Ploidacot: Difference between revisions

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The '''ploidacot''' system is a classification of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] based on how a temperament ~~can be thought~~ of as a ~~union of copies~~ of [[~~Pythagorean tuning~~]].

The '''ploidacot''' system is a classification of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] based on how a temperament divides the intervals of [[Pythagorean tuning]]. Ploidacots are written as ''m''-ploid ''s''-sheared ''n''-cot, with ''m''- and ''n''- often replaced by greek numeral prefixes, such as mono-, di-, tri-, etc. (and ''m''-ploid omitted entirely if the [[2/1|octave]] is not split), and "''s''-sheared" replaced by a greek letter, such as alpha-, beta-, etc. (or omitted entirely if ''s'' = 0). The "ploid" number of a temperament refers to how many equal parts, or "ploids" (known as [[period]]s in [[regular temperament theory]]), the octave is divided into, and the "cot" number refers to how many [[generator]] steps of the temperament, or "cots", is needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. However, stacking ''n'' cots often doesn't reach 3/2, but instead an interval ''s'' ploids above 3/2. There are infinitely many possible values of ''s'', but for the sake of ploidacot, ''s'' takes its residue modulo ''n'', and is an integer between 0 and ''n'' - 1 inclusive.

~~The simplest~~ and ~~most straightforward way to conceptualize this concept~~ is ~~simply; if your temperament can divide its~~ [[~~3/2~~]] ~~interval into N steps~~, ~~it can be called an N-cot tuning.~~

For example, [[meantone]] is monocot because it is does not split the octave, and is generated by the perfect fifth. [[Kleismic]] is alpha-hexacot, since does not split the octave, but splits [[3/1]], which is one octave above 3/2, into six equal parts (~317{{c}} each). [[Pajara]] is diploid monocot, since it is generated by the fifth and splits the octave in two 600{{c}} halves. [[Shrutar]] is diploid alpha-dicot, since it splits the octave in half, and splits the interval 600{{c}} above 3/2 (~1300{{c}}) into two ~650{{c}} halves. Note that in shrutar the interval one ploid above 3/2 is ~1300{{c}} and not 3/1, since the octave is split into two 600{{c}} ploids.

~~For example;~~ since [[~~34edo~~]]'s 3/2 ~~maps to 20\34~~, ~~and 20~~ is ~~divisible~~ by ~~both 4~~ and 5, so it ~~can be called tetracot~~ (~~4-cot) or pentacot (5-cot~~).

It is similar to the [[pergen]], and is a canonical naming scheme for pergens of rank-2 temperaments of the 2.3.… [[subgroup]] in that every such pergen corresponds to a unique name in the ploidacot system.

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-The '''ploidacot''' system is a classification of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] based on how a temperament can be thought of as a union of copies of [[Pythagorean tuning]].
+The '''ploidacot''' system is a classification of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] based on how a temperament divides the intervals of [[Pythagorean tuning]]. Ploidacots are written as ''m''-ploid ''s''-sheared ''n''-cot, with ''m''- and ''n''- often replaced by greek numeral prefixes, such as mono-, di-, tri-, etc. (and ''m''-ploid omitted entirely if the [[2/1|octave]] is not split), and "''s''-sheared" replaced by a greek letter, such as alpha-, beta-, etc. (or omitted entirely if ''s'' = 0). The "ploid" number of a temperament refers to how many equal parts, or "ploids" (known as [[period]]s in [[regular temperament theory]]), the octave is divided into, and the "cot" number refers to how many [[generator]] steps of the temperament, or "cots", is needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. However, stacking ''n'' cots often doesn't reach 3/2, but instead an interval ''s'' ploids above 3/2. There are infinitely many possible values of ''s'', but for the sake of ploidacot, ''s'' takes its residue modulo ''n'', and is an integer between 0 and ''n'' - 1 inclusive.
-The simplest and most straightforward way to conceptualize this concept is simply; if your temperament can divide its [[3/2]] interval into N steps, it can be called an N-cot tuning.
+For example, [[meantone]] is monocot because it is does not split the octave, and is generated by the perfect fifth. [[Kleismic]] is alpha-hexacot, since does not split the octave, but splits [[3/1]], which is one octave above 3/2, into six equal parts (~317{{c}} each). [[Pajara]] is diploid monocot, since it is generated by the fifth and splits the octave in two 600{{c}} halves. [[Shrutar]] is diploid alpha-dicot, since it splits the octave in half, and splits the interval 600{{c}} above 3/2 (~1300{{c}}) into two ~650{{c}} halves. Note that in shrutar the interval one ploid above 3/2 is ~1300{{c}} and not 3/1, since the octave is split into two 600{{c}} ploids.
-For example; since [[34edo]]'s 3/2 maps to 20\34, and 20 is divisible by both 4 and 5, so it can be called tetracot (4-cot) or pentacot (5-cot).
 It is similar to the [[pergen]], and is a canonical naming scheme for pergens of rank-2 temperaments of the 2.3.… [[subgroup]] in that every such pergen corresponds to a unique name in the ploidacot system.