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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.
{{Mathematical interest}}


== Correspondence of EDN to EDO ==
'''Equal division of the [[natave]]''' ('''EDe''' or '''EDN''') is the equal division of [[Acoustic e|acoustic ''e'']] (where ''e'' is treated as a musical interval in the same way as ''2'' is an octave or ''1.5'' is a perfect fifth).
 
''e'' is of particular interest because of its relationship with logarithms, given the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.
 
Sometimes it is convenient to treat [[equal-step tuning]]s as (possibly non-integer) EDes in mathematics and computer programs, since it makes the logarithm used in equations the natural logarithm.
 
== Correspondence of EDe to EDO ==
{| class="wikitable"
{| class="wikitable"
|+
!Tuning
!Equivalent edo
!Comment
|-
|-
|2edn
! Tuning
|
! Equivalent edo
|A stack of two major sixths
! Comment
|-
|-
|3edn
| 2EDe
|[[2edo]]
|  
|
| A stack of two major sixths
|-
|-
|4edn
| 3EDe
|
| [[2edo]]
|
|-
| 4EDe
|  
| rowspan="2" |Neither are equivalent with [[3edo]]
| rowspan="2" |Neither are equivalent with [[3edo]]
|-
|-
|5edn
| 5EDe
|
|  
|-
|-
|6edn
| 6EDe
|[[4edo]]
| [[4edo]]
|With a stretch
| With a stretch
|-
|-
|7edn
| 7EDe
|[[5edo]]
| [[5edo]]
|
|  
|-
|-
|8edn
| 8EDe
|
|  
|Entirely misses 2/1, falling halfway between 5edo and 6edo
| Entirely misses 2/1, falling halfway between 5edo and 6edo
|-
|-
|9edn
| 9EDe
|[[6edo]]
| [[6edo]]
|With a considerable stretch
| With a considerable stretch
|-
|-
|10edn
| 10EDe
|[[7edo]]
| [[7edo]]
|
|  
|-
|-
|11edn
| 11EDe
|
|  
| rowspan="2" |Neither are equivalent to 8edo
| rowspan="2" |Neither are equivalent to 8edo
|-
|-
|12edn
| 12EDe
|
|  
|-
|-
|13edn
| 13EDe
|[[9edo]]
| [[9edo]]
|
|  
|-
|-
|14edn
| 14EDe
|
|  
| rowspan="2" |Neither are equivalent to 10edo
| rowspan="2" |Neither are equivalent to 10edo
|-
|-
|15edn
| 15EDe
|
|  
|-
|-
|16edn
| 16EDe
|[[11edo]]
| [[11edo]]
|
|  
|-
|-
|17edn
| 17EDe
|[[12edo]]
| [[12edo]]
|With a noticeable stretch, given the dominance of 12edo this is more likely to sound like out of tune 12edo than it's own tuning
| With a noticeable stretch, given the dominance of 12edo this is more likely to sound like out of tune 12edo than it's own tuning
|-
|-
|18edn
| 18EDe
|
|  
|Entirely misses 2/1, falling halfway between 12 and 13edo
| Entirely misses 2/1, falling halfway between 12 and 13edo
|-
|-
|19edn
| 19EDe
|[[13edo]]
| [[13edo]]
|Noticeably compressed
| Noticeably compressed
|-
|-
|20edn
| 20EDe
|[[14edo]]
| [[14edo]]
|Noticeably stretched
| Noticeably stretched
|-
|-
|21edn
| 21EDe
|
|  
|Entirely misses 2/1, falling halfway between 14edo and 15edo
| Entirely misses 2/1, falling halfway between 14edo and 15edo
|-
|-
|22edn
| 22EDe
|
|  
|Cannot be considered equivalent to [[15edo]]
| Cannot be considered equivalent to [[15edo]]
|-
|-
|23edn
| 23EDe
|[[16edo]]
| [[16edo]]
|
|  
|-
|-
|24edn
| 24EDe
|[[17edo]]
| [[17edo]]
|Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%
| Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%
|}
|}


== Zeta function and tuning ==
== Zeta function and tuning ==
In [[Gene Ward Smith|Gene]]’s [[the Riemann zeta function and tuning#The Black Magic Formulas|black magic formulas]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).


In [[Gene Ward Smith|Gene]]’s [[The Riemann Zeta Function and Tuning#The Black Magic Formulas|black magic formulas]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).
The sequence of non-[[stretched and compressed tuning|stretched]] zeta peak EDe's are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to {{EDOs|1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308}}... edos.
 
The sequence of non-[[Stretched tuning|stretched]] zeta peak edns are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to {{EDOs|1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308}}... edos.


== Selected divisions ==
== Selected divisions ==


=== 10-EDN ===
=== 10-EDe ===
{| class="wikitable"
{| class="wikitable"
|+Intervals of 10-EDN
|+ style="font-size: 105%;" | Intervals of 10-EDe
!Step
!Cents
!Ratio
!JI approximation(s)
!Interval
|-
|-
|0
! Step
|0.0
! Cents
|1/1
! Ratio
|1/1
! JI approximation(s)
|unison
! Interval
|-
|-
|1
| 0
|173.12
| 0.0
|e<sup>1/10</sup>
| 1/1
|11/10
| 1/1
|flat whole tone
| unison
|-
|-
|2
| 1
|346.25
| 173.12
|e<sup>1/5</sup>
| e<sup>1/10</sup>
|11/9
| 11/10
|neutral third
| flat whole tone
|-
|-
|3
| 2
|519.37
| 346.25
|e<sup>3/10</sup>
| e<sup>1/5</sup>
|43/32
| 11/9
|sharp fourth
| neutral third
|-
|-
|4
| 3
|692.49
| 519.37
|e<sup>2/5</sup>
| e<sup>3/10</sup>
|3/2
| 43/32
|flat fifth
| sharp fourth
|-
|-
|5
| 4
|865.62
| 692.49
|e<sup>1/2</sup>
| e<sup>2/5</sup>
|5/3
| 3/2
|flat major sixth
| flat fifth
|-
|-
|6
| 5
|1038.74
| 865.62
|e<sup>3/5</sup>
| e<sup>1/2</sup>
|117/64
| 5/3
|neutral seventh
| flat major sixth
|-
|-
|7
| 6
|1211.86
| 1038.74
|e<sup>7/10</sup>
| e<sup>3/5</sup>
|2/1
| 117/64
|stretched octave
| neutral seventh
|-
|-
|8
| 7
|1384.99
| 1211.86
|e<sup>4/5</sup>
| e<sup>7/10</sup>
|20/9
| 2/1
|flat major ninth
| stretched octave
|-
|-
|9
| 8
|1558.11
| 1384.99
|e<sup>9/10</sup>
| e<sup>4/5</sup>
|22/9
| 20/9
|neutral tenth
| flat major ninth
|-
|-
|10
| 9
|1731.23
| 1558.11
|e
| e<sup>9/10</sup>
|43/16
| 22/9
|natave
| 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).
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.
10-EDe 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.
{{Harmonics in equal|10|1457|536|title=Approximation of harmonics in 10-EDe}}
 
=== 17-EDe ===
17-EDe 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.
{{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDe}}


20-EDN is a doubling of 10-EDN with intervals closer to semitones.
=== 20-EDe ===
20-EDe is a doubling of 10-EDe with intervals closer to semitones.
{{Harmonics in equal|20|1457|536|title=Approximation of harmonics in 20-EDe}}


=== 17-EDN ===
=== 24-EDe ===
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-EDe has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 3606.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents).
{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDe}}


=== 24-EDN ===
== See also ==
24-EDN has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 3606.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents)..
* [[Edϕ]]
* [[Acoustic pi]]
* [[User:Eliora/Phi to the phi]]


[[Category:Transcendental]]
[[Category:Transcendental]]
[[Category:Equal-step tuning]]
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