44edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave" Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
== Theory == | == Theory == | ||
44edo is a double of [[22edo]], to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the [[13/1|13th harmonic]] with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. | 44edo is a double of [[22edo]], to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the [[13/1|13th harmonic]] with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. | ||
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It is on the [[optimal ET sequence]] for 7-, 11- and 13-limit [[nautilus]] temperament, for 11-limit [[spell]] temperament, and for 13-limit [[cantrip]] temperament. In the [[13-limit]] it supplies the [[optimal patent val]] for [[vigin]] temperament. | It is on the [[optimal ET sequence]] for 7-, 11- and 13-limit [[nautilus]] temperament, for 11-limit [[spell]] temperament, and for 13-limit [[cantrip]] temperament. In the [[13-limit]] it supplies the [[optimal patent val]] for [[vigin]] temperament. | ||
The [[k* | The [[k*N subgroups|2*44]] subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for [[88edo]]. | ||
=== Harmonics === | === Harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
44edo has subsets {{EDOs|2, 4, 11, 22}}. | 44edo has subsets {{EDOs| 2, 4, 11, 22 }}. | ||
One step of 44edo is very close (only 0.0086 cents sharp) to [[64/63]] (the septimal comma). [[Ruthenium]] temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example [[1012edo]], [[1848edo]], and [[2684edo]]. | One step of 44edo is very close (only 0.0086 cents sharp) to [[64/63]] (the septimal comma). [[Ruthenium]] temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example [[1012edo]], [[1848edo]], and [[2684edo]]. | ||
== Intervals == | == Intervals == | ||
{{ | {{Todo|complete table|inline=1|text = Add a column of approximated JI ratios to the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error.}} | ||
In 44edo, sharps and flats alter pitch by 6 | In 44edo, sharps and flats alter pitch by 6 edosteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo). | ||
{| class="wikitable center-all right-2 left-3" | {| class="wikitable center-all right-2 left-3" | ||
|- | |- | ||
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=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | This notation uses the same sagittal sequence as edos [[23edo #Second-best fifth notation|23b]], [[30edo #Sagittal notation|30]], and [[37edo #Sagittal notation|37]], and is a superset of the notations for edos [[22edo #Sagittal notation|22]] and [[11edo #Sagittal notation|11]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
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</imagemap> | </imagemap> | ||
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 190.91 | | 190.91 | ||
| 9/8 | | 9/8 | ||
| [[Spell]]/[[cantrip]] | | [[Spell]] / [[cantrip]] | ||
|- | |- | ||
| 1 | | 1 | ||
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== Instrument layouts == | == Instrument layouts == | ||
[[Lumatone mapping for 44edo]] | * [[Lumatone mapping for 44edo]] | ||
* [[Skip fretting system 44 2 11]] | |||
[[Skip fretting system 44 2 11]] | |||
== Music == | == Music == | ||
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[[Category:44edo| ]] <!-- main article --> | [[Category:44edo| ]] <!-- main article --> | ||