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 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 '''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]]. A particularly simple case is if a temperament divides its [[3/2]] interval into ''n'' steps, it can be called an ''n''-cot tuning. More generally, 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. Temperaments where 3/2 is a whole number of ploids are written as ''acot''. However, stacking ''n'' cots sometimes 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'' (which is the same for all possible cots), and is an integer between 0 and ''n'' - 1 inclusive.

The "ploid" number of a temperament refers to how many equal parts, or [[period]]s the octave is divided into, and the "cot" number refers to how many [[generator]] steps of the temperament are needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. Temperaments where 3/2 is a whole number of ploids are written as ''acot''. However, stacking ''n'' cots sometimes 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'' (which is the same for all possible cots), and is an integer between 0 and {{nowrap| ''n'' - 1 }} inclusive.

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, [[meantone]] is monocot because it is does not split the octave, and is generated by the perfect fifth. [[Kleismic]] is alpha-hexacot, since it 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.

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.

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

The ploidacot system was developed by [[Praveen Venkataramana]].

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-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 '''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]]. A particularly simple case is if a temperament divides its [[3/2]] interval into ''n'' steps, it can be called an ''n''-cot tuning. More generally, 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. Temperaments where 3/2 is a whole number of ploids are written as ''acot''. However, stacking ''n'' cots sometimes 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'' (which is the same for all possible cots), and is an integer between 0 and ''n'' - 1 inclusive.
+The "ploid" number of a temperament refers to how many equal parts, or [[period]]s the octave is divided into, and the "cot" number refers to how many [[generator]] steps of the temperament are needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. Temperaments where 3/2 is a whole number of ploids are written as ''acot''. However, stacking ''n'' cots sometimes 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'' (which is the same for all possible cots), and is an integer between 0 and {{nowrap| ''n'' - 1 }} inclusive.
-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, [[meantone]] is monocot because it is does not split the octave, and is generated by the perfect fifth. [[Kleismic]] is alpha-hexacot, since it 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.
-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.
+It is similar to the [[pergen]], and is a canonical naming scheme for pergens of rank-2 temperaments of 2.3.(…) [[subgroup]]s in that every such pergen corresponds to a unique name in the ploidacot system.
 The ploidacot system was developed by [[Praveen Venkataramana]].