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]]. 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 [[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]]s 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 [[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 [[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 does not 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.

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== Specification ==

=== Ploids ===

Any rank-2 temperament of the 2.3.… subgroup has an octave, and it may split the octave into a number of parts, or [[period]]s, called '''ploids'''. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (''not'' *monoploid), which can be omitted.

Any rank-2 temperament of the 2.3.(…) [[subgroup]] has an octave, and it may split the octave into a number of parts, or [[period]]s, called '''ploids'''. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (''not'' *monoploid), which can be omitted.

=== Cots ===

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The idea of classifying temperaments by splitting a subgroup can be traced back to the introduction of the [[wedgie]], a mathematical construct that uniquely characterizes a rank-''r'' temperament by how the temperament splits each rank-''r'' subgroup of the original subgroup.

The ploidacot system comes more directly after [[Kite Giedraitis]]' [[pergen]] system, and may be considered 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 comes more directly after [[Kite Giedraitis]]' [[pergen]] system, and may be considered a canonical naming scheme for pergens of rank-2 temperaments of 2.3.(…) subgroups in that every such pergen corresponds to a unique name in the ploidacot system.

=== Relationship to pergens ===

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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]]. 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 [[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]]s 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 [[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 [[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 does not 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.
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 == Specification ==
 === Ploids ===
-Any rank-2 temperament of the 2.3.… subgroup has an octave, and it may split the octave into a number of parts, or [[period]]s, called '''ploids'''. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (''not'' *monoploid), which can be omitted.
+Any rank-2 temperament of the 2.3.(…) [[subgroup]] has an octave, and it may split the octave into a number of parts, or [[period]]s, called '''ploids'''. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (''not'' *monoploid), which can be omitted.
 === Cots ===
@@ Line 93: / Line 93: @@
 The idea of classifying temperaments by splitting a subgroup can be traced back to the introduction of the [[wedgie]], a mathematical construct that uniquely characterizes a rank-''r'' temperament by how the temperament splits each rank-''r'' subgroup of the original subgroup.
-The ploidacot system comes more directly after [[Kite Giedraitis]]' [[pergen]] system, and may be considered 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 comes more directly after [[Kite Giedraitis]]' [[pergen]] system, and may be considered a canonical naming scheme for pergens of rank-2 temperaments of 2.3.(…) subgroups in that every such pergen corresponds to a unique name in the ploidacot system.
 === Relationship to pergens ===