EDe: Difference between revisions
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| Line 18: | Line 18: | ||
|1 | |1 | ||
|173.12 | |173.12 | ||
|e | |e<sup>1/10</sup> | ||
|11/10 | |11/10 | ||
|flat whole tone | |flat whole tone | ||
| Line 24: | Line 24: | ||
|2 | |2 | ||
|346.25 | |346.25 | ||
|e | |e<sup>1/5</sup> | ||
|11/9 | |11/9 | ||
|neutral third | |neutral third | ||
| Line 30: | Line 30: | ||
|3 | |3 | ||
|519.37 | |519.37 | ||
|e | |e<sup>3/10</sup> | ||
|43/32 | |43/32 | ||
|sharp fourth | |sharp fourth | ||
| Line 36: | Line 36: | ||
|4 | |4 | ||
|692.49 | |692.49 | ||
|e | |e<sup>2/5</sup> | ||
|3/2 | |3/2 | ||
|flat fifth | |flat fifth | ||
| Line 42: | Line 42: | ||
|5 | |5 | ||
|865.62 | |865.62 | ||
|e | |e<sup>1/2</sup> | ||
|5/3 | |5/3 | ||
|flat major sixth | |flat major sixth | ||
| Line 48: | Line 48: | ||
|6 | |6 | ||
|1038.74 | |1038.74 | ||
|e | |e<sup>3/5</sup> | ||
|117/64 | |117/64 | ||
|neutral seventh | |neutral seventh | ||
| Line 54: | Line 54: | ||
|7 | |7 | ||
|1211.86 | |1211.86 | ||
|e | |e<sup>7/10</sup> | ||
|2/1 | |2/1 | ||
|stretched octave | |stretched octave | ||
| Line 60: | Line 60: | ||
|8 | |8 | ||
|1384.99 | |1384.99 | ||
|e | |e<sup>4/5</sup> | ||
|20/9 | |20/9 | ||
|flat major ninth | |flat major ninth | ||
| Line 66: | Line 66: | ||
|9 | |9 | ||
|1558.11 | |1558.11 | ||
|e | |e<sup>9/10</sup> | ||
|22/9 | |22/9 | ||
|neutral tenth | |neutral tenth | ||
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|10 | |10 | ||
|1731.23 | |1731.23 | ||
|e | |e | ||
|43/16 | |43/16 | ||
|natave | |natave | ||
| Line 83: | Line 83: | ||
== 17-EDN == | == 17-EDN == | ||
17-EDN is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222 cents) and gives it a rather pleasant sharp fifth of 712 cents. | 17-EDN is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents. | ||
== 24-EDN == | |||
24-EDN has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 2406.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents).. | |||
Revision as of 19:07, 23 September 2020
Equal divisions of the natave, which is the mathematical constant e used as a musical interval. e is of particular interest because of its relationship with logarithms, the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.
10-EDN
| Step | Cents | Ratio | JI approximation(s) | Interval |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | 1/1 | unison |
| 1 | 173.12 | e1/10 | 11/10 | flat whole tone |
| 2 | 346.25 | e1/5 | 11/9 | neutral third |
| 3 | 519.37 | e3/10 | 43/32 | sharp fourth |
| 4 | 692.49 | e2/5 | 3/2 | flat fifth |
| 5 | 865.62 | e1/2 | 5/3 | flat major sixth |
| 6 | 1038.74 | e3/5 | 117/64 | neutral seventh |
| 7 | 1211.86 | e7/10 | 2/1 | stretched octave |
| 8 | 1384.99 | e4/5 | 20/9 | flat major ninth |
| 9 | 1558.11 | e9/10 | 22/9 | neutral tenth |
| 10 | 1731.23 | e | 43/16 | natave |
Beyond the natave, some particularly pleasant JI intervals can be found: 11\10 is only 2 cents sharp from 3/1; 13\10 is very close to 11/2; and 23\10 is very close to 10/1. This last approximation in particular makes this equal division almost equivalent to 23-ed(10/1).
10-EDN is similar to 7-EDO in that its step size is roughly 1/7 of an octave, therefore roughly corresponding to the diatonic scale, but with warped, equal-size steps. However, the octave is stretched, which simultaneously helps the extremely flat fifth of 7-EDO.
20-EDN is a doubling of 10-EDN with intervals closer to semitones.
17-EDN
17-EDN is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.
24-EDN
24-EDN has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 2406.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents)..