Equal-step tuning: Difference between revisions
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When a tuning is called "n-tone equal temperament" (abbreviated n-tET or n-ET), this usually means "n divisions of 2/1, the octave, or some approximation thereof" but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, the page [[EDOs to ETs]] may help clarify. | When a tuning is called "n-tone equal temperament" (abbreviated n-tET or n-ET), this usually means "n divisions of 2/1, the octave, or some approximation thereof" but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, the page [[EDOs to ETs]] may help clarify. | ||
There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term ''EDO''(occasionally written ED2), meaning "equal divisions of the octave" (or "equal divisions of 2/1"), leaves comparison to JI out of the picture, aside from the octave itself (which is assumed to be Just). There are other less standard terms, many in the [http://www.tonalsoft.com/enc/encyclopedia.aspx Tonalsoft Encyclopedia]. More generally, the term ''EDn'' can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic). | There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term ''EDO'' (occasionally written ED2), meaning "equal divisions of the octave" (or "equal divisions of 2/1"), leaves comparison to JI out of the picture, aside from the octave itself (which is assumed to be Just). There are other less standard terms, many in the [http://www.tonalsoft.com/enc/encyclopedia.aspx Tonalsoft Encyclopedia]. More generally, the term ''EDn'' can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic). | ||
'''As the steps are tuned to be equal, equal scales may be taken to close anywhere composers wish them to.''' Barring the convention of closing equal divisions of particular just intervals at those stated just intervals, there are infinite synonymous names for each equal scale. Barring further the large number of names which would be avoided in discourses on comparative modality and tonality, there is still a a great width to the universe of modes and keys which modal and tonal compositional art can access. | '''As the steps are tuned to be equal, equal scales may be taken to close anywhere composers wish them to.''' Barring the convention of closing equal divisions of particular just intervals at those stated just intervals, there are infinite synonymous names for each equal scale. Barring further the large number of names which would be avoided in discourses on comparative modality and tonality, there is still a a great width to the universe of modes and keys which modal and tonal compositional art can access. | ||
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'''As there are infinite intervals, there are infinite 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_surveys|sequentially]] or [[Polymicrotonality|simultaneously]]. | '''As there are infinite intervals, there are infinite 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_surveys|sequentially]] or [[Polymicrotonality|simultaneously]]. | ||
== Simultaneous equal divisions == | ==Simultaneous equal divisions== | ||
What do | What do 12ED2, 19ED3, and 28ED5 all have in common? They're 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. | ||
The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it's a good equal division of 6/1 as well. | The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it's a good equal division of 6/1 as well. | ||
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For the mathematically inclined, this kind of diagram is closely related to [[The_Riemann_Zeta_Function_and_Tuning|the Riemann zeta function]]. | For the mathematically inclined, this kind of diagram is closely related to [[The_Riemann_Zeta_Function_and_Tuning|the Riemann zeta function]]. | ||
== Gallery of equal divisions == | ==Gallery of equal divisions== | ||
===[[EdVI through edX|…of sixths through tenths]]=== | ===[[EdVI through edX|…of sixths through tenths]]=== | ||
* [[EdVI|'''<big>…of sixths (e. g. 5/3 and 11/7)</big>''']] | * [[EdVI|'''<big>…of sixths (e. g. 5/3 and 11/7)</big>''']] | ||
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* '''<big>[[EdXIX|…of nineteenths (e. g. 6/1)]]</big>''' | * '''<big>[[EdXIX|…of nineteenths (e. g. 6/1)]]</big>''' | ||
** [[Ed6|'''<big>...of the hexatave (6/1)</big>''']] | ** [[Ed6|'''<big>...of the hexatave (6/1)</big>''']] | ||
* [[Ed7|'''<big>...of the 7th Natural (7/1)</big>''']] | * [[Ed7|'''<big>...of the 7th Natural (7/1)</big>''']] | ||
* [[Ed8|'''<big>...of the Triple Octave (8/1)</big>''']] | * [[Ed8|'''<big>...of the Triple Octave (8/1)</big>''']] | ||
* [[Ed9|'''<big>...of the Double Tritave (9/1)</big>''']] | * [[Ed9|'''<big>...of the Double Tritave (9/1)</big>''']] | ||
* [[Ed10|'''<big>...of the Just Major 24th (10/1)</big>''']] | * [[Ed10|'''<big>...of the Just Major 24th (10/1)</big>''']] | ||
* [[Ed11|'''<big>...of the 11th Natural (11/1)</big>''']] | * [[Ed11|'''<big>...of the 11th Natural (11/1)</big>''']] | ||
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===[[Equal divisions of the whole tone|...of whole tones (e. g. 8/7, 9/8, and 10/9)]]=== | ===[[Equal divisions of the whole tone|...of whole tones (e. g. 8/7, 9/8, and 10/9)]]=== | ||
* '''<big>...of the Septimal Whole Tone (8/7)</big>''' | |||
** 5 - [[5ed8/7|Fifth root of 8/7]] | |||
* '''<big>...of the Pythagorean Whole Tone (9/8)</big>''' | * '''<big>...of the Pythagorean Whole Tone (9/8)</big>''' | ||
** 9 - [[9ed9/8|Ninth root of 9/8]] | ** 9 - [[9ed9/8|Ninth root of 9/8]] | ||
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** 8 - [[Delta scale]] | ** 8 - [[Delta scale]] | ||
== Equal multiplications == | ==Equal multiplications== | ||
=== ...of a given cents value === | === ...of a given cents value === | ||
* [[88cET]] | * [[88cET]] | ||
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** [[22/21s equal temperament]] - multiples of [[22/21]] | ** [[22/21s equal temperament]] - multiples of [[22/21]] | ||
** [[25/24s equal temperament]] - multiples of [[25/24]] | ** [[25/24s equal temperament]] - multiples of [[25/24]] | ||
== See also == | ==See also== | ||
* [[Edo anatomy]] | * [[Edo anatomy]] | ||
* [[Macrotonal edos]] | * [[Macrotonal edos]] | ||
* [[Maximal evenness]] | * [[Maximal evenness]] | ||
* [[Toctave]] | * [[Toctave]] | ||
* [[The Riemann Zeta Function and Tuning]] | * [[The Riemann Zeta Function and Tuning]] | ||
[[Category:Equal-step tuning| ]] <!-- main article --> | |||
[[Category:Tuning]] | [[Category:Tuning]] | ||
[[Category:Temperament]] | [[Category:Temperament]] | ||