Equal-step tuning: Difference between revisions
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| 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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''As there are infinitely many intervals, there are infinitely many equal scales.'' Barring technicalities, there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings [[ET survey|sequentially]] or [[Polymicrotonality|simultaneously]]. | ''As there are infinitely many intervals, there are infinitely many equal scales.'' Barring technicalities, there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings [[ET survey|sequentially]] or [[Polymicrotonality|simultaneously]]. | ||
An equal-step tuning is an [[Arithmetic tuning|arithmetic]] and [[harmonotonic tuning]]. In terms of what musical resource is divided, it divides pitch, so | An equal-step tuning is an [[Arithmetic tuning|arithmetic]] and [[harmonotonic tuning]]. In terms of what musical resource is divided, it divides pitch, so another term for equal-step tuning is '''equal pitch division (EPD)'''. Since pitch is the most commonly divided resource, this can be shortened to just '''equal division (ED)'''. | ||
== Formula == | == Formula == | ||
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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 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) | |||
* [[Ed6/5]] (… of the classic minor third) | * [[Ed6/5]] (… of the classic minor third) | ||
* [[Ed5/4]] (… of the classic major third) | * [[Ed5/4]] (… of the classic major third) | ||
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* '''[[EDF]] (… of the perfect fifth, [[3/2]])''' | * '''[[EDF]] (… of the perfect fifth, [[3/2]])''' | ||
** most famously [[Carlos Alpha]], [[Carlos Beta|Beta]] and [[Carlos Gamma|Gamma]], but lots of others, too | ** most famously approximates [[Carlos Alpha]], [[Carlos Beta|Beta]] and [[Carlos Gamma|Gamma]], but lots of others, too | ||
* [[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) | |||
* [[Ede|Ed''e'']] (… of [[acoustic e|acoustic ''e'']]) | |||
* '''[[EDT]] (… of the [[tritave]]/twelfth, 3/1)''' | * '''[[EDT]] (… of the [[tritave]]/twelfth, 3/1)''' | ||
** most famously the [[Bohlen–Pierce scale]], but lots of others, too | ** most famously approximates the [[Bohlen–Pierce scale]], but lots of others, too | ||
* [[Ed7/2]] (… of the septimal minor fourteenth) | * [[Ed7/2]] (… of the septimal minor fourteenth) | ||
* [[Ed4]] (… of the double octave, [[4/1]]) | |||
* | |||
* [[Ed5]] (… of the 5th harmonic) | * [[Ed5]] (… of the 5th harmonic) | ||
* [[Ed6]] (… of the 6th harmonic) | * [[Ed6]] (… of the 6th harmonic) | ||
* [[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) | * [[Ed12]] (… of the 12th harmonic) | ||
=== Equal multiplications === | === Equal multiplications === | ||
An equal multiplication of a ''rational'' interval can also be called an | An equal multiplication of a ''rational'' interval can also be called an '''ambitonal sequence (AS)'''. For example, the 25/24 equal-step tuning could also be written AS25/24. | ||
An equal multiplication of an ''irrational'' interval can also be called an | An equal multiplication of an ''irrational'' interval can also be called an '''arithmetic pitch sequence (APS)'''. For example, the 65-[[cents]] equal-step tuning could also be written APS65¢. | ||
The union of both is equivalent to the unity division of a target interval. For example, AS25/24 is 1ed25/24, and APS65¢ is 1ed65¢. | The union of both is equivalent to the unity division of a target interval. For example, AS25/24 is 1ed25/24, and APS65¢ is 1ed65¢. | ||
==== List of 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 ==== | |||
* APS13.94—13.97¢, tunings for the [[8ed16/15|Delta scale]] | |||
==== List of APS ==== | |||
* APS35.099¢, tuning of [[Carlos Gamma]] | * APS35.099¢, tuning of [[Carlos Gamma]] | ||
* APS63.59—63.82¢, [[Phoenix]] tunings | |||
* APS63.833¢, tuning of [[Carlos Beta]] | * APS63.833¢, tuning of [[Carlos Beta]] | ||
* [[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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** [[3zpi]] = 470.7997…cET | ** [[3zpi]] = 470.7997…cET | ||
** … | ** … | ||
== 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 | 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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* [[Families of scales]] | * [[Families of scales]] | ||
* [[Maximal evenness]] | * [[Maximal evenness]] | ||
* | * Some pages about the same thing as equal-step tuning, but framed differently: | ||
** [[User:Cmloegcmluin/EPD]] (ED for short) | |||
** [[User:Cmloegcmluin/AS]] | |||
** [[User:Cmloegcmluin/APS]] | |||
== External links == | == External links == | ||
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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]] | ||