EDe: Difference between revisions
{{Mathematical interest}} |
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== Correspondence of EDN to EDO == | == Correspondence of EDN to EDO == | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Tuning | |||
! Equivalent edo | |||
! Comment | |||
|- | |- | ||
| | | 2edn | ||
| | | | ||
| | | 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). | |||
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-[[ | |||
== Selected divisions == | == Selected divisions == | ||
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=== 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). | ||
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{{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]] | |||