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冒頭残り2段落は保留；平均律をtemperamentの訳としている文献の例を挙げる
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''ステップが等しくなるように調律されているため、作曲家はスケールのどこでも好きなところにアプローチできる。'' 通常は簡約化されるものではあるが、均等な分割のスケールは無限に別名を持っている（12edo = 24ed4 = 36ed8 = …など）。さらにモード音楽と調性音楽の議論において避けられる様々な名前を除外しても、作曲活動から利用可能なモードとキーはまだとても幅広く存在する。
''ステップが等しくなるように調律されているため、作曲家はスケールのどこでも好きなところにアプローチできる。'' 通常は簡約化されるものではあるが、均等な分割のスケールは無限に別名を持っている（12edo = 24ed4 = 36ed8 = …など）。さらにモード音楽と調性音楽の議論において避けられる様々な名前を除外しても、作曲活動から利用可能なモードとキーはまだとても幅広く存在する。


An '''equal-step tuning''', '''equal tuning''', or '''equal division''' ('''ED''') is a [[period]]ic [[tuning system]] where the distance between adjacent steps is of constant [[Interval size|size]]. The size of this single step is given explicitly (e.g. [[88cET|88-cent equal tuning]]) or as a fraction of a larger interval (e.g. [[13edo|13 equal divisions of the octave]]). Any interval, rational/just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common, leading to [[edo]] systems. When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that [[root]].
<!--''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]].


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 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.
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''). Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division.
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歴史的には、純正律を実用的に修正した音律を平均律といった<ref>改訂新版　世界大百科事典『[https://kotobank.jp/word/%E5%B9%B3%E5%9D%87%E5%BE%8B|平均律]』 - コトバンク</ref>。これには中全音律やウェル・テンペラメントを含む。つまり、純正律からの矛盾した要求を何とかするために、音程を平均化するという方法をとったもの（≒temperament）をすべて平均律と呼んでいたのである。
 
== Formula ==
To find the step size of ''n''-ed-''p'' in terms of [[cent]]s, divide the cents of ''p'' by ''n''. The size ''s'' of ''k'' steps of ''n''-ed-''p'' (''k''\''n'' <''p''>) is
 
<math>\displaystyle s = 1200 \log_2 (p) \cdot k/n</math>
 
To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of ''p''. For example, the step of 12edo is 2<sup>1/12</sup> (≈ 1.059). So the ratio ''c'' of ''k'' steps of ''n''-ed-''p'' is


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 '''ed-''p''''' can be used, where ''p'' is any frequency ratio. For example, the equal-tempered [[Bohlen-Pierce]] scale may also be referred to as 13ed3, for 13 equal divisions of 3/1 (the 3rd harmonic).
<math>\displaystyle c = p^{k/n}</math>


''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.
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.


''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]].
== Simultaneous 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.


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''). Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division.  
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 is a good equal division of 6/1 as well.
 
[[File:equal.png|alt=equal.png|800x69px|equal.png]]


(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]].


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