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== Correspondence of EDN to EDO ==
== Correspondence of EDN to EDO ==
{| class="wikitable"
{| class="wikitable"
|+
!Tuning
!Equivalent edo
!Comment
|-
|-
|2edn
! Tuning
|
! Equivalent edo
|A stack of two major sixths
! Comment
|-
|-
|3edn
| 2edn
|[[2edo]]
|  
|
| A stack of two major sixths
|-
|-
|4edn
| 3edn
|
| [[2edo]]
|
|-
| 4edn
|  
| rowspan="2" |Neither are equivalent with [[3edo]]
| rowspan="2" |Neither are equivalent with [[3edo]]
|-
|-
|5edn
| 5edn
|
|  
|-
|-
|6edn
| 6edn
|[[4edo]]
| [[4edo]]
|With a stretch
| With a stretch
|-
|-
|7edn
| 7edn
|[[5edo]]
| [[5edo]]
|
|  
|-
|-
|8edn
| 8edn
|
|  
|Entirely misses 2/1, falling halfway between 5edo and 6edo
| Entirely misses 2/1, falling halfway between 5edo and 6edo
|-
|-
|9edn
| 9edn
|[[6edo]]
| [[6edo]]
|With a considerable stretch
| With a considerable stretch
|-
|-
|10edn
| 10edn
|[[7edo]]
| [[7edo]]
|
|  
|-
|-
|11edn
| 11edn
|
|  
| rowspan="2" |Neither are equivalent to 8edo
| rowspan="2" |Neither are equivalent to 8edo
|-
|-
|12edn
| 12edn
|
|  
|-
|-
|13edn
| 13edn
|[[9edo]]
| [[9edo]]
|
|  
|-
|-
|14edn
| 14edn
|
|  
| rowspan="2" |Neither are equivalent to 10edo
| rowspan="2" |Neither are equivalent to 10edo
|-
|-
|15edn
| 15edn
|
|  
|-
|-
|16edn
| 16edn
|[[11edo]]
| [[11edo]]
|
|  
|-
|-
|17edn
| 17edn
|[[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
| 18edn
|
|  
|Entirely misses 2/1, falling halfway between 12 and 13edo
| Entirely misses 2/1, falling halfway between 12 and 13edo
|-
|-
|19edn
| 19edn
|[[13edo]]
| [[13edo]]
|Noticeably compressed
| Noticeably compressed
|-
|-
|20edn
| 20edn
|[[14edo]]
| [[14edo]]
|Noticeably stretched
| Noticeably stretched
|-
|-
|21edn
| 21edn
|
|  
|Entirely misses 2/1, falling halfway between 14edo and 15edo
| Entirely misses 2/1, falling halfway between 14edo and 15edo
|-
|-
|22edn
| 22edn
|
|  
|Cannot be considered equivalent to [[15edo]]
| Cannot be considered equivalent to [[15edo]]
|-
|-
|23edn
| 23edn
|[[16edo]]
| [[16edo]]
|
|  
|-
|-
|24edn
| 24edn
|[[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 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.
 
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 ==
Line 110: Line 109:
=== 10-EDN ===
=== 10-EDN ===
{| class="wikitable"
{| class="wikitable"
|+Intervals of 10-EDN
|+ style="font-size: 105%;" | Intervals of 10-EDN
!Step
|-
!Cents
! Step
!Ratio
! Cents
!JI approximation(s)
! Ratio
!Interval
! JI approximation(s)
! Interval
|-
|-
|0
| 0
|0.0
| 0.0
|1/1
| 1/1
|1/1
| 1/1
|unison
| unison
|-
|-
|1
| 1
|173.12
| 173.12
|e<sup>1/10</sup>
| e<sup>1/10</sup>
|11/10
| 11/10
|flat whole tone
| flat whole tone
|-
|-
|2
| 2
|346.25
| 346.25
|e<sup>1/5</sup>
| e<sup>1/5</sup>
|11/9
| 11/9
|neutral third
| neutral third
|-
|-
|3
| 3
|519.37
| 519.37
|e<sup>3/10</sup>
| e<sup>3/10</sup>
|43/32
| 43/32
|sharp fourth
| sharp fourth
|-
|-
|4
| 4
|692.49
| 692.49
|e<sup>2/5</sup>
| e<sup>2/5</sup>
|3/2
| 3/2
|flat fifth
| flat fifth
|-
|-
|5
| 5
|865.62
| 865.62
|e<sup>1/2</sup>
| e<sup>1/2</sup>
|5/3
| 5/3
|flat major sixth
| flat major sixth
|-
|-
|6
| 6
|1038.74
| 1038.74
|e<sup>3/5</sup>
| e<sup>3/5</sup>
|117/64
| 117/64
|neutral seventh
| neutral seventh
|-
|-
|7
| 7
|1211.86
| 1211.86
|e<sup>7/10</sup>
| e<sup>7/10</sup>
|2/1
| 2/1
|stretched octave
| stretched octave
|-
|-
|8
| 8
|1384.99
| 1384.99
|e<sup>4/5</sup>
| e<sup>4/5</sup>
|20/9
| 20/9
|flat major ninth
| flat major ninth
|-
|-
|9
| 9
|1558.11
| 1558.11
|e<sup>9/10</sup>
| e<sup>9/10</sup>
|22/9
| 22/9
|neutral tenth
| neutral tenth
|-
|-
|10
| 10
|1731.23
| 1731.23
|e
| e
|43/16
| 43/16
|natave
| 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).
Line 200: Line 200:
{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDN}}
{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDN}}


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

Revision as of 13:16, 10 April 2025

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 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.

Correspondence of EDN to EDO

Tuning Equivalent edo Comment
2edn A stack of two major sixths
3edn 2edo
4edn Neither are equivalent with 3edo
5edn
6edn 4edo With a stretch
7edn 5edo
8edn Entirely misses 2/1, falling halfway between 5edo and 6edo
9edn 6edo With a considerable stretch
10edn 7edo
11edn Neither are equivalent to 8edo
12edn
13edn 9edo
14edn Neither are equivalent to 10edo
15edn
16edn 11edo
17edn 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
18edn Entirely misses 2/1, falling halfway between 12 and 13edo
19edn 13edo Noticeably compressed
20edn 14edo Noticeably stretched
21edn Entirely misses 2/1, falling halfway between 14edo and 15edo
22edn Cannot be considered equivalent to 15edo
23edn 16edo
24edn 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 edns 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-EDN

Intervals of 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.

Approximation of harmonics in 10-EDN
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-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.

Approximation of harmonics in 17-EDN
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-EDN

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

Approximation of harmonics in 20-EDN
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-EDN

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).

Approximation of harmonics in 24-EDN
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)
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