Equal-step tuning: Difference between revisions

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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]].

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 [[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.

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 [[temperament]] – that is, of a [[just intonation|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 '''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).

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-tempering of the [[Bohlen–Pierce scale]] may be referred to as 13ed3, 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.

''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]]'')~~. Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division~~.

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]]'').

== Formula ==

@@ Line 7: / Line 7: @@
 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]].
-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 [[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.
+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 [[temperament]] – that is, of a [[just intonation|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 '''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).
+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-tempering of the [[Bohlen–Pierce scale]] may be referred to as 13ed3, 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.
 ''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]]''). Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division.
+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]]'').
 == Formula ==

Equal-step tuning: Difference between revisions

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