7ed5: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''[[Ed5|Division of the 5th harmonic]] into 7 equal parts''' (7ed5) is related to [[3edo|3 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 398.0448 cents. It is | '''[[Ed5|Division of the 5th harmonic]] into 7 equal parts''' (7ed5) is related to [[3edo|3 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 398.0448 cents. It is present (though possibly tempered) in any [[regular temperament]] which tempers out [[441/440]] and 244515348/244140625 in the [[11-limit]], such as [[equal temperament]]s [[3edo|3]], [[12edo|12]], [[15edo|15]], [[175edo|175]], [[190edo|190]], [[202edo|202]], and [[217edo]]. | ||
Due to [[Kirnberger's atom]], its step is 100.0002¢ flat of [[4/3]]. | |||
== Harmonics == | == Harmonics == | ||
Revision as of 04:38, 23 January 2025
| ← 6ed5 | 7ed5 | 8ed5 → |
(convergent)
(semiconvergent)
Division of the 5th harmonic into 7 equal parts (7ed5) is related to 3 edo, but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 398.0448 cents. It is present (though possibly tempered) in any regular temperament which tempers out 441/440 and 244515348/244140625 in the 11-limit, such as equal temperaments 3, 12, 15, 175, 190, 202, and 217edo.
Due to Kirnberger's atom, its step is 100.0002¢ flat of 4/3.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6 | +88 | -12 | +0 | +82 | -184 | -18 | +177 | -6 | -171 | +77 |
| Relative (%) | -1.5 | +22.2 | -2.9 | +0.0 | +20.7 | -46.3 | -4.4 | +44.4 | -1.5 | -42.9 | +19.2 | |
| Steps (reduced) |
3 (3) |
5 (5) |
6 (6) |
7 (0) |
8 (1) |
8 (1) |
9 (2) |
10 (3) |
10 (3) |
10 (3) |
11 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -62 | -190 | +88 | -23 | -128 | +171 | +77 | -12 | -96 | -177 | +144 |
| Relative (%) | -15.6 | -47.8 | +22.2 | -5.9 | -32.3 | +42.9 | +19.4 | -2.9 | -24.2 | -44.4 | +36.3 | |
| Steps (reduced) |
11 (4) |
11 (4) |
12 (5) |
12 (5) |
12 (5) |
13 (6) |
13 (6) |
13 (6) |
13 (6) |
13 (6) |
14 (0) | |
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 398.0448 | 34/27 | pseudo-5/4 |
| 2 | 796.0896 | 19/12 | |
| 3 | 1194.1344 | 255/128 | pseudo-octave |
| 4 | 1592.1793 | 128/51 | pseudo-5/2 |
| 5 | 1990.2241 | 60/19 | |
| 6 | 2388.2689 | 135/34 | pseudo-4/1 |
| 7 | 2786.3137 | exact 5/1 | just major third plus two octaves |