22ed5/2: Difference between revisions
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== Theory == | == Theory == | ||
22ed5/2 has a vague similarity to [[17edo]] but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in [[primodality]] terms) ratios well, in particular it has decent approximations of [[7/2]], [[11/2]] and [[13/2]], and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports [[frosthog]] temperament, equating [[11/7]] to the square root of 5/2. | 22ed5/2 has a vague similarity to [[17edo]] but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in [[primodality]] terms) ratios well, in particular it has decent approximations of [[7/2]], [[11/2]] and [[13/2]], and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports [[frosthog]] temperament, equating [[11/7]] to the square root of 5/2. | ||
=== Harmonics === | |||
{{Harmonics in equal|22|5|2|columns=11}} | |||
{{Harmonics in equal|22|5|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22ed5/2 (continued)}} | |||
== Intervals == | == Intervals == | ||
{ | {|class="wikitable center-1 right-2" | ||
|- | |||
! # | |||
! Cents | |||
| | ! Approximate ratios* | ||
| | |- | ||
| | | 0 | ||
| 0.0 | |||
| [[1/1]] | |||
| | |- | ||
| | | 1 | ||
| | | 72.1 | ||
| | | [[26/25]] | ||
| | |- | ||
} | | 2 | ||
| 144.2 | |||
| [[625/572]], [[676/625]] | |||
|- | |||
| 3 | |||
| 216.3 | |||
| [[25/22]] | |||
|- | |||
| 4 | |||
| 288.4 | |||
| [[13/11]] | |||
|- | |||
| 5 | |||
| 360.5 | |||
| [[154/125]] | |||
|- | |||
| 6 | |||
| 432.6 | |||
| [[125/98]], [[625/484]], [[875/676]] | |||
|- | |||
| 7 | |||
| 504.7 | |||
| [[35/26]], [[121/91]] | |||
|- | |||
| 8 | |||
| 576.8 | |||
| [[7/5]] | |||
|- | |||
| 9 | |||
| 648.9 | |||
| [[182/125]] | |||
|- | |||
| 10 | |||
| 721.1 | |||
| [[275/182]], [[875/572]] | |||
|- | |||
| 11 | |||
| 793.2 | |||
| [[11/7]], [[35/22]] | |||
|- | |||
| 12 | |||
| 865.3 | |||
| [[91/55]] | |||
|- | |||
| 13 | |||
| 937.4 | |||
| [[625/364]] | |||
|- | |||
| 14 | |||
| 1009.5 | |||
| [[25/14]] | |||
|- | |||
| 15 | |||
| 1081.6 | |||
| [[13/7]] | |||
|- | |||
| 16 | |||
| 1153.7 | |||
| [[49/25]], [[242/125]], [[338/175]] | |||
|- | |||
| 17 | |||
| 1225.8 | |||
| [[625/308]] | |||
|- | |||
| 18 | |||
| 1297.9 | |||
| [[55/26]] | |||
|- | |||
| 19 | |||
| 1370.0 | |||
| [[11/5]] | |||
|- | |||
| 20 | |||
| 1442.1 | |||
| [[286/125]], [[3125/1352]] | |||
|- | |||
| 21 | |||
| 1514.2 | |||
| [[125/52]] | |||
|- | |||
| 22 | |||
| 1586.3 | |||
| [[5/2]] | |||
|} | |||
<nowiki/>* Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2-subgroup temperament. | |||
Latest revision as of 07:38, 11 November 2026
| ← 21ed5/2 | 22ed5/2 | 23ed5/2 → |
22 equal divisions of 5/2 (abbreviated 22ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 22 equal parts of about 72.1 ¢ each. Each step represents a frequency ratio of (5/2)1/22, or the 22nd root of 5/2.
Theory
22ed5/2 has a vague similarity to 17edo but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in primodality terms) ratios well, in particular it has decent approximations of 7/2, 11/2 and 13/2, and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports frosthog temperament, equating 11/7 to the square root of 5/2.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +25.8 | -27.2 | -20.5 | +25.8 | -1.4 | +20.1 | +5.3 | +17.7 | -20.5 | +30.8 | +24.4 |
| Relative (%) | +35.8 | -37.8 | -28.5 | +35.8 | -2.0 | +27.9 | +7.3 | +24.5 | -28.5 | +42.7 | +33.8 | |
| Steps (reduced) |
17 (17) |
26 (4) |
33 (11) |
39 (17) |
43 (21) |
47 (3) |
50 (6) |
53 (9) |
55 (11) |
58 (14) |
60 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +30.0 | -26.2 | -1.4 | +31.0 | -1.8 | -28.7 | +22.0 | +5.3 | -7.1 | -15.5 | -20.4 | -22.0 |
| Relative (%) | +41.6 | -36.3 | -2.0 | +43.1 | -2.5 | -39.7 | +30.4 | +7.3 | -9.9 | -21.5 | -28.3 | -30.5 | |
| Steps (reduced) |
62 (18) |
63 (19) |
65 (21) |
67 (1) |
68 (2) |
69 (3) |
71 (5) |
72 (6) |
73 (7) |
74 (8) |
75 (9) |
76 (10) | |
Intervals
| # | Cents | Approximate ratios* |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 72.1 | 26/25 |
| 2 | 144.2 | 625/572, 676/625 |
| 3 | 216.3 | 25/22 |
| 4 | 288.4 | 13/11 |
| 5 | 360.5 | 154/125 |
| 6 | 432.6 | 125/98, 625/484, 875/676 |
| 7 | 504.7 | 35/26, 121/91 |
| 8 | 576.8 | 7/5 |
| 9 | 648.9 | 182/125 |
| 10 | 721.1 | 275/182, 875/572 |
| 11 | 793.2 | 11/7, 35/22 |
| 12 | 865.3 | 91/55 |
| 13 | 937.4 | 625/364 |
| 14 | 1009.5 | 25/14 |
| 15 | 1081.6 | 13/7 |
| 16 | 1153.7 | 49/25, 242/125, 338/175 |
| 17 | 1225.8 | 625/308 |
| 18 | 1297.9 | 55/26 |
| 19 | 1370.0 | 11/5 |
| 20 | 1442.1 | 286/125, 3125/1352 |
| 21 | 1514.2 | 125/52 |
| 22 | 1586.3 | 5/2 |
* Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2-subgroup temperament.