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Equal-step tuning: Difference between revisions

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{{interwiki
{{interwiki
| de = Gleichstufige_Tonsysteme
| de = Gleichstufige_Tonsysteme
| en = Equal-step_tuning
| 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 it is an ''equal pitch division'' (''[[EPD]]'').
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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In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when ''k'' = ''n'', ''c'' is simply ''p'', because any number to the 1st power is itself.
In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when ''k'' = ''n'', ''c'' is simply ''p'', because any number to the 1st power is itself.


== Simultaneous equal divisions ==
== Lookalike equal divisions ==
What do 12ed2, 19ed3, and 28ed5 all have in common? They are all approximately the same scale. This happens because 12ed2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ed2 does not correspond closely to any equal division of 3/1 or 5/1.
What do 12ed2, 19ed3, and 28ed5 all have in common? They are all approximately the same scale. This happens because 12ed2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ed2 does not correspond closely to any equal division of 3/1 or 5/1.


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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 [[The Riemann zeta function and tuning|the Riemann zeta function]].
(Version that goes to 100ed2: [[User:BudjarnLambeth/ET alignment chart]])
 
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 ===
* [[Ed16/15]] (… of the classic diatonic semitone)
: <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>
** most famously the [[Delta scale]], but others too
 
* [[Ed9/8]] (… of the major whole tone)
* [[Ed6/5]] (… of the classic minor third)
* [[Ed5/4]] (… of the classic major third)
* [[Ed5/4]] (… of the classic major third)
* [[Ed4/3]] (… of the perfect fourth)
* [[Ed4/3]] (… of the perfect fourth)
* '''[[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)
* '''[[EDO]] (… of the [[octave]], 2/1)'''
* '''[[EDO]] (… of the [[octave]], 2/1)'''
* [[Ed9/4]] (… of the 3-limit major ninth)
* [[Ed7/3]] (… of the septimal minor tenth)
* [[Ed5/2]] (… of the classic major tenth)
* [[Ed5/2]] (… of the classic major tenth)
* [[EDN]] (… of the [[natave]], e/1)
* [[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)
* [[Ed4]] (… of the double octave, [[4/1]])
* [[Ed4]] (… of the double octave, [[4/1]])
* [[Ed5]] (… of the 5th harmonic)
* [[Ed5]] (… of the 5th harmonic)
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* [[Ed7]] (… of the 7th harmonic)
* [[Ed7]] (… of the 7th harmonic)
* [[Ed8]] (… of the 8th harmonic)
* [[Ed8]] (… of the 8th harmonic)
* [[Ed9]] (… of the 9th harmonic)
* [[Ed10]] (… of the 10th harmonic)
* [[Ed11]] (… of the 11th harmonic)
* [[Ed12]] (… of the 12th harmonic)
* [[Ed12]] (… of the 12th harmonic)
* [[Ed13]] (… of the 13th harmonic)
 


=== Equal multiplications ===
=== Equal multiplications ===
An equal multiplication of a ''rational'' interval can also be called an [[AS|ambitonal sequence (AS)]]. For example, the 25/24 equal-step tuning could also be written AS25/24.  
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 [[APS|arithmetic pitch sequence (APS)]]. For example, the 65-[[cents]] equal-step tuning could also be written APS65¢.  
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¢.  


* [[AS #List of ASs]]
==== List of notable AS ====
* [[APS #List of APSs]]
* [[1ed9/8|AS9/8]]
* [[21/20|AS21/20]]
 
==== List of notable APS ====
* APS13.94—13.97¢, tunings for the [[8ed16/15|Delta scale]]
* APS35.099¢, tuning of [[Carlos Gamma]]
* APS63.59—63.82¢, [[Phoenix]] tunings
* APS63.833¢, tuning of [[Carlos Beta]]
* [[1ed69c|APS69¢]]
* APS77.965¢, tuning of [[Carlos Alpha]]
* [[1ed86.4c|APS86.4¢]], a.k.a. "13.888edo"
* [[88cET|APS88¢]]
* [[1ed97.5c|APS97.5¢]]
* [[1ed125c|APS125¢]]
* [[Zeta peak index]] tunings
* [[Zeta peak index]] tunings
** [[1zpi]] = 1065.1767…cET
** [[1zpi]] = 1065.1767…cET
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** [[3zpi]] = 470.7997…cET
** [[3zpi]] = 470.7997…cET
** …
** …
== 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 [[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 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]] – this might necessitate special attention.


== See also ==  
== See also ==  
* [[Equal division of the octave]]
* [[Equal division of the octave]]
* [[Edonoi|Equal division of a non-octave interval]]
* [[Families of scales]]
* [[Families of scales]]
* [[Maximal evenness]]
* [[Maximal evenness]]
* [[:Category:Equal-step tuning]]
* 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 ==
* [http://www.nonoctave.com/tuning/quintave.html X. J. Scott's Equal Divisions of Rational Intervals]


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[[Category:Equal-step tuning| ]] <!-- main article -->
[[Category:Terms]]
[[Category:Acronyms]]
[[Category:Tuning]]
[[Category:Tuning]]
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