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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}}


== 10-EDN ==
'''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"
|+Intervals of 10-EDN
!Step
!Cents
!Ratio
!JI approximation(s)
!Interval
|-
|-
|0
! Tuning
|0.0
! Equivalent edo
|1/1
! Comment
|1/1
|-
|unison
| 2EDe
|
| A stack of two major sixths
|-
| 3EDe
| [[2edo]]
|  
|-
| 4EDe
|
| rowspan="2" |Neither are equivalent with [[3edo]]
|-
|-
|1
| 5EDe
|173.12
|  
|e^(1/10)
|11/10
|flat whole tone
|-
|-
|2
| 6EDe
|346.25
| [[4edo]]
|e^(1/5)
| With a stretch
|11/9
|neutral third
|-
|-
|3
| 7EDe
|519.37
| [[5edo]]
|e^(3/10)
|  
|43/32
|sharp fourth
|-
|-
|4
| 8EDe
|692.49
|  
|e^(2/5)
| Entirely misses 2/1, falling halfway between 5edo and 6edo
|3/2
|flat fifth
|-
|-
|5
| 9EDe
|865.62
| [[6edo]]
|e^(1/2)
| With a considerable stretch
|5/3
|flat major sixth
|-
|-
|6
| 10EDe
|1038.74
| [[7edo]]
|e^(3/5)
|  
|117/64
|neutral seventh
|-
|-
|7
| 11EDe
|1211.86
|  
|e^(7/10)
| rowspan="2" |Neither are equivalent to 8edo
|2/1
|stretched octave
|-
|-
|8
| 12EDe
|1384.99
|  
|e^(4/5)
|20/9
|flat major ninth
|-
|-
|9
| 13EDe
|1558.11
| [[9edo]]
|e^(9/10)
|  
|22/9
|neutral tenth
|-
|-
|10
| 14EDe
|1731.23
|
|e/1
| rowspan="2" |Neither are equivalent to 10edo
|43/16
|-
|natave
| 15EDe
|
|-
| 16EDe
| [[11edo]]
|
|-
| 17EDe
| [[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
|-
| 18EDe
|
| Entirely misses 2/1, falling halfway between 12 and 13edo
|-
| 19EDe
| [[13edo]]
| Noticeably compressed
|-
| 20EDe
| [[14edo]]
| Noticeably stretched
|-
| 21EDe
|
| Entirely misses 2/1, falling halfway between 14edo and 15edo
|-
| 22EDe
|
| Cannot be considered equivalent to [[15edo]]
|-
| 23EDe
| [[16edo]]
|
|-
| 24EDe
| [[17edo]]
| Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%
|}
 
== 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).
 
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.
 
== Selected divisions ==
 
=== 10-EDe ===
{| class="wikitable"
|+ style="font-size: 105%;" | Intervals of 10-EDe
|-
! Step
! Cents
! Ratio
! JI approximation(s)
! Interval
|-
| 0
| 0.0
| 1/1
| 1/1
| unison
|-
| 1
| 173.12
| e<sup>1/10</sup>
| 11/10
| flat whole tone
|-
| 2
| 346.25
| e<sup>1/5</sup>
| 11/9
| neutral third
|-
| 3
| 519.37
| e<sup>3/10</sup>
| 43/32
| sharp fourth
|-
| 4
| 692.49
| e<sup>2/5</sup>
| 3/2
| flat fifth
|-
| 5
| 865.62
| e<sup>1/2</sup>
| 5/3
| flat major sixth
|-
| 6
| 1038.74
| e<sup>3/5</sup>
| 117/64
| neutral seventh
|-
| 7
| 1211.86
| e<sup>7/10</sup>
| 2/1
| stretched octave
|-
| 8
| 1384.99
| e<sup>4/5</sup>
| 20/9
| flat major ninth
|-
| 9
| 1558.11
| e<sup>9/10</sup>
| 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).
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-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}}
 
=== 24-EDe ===
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}}


20-EDN is a doubling of 10-EDN with intervals closer to semitones.
== See also ==
* [[Edϕ]]
* [[Acoustic pi]]
* [[User:Eliora/Phi to the phi]]


== 17-EDN ==
[[Category:Transcendental]]
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.
[[Category:Equal-step tuning]]

Latest revision as of 19:37, 23 February 2026

This page presents a topic of primarily mathematical interest.

While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

Equal division of the natave (EDe or EDN) is the equal division of 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 tunings 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

Tuning Equivalent edo Comment
2EDe A stack of two major sixths
3EDe 2edo
4EDe Neither are equivalent with 3edo
5EDe
6EDe 4edo With a stretch
7EDe 5edo
8EDe Entirely misses 2/1, falling halfway between 5edo and 6edo
9EDe 6edo With a considerable stretch
10EDe 7edo
11EDe Neither are equivalent to 8edo
12EDe
13EDe 9edo
14EDe Neither are equivalent to 10edo
15EDe
16EDe 11edo
17EDe 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
18EDe Entirely misses 2/1, falling halfway between 12 and 13edo
19EDe 13edo Noticeably compressed
20EDe 14edo Noticeably stretched
21EDe Entirely misses 2/1, falling halfway between 14edo and 15edo
22EDe Cannot be considered equivalent to 15edo
23EDe 16edo
24EDe 17edo Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%

Zeta function and tuning

In Gene’s 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 zeta peak EDe's are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to 1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308... edos.

Selected divisions

10-EDe

Intervals of 10-EDe
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-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.

Approximation of harmonics in 10-EDe
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +11.9 +2.4 +23.7 -16.3 +14.3 -79.5 +35.6 +4.8 -4.5 +3.6 +26.1
Relative (%) +6.9 +1.4 +13.7 -9.4 +8.2 -45.9 +20.6 +2.8 -2.6 +2.1 +15.1
Steps
(reduced) 		7
(7) 		11
(1) 		14
(4) 		16
(6) 		18
(8) 		19
(9) 		21
(1) 		22
(2) 		23
(3) 		24
(4) 		25
(5)

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.

Approximation of harmonics in 17-EDe
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +22.0 +33.0 +44.1 -36.7 -46.8 -8.2 -35.7 -35.9 -14.7 +24.0 -24.8
Relative (%) +21.7 +32.4 +43.3 -36.0 -46.0 -8.0 -35.0 -35.3 -14.4 +23.6 -24.3
Steps
(reduced) 		12
(12) 		19
(2) 		24
(7) 		27
(10) 		30
(13) 		33
(16) 		35
(1) 		37
(3) 		39
(5) 		41
(7) 		42
(8)

20-EDe

20-EDe is a doubling of 10-EDe with intervals closer to semitones.

Approximation of harmonics in 20-EDe
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +11.9 +2.4 +23.7 -16.3 +14.3 +7.1 +35.6 +4.8 -4.5 +3.6 +26.1
Relative (%) +13.7 +2.8 +27.4 -18.9 +16.5 +8.2 +41.1 +5.6 -5.2 +4.2 +30.2
Steps
(reduced) 		14
(14) 		22
(2) 		28
(8) 		32
(12) 		36
(16) 		39
(19) 		42
(2) 		44
(4) 		46
(6) 		48
(8) 		50
(10)

24-EDe

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).

Approximation of harmonics in 24-EDe
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +26.3 -26.5 -19.6 +26.9 -0.2 +21.5 +6.7 +19.2 -18.9 +32.5 +26.1
Relative (%) +36.4 -36.7 -27.1 +37.4 -0.2 +29.8 +9.3 +26.7 -26.2 +45.1 +36.2
Steps
(reduced) 		17
(17) 		26
(2) 		33
(9) 		39
(15) 		43
(19) 		47
(23) 		50
(2) 		53
(5) 		55
(7) 		58
(10) 		60
(12)

See also

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