Equal-step tuning: Difference between revisions
→Catalog of equal-step tunings: remove ed16/15, add 2 notable aps |
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{{interwiki | {{interwiki | ||
| de = Gleichstufige_Tonsysteme | | de = Gleichstufige_Tonsysteme | ||
| en = Equal- | | en = Equal-step tuning | ||
| ja = 平均律 | |||
}} | }} | ||
{{Wikipedia|Equal temperament}} | {{Wikipedia|Equal temperament}} | ||
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(Unlimited resolution version: [[:File:equal.svg|equal.svg]]) | (Unlimited resolution version: [[:File:equal.svg|equal.svg]]) | ||
For the mathematically inclined, this kind of diagram is closely related to [[ | For the mathematically inclined, this kind of diagram is closely related to the [[Riemann zeta function]]. | ||
== Catalog of equal-step tunings == | == Catalog of equal-step tunings == | ||
=== Equal divisions === | === Equal divisions === | ||
<small>''Includes | : <small>''Includes Ed''p'' for ''p'' with a [[Wilson height]] ≤ 10 and integer limit ≤ 8 (plus some extras due to strong consensus for their inclusion).''</small> | ||
* [[Ed9/8]] (… of the major whole tone) | * [[Ed9/8]] (… of the major whole tone) | ||
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* [[Edφ]] (… of [[acoustic phi]]) | * [[Edφ|Ed''φ'']] (… of [[acoustic phi]]) | ||
* [[Ed5/3]] (… of the classic major sixth) | * [[Ed5/3]] (… of the classic major sixth) | ||
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* [[Ed9/4]] (… of the 3-limit major ninth) | |||
* [[Ed7/3]] (… of the septimal minor tenth) | * [[Ed7/3]] (… of the septimal minor tenth) | ||
* [[Ed5/2]] (… of the classic major tenth) | * [[Ed5/2]] (… of the classic major tenth) | ||
* [[Ed8/3]] (… of the perfect eleventh) | * [[Ed8/3]] (… of the perfect eleventh) | ||
* [[ | * [[Ede|Ed''e'']] (… of [[acoustic e|acoustic ''e'']]) | ||
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* [[Ed7]] (… of the 7th harmonic) | * [[Ed7]] (… of the 7th harmonic) | ||
* [[Ed8]] (… of the 8th harmonic) | * [[Ed8]] (… of the 8th harmonic) | ||
* [[Ed12]] (… of the 12th harmonic) | |||
=== Equal multiplications === | === Equal multiplications === | ||
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==== List of notable AS ==== | ==== List of notable AS ==== | ||
* [[1ed9/8|AS9/8]] | * [[1ed9/8|AS9/8]] | ||
* [[ | * [[21/20|AS21/20]] | ||
* [[1ed33/32|AS33/32]] | * [[1ed33/32|AS33/32]] | ||
==== List of notable APS ==== | ==== List of notable APS ==== | ||
* APS13.94—13.97¢, tunings for the [[Delta scale]] | * APS13.94—13.97¢, tunings for the [[8ed16/15|Delta scale]] | ||
* APS35.099¢, tuning of [[Carlos Gamma]] | * APS35.099¢, tuning of [[Carlos Gamma]] | ||
* APS63.59—63.82¢, [[Phoenix]] tunings | * APS63.59—63.82¢, [[Phoenix]] tunings | ||
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* [[1ed69c|APS69¢]] | * [[1ed69c|APS69¢]] | ||
* APS77.965¢, tuning of [[Carlos Alpha]] | * APS77.965¢, tuning of [[Carlos Alpha]] | ||
* [[1ed86.4c|APS86.4¢]] | * [[1ed86.4c|APS86.4¢]], a.k.a. "13.888edo" | ||
* [[88cET|APS88¢]] | * [[88cET|APS88¢]] | ||
* [[1ed97.5c|APS97.5¢]] | * [[1ed97.5c|APS97.5¢]] | ||
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== Edonoi == | == Edonoi == | ||
An '''equal division of a non-octave interval''' ('''EDONOI''' or '''edonoi''') is a [[tuning]] obtained by dividing a [[non-octave]] [[interval]] in a certain number of equal steps. In a broader sense, any equal-step tuning that is not an integer [[edo]] is an edonoi. | An '''equal division of a non-octave interval''' ('''EDONOI''' or '''edonoi''') is a [[tuning]] obtained by dividing a [[non-octave]] [[interval]] in a certain number of equal steps. In a broader sense, any equal-step tuning that is not an integer [[edo]] is an edonoi. | ||
The most often used edonoi include the equal-tempering of the [[ | The most often used edonoi include the equal-tempering of the [[Bohlen–Pierce scale]] (i.e. [[13edt|13 equal divisions of 3]]), the [[Phoenix]] tuning, tunings of [[Carlos Alpha]], [[Carlos Beta|Beta]], and [[Carlos Gamma|Gamma]], the [[19edt|19 equal divisions of 3]], the [[6edf|6 equal divisions of 3/2]], the [[2ed13/10|2 equal divisions of 13/10]], and [[88cET]]. Other strong edonoi include [[69ed7]] and [[143ed11]], respectively very accurate in the 3.5.7.11.13 subgroup and the 5.7.8.9.11.13.17.23 subgroup. For a more extensive gallery, see the [[#Equal divisions]] section above. | ||
Some edonoi contain an interval close to [[2/1]] that might function like a stretched or squashed octave | Some edonoi contain an interval close to [[2/1]] that might function like a [[stretched and compressed tuning|stretched or squashed]] octave – those edonoi can thus be considered variations on edos. | ||
Other edonoi contain no approximation of an octave or a compound octave (at least, not for a while), and continue generating new tones as they continue upward or downward. Such scales lack a very familiar compositional redundancy, that of [[octave equivalence]] | Other edonoi contain no approximation of an octave or a compound octave (at least, not for a while), and continue generating new tones as they continue upward or downward. Such scales lack a very familiar compositional redundancy, that of [[octave equivalence]] – this might necessitate special attention. | ||
== See also == | == See also == | ||
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[[Category:Equal-step tuning| ]] <!-- main article --> | [[Category:Equal-step tuning| ]] <!-- main article --> | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Acronyms]] | [[Category:Acronyms]] | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||