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~~]]. 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.

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 [[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 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 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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Even if 3 is included in a given temperament, the ploidaseph framework may occasionally be more useful than the ploidacot framework, in cases where the mapping of 3 is very complex and the structure of the temperament therefore deprioritizes prime 3. [[Hemiwürschmidt]], a [[strong extension]] of the aforementioned didacus, has a ploidacot of beta-hexadecacot as it divides 6/1 into sixteen generators; while [[trismegistus]] has a ploidacot of epsilon-pentadecacot as it maps [[96/1]] to fifteen generators. Each of these has a more intuitizable expression in terms of 2.5 intervals, which are much simpler in the respective temperaments: hemiwürschmidt is diseph and trismegistus is alpha-triseph (one-third 5/2).

Combining ploidacots and ploidasephs determines its [[5-limit]] properties; for instance, meantone can be labeled as "monocot beta-tetraseph" because four generators make up [[5/1]] while the generator represents [[3/2]], and valentine can be labeled as "enneacot pentaseph" because five generators make up [[5/4]] and nine of them make up [[3/2]].

== Examples ==

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* [[Blackwood]] is pentaploid acot

* [[Whitewood]] is heptaploid acot

* [[Ennealimmal]] is enneaploid

* [[Compton]] is dodecaploid acot

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== List of ploidacots ==

=== Acot ===

* Pentaploid acot ([[blackwood]], [[5edo]])

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* [[Ploidacot/Triploid monocot|Triploid monocot]]

* [[Ploidacot/Tetraploid monocot|Tetraploid monocot]]

* [[Ploidacot/Pentaploid monocot|Pentaploid monocot]]

* [[Ploidacot/Hexaploid monocot|Hexaploid monocot]]

* [[Ploidacot/Heptaploid monocot|Heptaploid monocot]]

=== Dicot ===

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* [[Ploidacot/Diploid dicot|Diploid dicot]]

* [[Ploidacot/Diploid alpha-dicot|Diploid alpha-dicot]]

* [[Ploidacot/Triploid dicot|Triploid dicot]]

* [[Ploidacot/Triploid alpha-dicot|Triploid alpha-dicot]]

=== Tricot ===

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* [[Ploidacot/Alpha-tricot|Alpha-tricot]]

* [[Ploidacot/Omega-tricot|Omega-tricot]]

* [[Ploidacot/Diploid tricot|Diploid tricot]]

* [[Ploidacot/Diploid alpha-tricot|Diploid alpha-tricot]]

* [[Ploidacot/Diploid beta-tricot|Diploid beta-tricot]]

* [[Ploidacot/Triploid tricot|Triploid tricot]]

=== Tetracot ===

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* [[Ploidacot/Pentacot|Pentacot]]

* [[Ploidacot/Alpha-pentacot|Alpha-pentacot]]

* [[Ploidacot/Beta-pentacot|Beta-pentacot]]

* [[Ploidacot/Gamma-pentacot|Gamma-pentacot]]

* [[Ploidacot/Omega-pentacot|Omega-pentacot]]

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* [[Ploidacot/Hexacot|Hexacot]]

* [[Ploidacot/Alpha-hexacot|Alpha-hexacot]]

* [[Ploidacot/Beta-hexacot|Beta-hexacot]]

* [[Ploidacot/Gamma-hexacot|Gamma-hexacot]]

* [[Ploidacot/Delta-hexacot|Delta-hexacot]]

* [[Ploidacot/Omega-hexacot|Omega-hexacot]]

=== Heptacot ===

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* [[Ploidacot/Alpha-heptacot|Alpha-heptacot]]

* [[Ploidacot/Beta-heptacot|Beta-heptacot]]

* [[Ploidacot/Gamma-heptacot|Gamma-heptacot]]

* [[Ploidacot/Delta-heptacot|Delta-heptacot]]

* [[Ploidacot/Epsilon-heptacot|Epsilon-heptacot]]

* [[Ploidacot/Omega-heptacot|Omega-heptacot]]

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* [[Ploidacot/Octacot|Octacot]]

* [[Ploidacot/Beta-octacot|Beta-octacot]]

* [[Ploidacot/Epsilon-octacot|Epsilon-octacot]]

* [[Ploidacot/Omega-octacot|Omega-octacot]]

=== Enneacot ===

* [[Ploidacot/Enneacot|Enneacot]]

* [[Ploidacot/Delta-enneacot|Delta-enneacot]]

* [[Ploidacot/Omega-enneacot|Omega-enneacot]]

=== Decacot ===

* [[Ploidacot/Decacot|Decacot]]

* [[Ploidacot/Beta-decacot|Beta-decacot]]

* [[Ploidacot/Epsilon-decacot|Epsilon-decacot]]

=== >10 cots ===

* [[Ploidacot/Hendecacot|Hendecacot]]

* [[Ploidacot/Icosacot|Icosacot]]

== See also ==

* [[Wedgie]] – a mathematical generalization of the concept of ploidacots that uniquely characterizes a temperament

[[Category:Temperament naming]]

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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]]. 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.
+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 [[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 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 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]].
@@ Line 83: / Line 89: @@
 Even if 3 is included in a given temperament, the ploidaseph framework may occasionally be more useful than the ploidacot framework, in cases where the mapping of 3 is very complex and the structure of the temperament therefore deprioritizes prime 3. [[Hemiwürschmidt]], a [[strong extension]] of the aforementioned didacus, has a ploidacot of beta-hexadecacot as it divides 6/1 into sixteen generators; while [[trismegistus]] has a ploidacot of epsilon-pentadecacot as it maps [[96/1]] to fifteen generators. Each of these has a more intuitizable expression in terms of 2.5 intervals, which are much simpler in the respective temperaments: hemiwürschmidt is diseph and trismegistus is alpha-triseph (one-third 5/2).
+Combining ploidacots and ploidasephs determines its [[5-limit]] properties; for instance, meantone can be labeled as "monocot beta-tetraseph" because four generators make up [[5/1]] while the generator represents [[3/2]], and valentine can be labeled as "enneacot pentaseph" because five generators make up [[5/4]] and nine of them make up [[3/2]].
 == Examples ==
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 * [[Blackwood]] is pentaploid acot
 * [[Whitewood]] is heptaploid acot
-* [[Ennealimmal]] is enneaploid
 * [[Compton]] is dodecaploid acot
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 == List of ploidacots ==
 === Acot ===
 * Pentaploid acot ([[blackwood]], [[5edo]])
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 * [[Ploidacot/Triploid monocot|Triploid monocot]]
 * [[Ploidacot/Tetraploid monocot|Tetraploid monocot]]
+* [[Ploidacot/Pentaploid monocot|Pentaploid monocot]]
+* [[Ploidacot/Hexaploid monocot|Hexaploid monocot]]
+* [[Ploidacot/Heptaploid monocot|Heptaploid monocot]]
 === Dicot ===
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 * [[Ploidacot/Diploid dicot|Diploid dicot]]
 * [[Ploidacot/Diploid alpha-dicot|Diploid alpha-dicot]]
+* [[Ploidacot/Triploid dicot|Triploid dicot]]
+* [[Ploidacot/Triploid alpha-dicot|Triploid alpha-dicot]]
 === Tricot ===
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 * [[Ploidacot/Alpha-tricot|Alpha-tricot]]
 * [[Ploidacot/Omega-tricot|Omega-tricot]]
+* [[Ploidacot/Diploid tricot|Diploid tricot]]
 * [[Ploidacot/Diploid alpha-tricot|Diploid alpha-tricot]]
+* [[Ploidacot/Diploid beta-tricot|Diploid beta-tricot]]
+* [[Ploidacot/Triploid tricot|Triploid tricot]]
 === Tetracot ===
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 * [[Ploidacot/Pentacot|Pentacot]]
 * [[Ploidacot/Alpha-pentacot|Alpha-pentacot]]
+* [[Ploidacot/Beta-pentacot|Beta-pentacot]]
 * [[Ploidacot/Gamma-pentacot|Gamma-pentacot]]
 * [[Ploidacot/Omega-pentacot|Omega-pentacot]]
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 * [[Ploidacot/Hexacot|Hexacot]]
 * [[Ploidacot/Alpha-hexacot|Alpha-hexacot]]
+* [[Ploidacot/Beta-hexacot|Beta-hexacot]]
+* [[Ploidacot/Gamma-hexacot|Gamma-hexacot]]
+* [[Ploidacot/Delta-hexacot|Delta-hexacot]]
+* [[Ploidacot/Omega-hexacot|Omega-hexacot]]
 === Heptacot ===
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 * [[Ploidacot/Alpha-heptacot|Alpha-heptacot]]
 * [[Ploidacot/Beta-heptacot|Beta-heptacot]]
+* [[Ploidacot/Gamma-heptacot|Gamma-heptacot]]
+* [[Ploidacot/Delta-heptacot|Delta-heptacot]]
+* [[Ploidacot/Epsilon-heptacot|Epsilon-heptacot]]
 * [[Ploidacot/Omega-heptacot|Omega-heptacot]]
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 * [[Ploidacot/Octacot|Octacot]]
 * [[Ploidacot/Beta-octacot|Beta-octacot]]
+* [[Ploidacot/Epsilon-octacot|Epsilon-octacot]]
+* [[Ploidacot/Omega-octacot|Omega-octacot]]
 === Enneacot ===
 * [[Ploidacot/Enneacot|Enneacot]]
+* [[Ploidacot/Delta-enneacot|Delta-enneacot]]
 * [[Ploidacot/Omega-enneacot|Omega-enneacot]]
+=== Decacot ===
+* [[Ploidacot/Decacot|Decacot]]
+* [[Ploidacot/Beta-decacot|Beta-decacot]]
+* [[Ploidacot/Epsilon-decacot|Epsilon-decacot]]
 === >10 cots ===
+* [[Ploidacot/Hendecacot|Hendecacot]]
 * [[Ploidacot/Icosacot|Icosacot]]
 == See also ==
 * [[Wedgie]] – a mathematical generalization of the concept of ploidacots that uniquely characterizes a temperament
 [[Category:Temperament naming]]