Xenharmonic Wiki

EDe: Difference between revisions

← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,263 edits
m BudjarnLambeth moved page EDN to EDe: More intuitive name (discussed on Discord)
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,263 edits
Reword intro, replace EDN instances w EDe
Line 1: Line 1:
{{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 the constant ''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.
 
== Correspondence of EDe to EDO ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 10: Line 12:
! Comment
! Comment
|-
|-
| 2edn
| 2EDe
|  
|  
| A stack of two major sixths
| 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%
Line 103: Line 105:
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 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 ==
== Selected divisions ==


=== 10-EDN ===
=== 10-EDe ===
{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Intervals of 10-EDN
|+ style="font-size: 105%;" | Intervals of 10-EDN
Line 185: Line 187:
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-EDN ===
=== 17-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.
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.
{{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDN}}
{{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDe}}


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


=== 24-EDN ===
=== 24-EDe ===
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).
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-EDN}}
{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDe}}


[[Category:Transcendental]]
[[Category:Transcendental]]
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
Retrieved from "https://en.xen.wiki/w/EDe"