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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. | 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 == | |||
{| class="wikitable" | |||
|+Intervals of 10-EDN | |||
!Step | |||
!Cents | |||
!Ratio | |||
!JI approximation(s) | |||
!Interval | |||
|- | |||
|0 | |||
|0.0 | |||
|1/1 | |||
|1/1 | |||
|unison | |||
|- | |||
|1 | |||
|173.12 | |||
|e^(1/10) | |||
|11/10 | |||
|flat whole tone | |||
|- | |||
|2 | |||
|346.25 | |||
|e^(1/5) | |||
|11/9 | |||
|neutral third | |||
|- | |||
|3 | |||
|519.37 | |||
|e^(3/10) | |||
|43/32 | |||
|sharp fourth | |||
|- | |||
|4 | |||
|692.49 | |||
|e^(2/5) | |||
|3/2 | |||
|flat fifth | |||
|- | |||
|5 | |||
|865.62 | |||
|e^(1/2) | |||
|5/3 | |||
|flat major sixth | |||
|- | |||
|6 | |||
|1038.74 | |||
|e^(3/5) | |||
|117/64 | |||
|neutral seventh | |||
|- | |||
|7 | |||
|1211.86 | |||
|e^(7/10) | |||
|2/1 | |||
|stretched octave | |||
|- | |||
|8 | |||
|1384.99 | |||
|e^(4/5) | |||
|20/9 | |||
|flat major ninth | |||
|- | |||
|9 | |||
|1558.11 | |||
|e^(9/10) | |||
|22/9 | |||
|neutral tenth | |||
|- | |||
|10 | |||
|1731.23 | |||
|e/1 | |||
|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 cents) and gives it a rather pleasant sharp fifth of 712 cents. | |||