5ed5/2: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
5ed5/2 corresponds to 3.7824…[[edo]]. It can function as a generator chain for the [[hanson]] temperament. Its [[patent val]] tempers out the same 7-limit commas as [[4edo]], but the mapping of 11 and higher prime harmonics differs. | |||
=== Harmonics === | |||
{{Harmonics in equal|5|5|2|columns=11}} | |||
{{Harmonics in equal|5|5|2|columns=12|start=12|collapsed=true|Approximation of harmonics in 5ed5/2 (continued)}} | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
Latest revision as of 13:47, 27 May 2026
| ← 4ed5/2 | 5ed5/2 | 6ed5/2 → |
(semiconvergent)
(convergent)
5 equal divisions of 5/2 (abbreviated 5ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 5 equal parts of about 317 ¢ each. Each step represents a frequency ratio of (5/2)1/5, or the 5th root of 5/2.
Theory
5ed5/2 corresponds to 3.7824…edo. It can function as a generator chain for the hanson temperament. Its patent val tempers out the same 7-limit commas as 4edo, but the mapping of 11 and higher prime harmonics differs.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +69 | +2 | +138 | +69 | +71 | +121 | -110 | +3 | +138 | -27 | +140 |
| Relative (%) | +21.8 | +0.5 | +43.5 | +21.8 | +22.3 | +38.2 | -34.7 | +1.0 | +43.5 | -8.5 | +44.0 | |
| Steps (reduced) |
4 (4) |
6 (1) |
8 (3) |
9 (4) |
10 (0) |
11 (1) |
11 (1) |
12 (2) |
13 (3) |
13 (3) |
14 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1 | -127 | +71 | -41 | -146 | +72 | -21 | -110 | +123 | +42 | -35 | -108 |
| Relative (%) | +0.4 | -40.1 | +22.3 | -12.9 | -46.0 | +22.8 | -6.7 | -34.7 | +38.7 | +13.3 | -11.0 | -34.2 | |
| Steps (reduced) |
14 (4) |
14 (4) |
15 (0) |
15 (0) |
15 (0) |
16 (1) |
16 (1) |
16 (1) |
17 (2) |
17 (2) |
17 (2) |
17 (2) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 317.3 | 5/4, 6/5, 7/6, 11/9, 13/11, 15/13 |
| 2 | 634.5 | 3/2, 7/5, 10/7, 13/9, 18/13, 19/13 |
| 3 | 951.8 | 5/3, 7/4, 9/5, 12/7, 19/11 |
| 4 | 1269.1 | 2/1, 13/6, 15/7, 19/9, 21/10 |
| 5 | 1586.3 | 5/2, 12/5, 13/5, 18/7 |