Equal-step tuning: Difference between revisions

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The '''equal-step tuning''', '''equal tuning''', or '''equal pitch division''' is the [[tuning system]] where the distance between adjacent steps is of constant 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]].

The '''equal-step tuning''', '''equal tuning''', or '''equal pitch division''' ('''EPD''') is the [[tuning system]] where the distance between adjacent steps is of constant 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 [[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.

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''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 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 survey|sequentially]] or [[Polymicrotonality|simultaneously]].

An equal step tuning is an [[Arithmetic tuning|arithmetic]] and [[harmonotonic tuning]].

== Formula ==

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 {{Wikipedia|Equal temperament}}
-The '''equal-step tuning''', '''equal tuning''', or '''equal pitch division''' is the [[tuning system]] where the distance between adjacent steps is of constant 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]].
+The '''equal-step tuning''', '''equal tuning''', or '''equal pitch division''' ('''EPD''') is the [[tuning system]] where the distance between adjacent steps is of constant 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 [[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.
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 ''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 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 survey|sequentially]] or [[Polymicrotonality|simultaneously]].
+An equal step tuning is an [[Arithmetic tuning|arithmetic]] and [[harmonotonic tuning]].
 == Formula ==