.4581 | 44/17 | -7.8 cents from 13/5 |
| 14 | 1773.1087 | 39/14 | |
| 15 | 1899.7593 | 3/1 | |
| 16 | 2026.4100 | ||
| 17 | 2153.0606 | 52/15 | +34.4 cents from 17/5 |
| 18 | 2279.7112 | 56/15 | -31.5 cents from 19/5 |
| 19 | 2406.3618 | 225/56 | |
| 20 | 2533.0125 | 95/22 | +48.5 cents from 21/5 |
| 21 | 2659.6631 | 65/14 | +17.7 cents from 23/5 |
| 22 | 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 (¢) | -60.1 | -2.2 | +6.4 | +0.0 | -62.3 | +50.7 | -53.8 | -4.4 | -60.1 | +28.2 | +4.2 |
| Relative (%) | -47.5 | -1.7 | +5.0 | +0.0 | -49.2 | +40.1 | -42.5 | -3.5 | -47.5 | +22.2 | +3.3 | |
| Steps (reduced) |
9 (9) |
15 (15) |
19 (19) |
22 (0) |
24 (2) |
27 (5) |
28 (6) |
30 (8) |
31 (9) |
33 (11) |
34 (12) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.8 | -9.4 | -2.2 | +12.7 | +34.4 | +62.1 | -31.5 | +6.4 | +48.5 | -32.0 | +17.7 |
| Relative (%) | -6.1 | -7.4 | -1.7 | +10.0 | +27.2 | +49.0 | -24.9 | +5.0 | +38.3 | -25.3 | +14.0 | |
| Steps (reduced) |
35 (13) |
36 (14) |
37 (15) |
38 (16) |
39 (17) |
40 (18) |
40 (18) |
41 (19) |
42 (20) |
42 (20) |
43 (21) | |
22ED5 as a generator
22ED5 can also be thought of as a generator of the 19-limit mowglic temperament, which tempers out 351/350, 476/475, 495/494, 513/512, 540/539, and 1701/1690, which is an extension of the mowgli temperament. This temperament is supported by 19EDO, 161EDO, and 180EDO among others.