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{{Mathematical interest}}
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{{Mathematical interest}}
{{Mathematical interest}}


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 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).


== Correspondence of EDN to EDO ==
''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
|-
| 2EDe
|  
| A stack of two major sixths
|-
|-
|3edn
| 3EDe
|[[2edo]]
| [[2edo]]
|
|  
|-
|-
|4edn
| 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-EDN}}
{{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}}


=== 17-EDN ===
=== 20-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.
20-EDe is a doubling of 10-EDe with intervals closer to semitones.
{{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDN}}
{{Harmonics in equal|20|1457|536|title=Approximation of harmonics in 20-EDe}}


=== 20-EDN ===
=== 24-EDe ===
20-EDN is a doubling of 10-EDN with intervals closer to semitones.
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|20|1457|536|title=Approximation of harmonics in 20-EDN}}
{{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]]
{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDN}}
* [[Acoustic pi]]
* [[User:Eliora/Phi to the phi]]


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