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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]].
The ploidacot system was developed by [[Praveen Venkataramana]].
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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]].
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]].
== Relationship to pergens ==
Each ploidacot has one pergen. The numbers of ploid (''p''), shear (''s''), and cot (''c'') are given, its pergen form has following features:
* Every ''p''-ploid has a form of (P8/''p'', X).
* Haploids (''p'' = 1) are of the form (P8, X/''c'') since the octave is unsplit.
* Monocots (''c'' = 1) are of the form (P8/''p'', P5) since the fifth and its compounds are unsplit.
* If ''p'' and ''c'' are coprime, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* If ''s'' mod GCD(''p'', ''c'') = 0, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* If ''s'' mod GCD(''p'', ''c'') is not 0, the ploidacot has an imperfect pergen.
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Pergen forms of ploidacot
|-
! colspan="2" | Ploids
! Haploid
! Diploid
! Triploid
! Tetraploid
! Pentaploid
! Hexaploid
! Heptaploid
! Octaploid
|-
! colspan="2" | Monocot
| (P8, P5)
| (P8/2, P5)
| (P8/3, P5)
| (P8/4, P5)
| (P8/5, P5)
| (P8/6, P5)
| (P8/7, P5)
| (P8/8, P5)
|-
! rowspan="2" | Dicots
! dicot
| (P8, P5/2)
| (P8/2, P4/2)
| (P8/3, P5/2)
| (P8/4, P4/2)
| (P8/5, P5/2)
| (P8/6, P4/2)
| (P8/7, P5/2)
| (P8/8, P4/2)
|-
! alpha-dicot
| (P8, P4/2)
| (P8/2, M2/4)
| (P8/3, P4/2)
| (P8/4, m6/8)
| (P8/5, P4/2)
| (P8/6, M2/4)
| (P8/7, P4/2)
| (P8/8, d4/16)
|-
! rowspan="3" | Tricots
! tricot
| (P8, P5/3)
| (P8/2, P5/3)
| (P8/3, P4/3)
| (P8/4, P5/3)
| (P8/5, P5/3)
| (P8/6, P4/3)
| (P8/7, P5/3)
| (P8/8, P5/3)
|-
! alpha-tricot
| (P8, P11/3)
| (P8/2, P4/3)
| (P8/3, m3/9)
| (P8/4, P11/3)
| (P8/5, P4/3)
| (P8/6, M6/9)
| (P8/7, P11/3)
| (P8/8, P4/3)
|-
! beta-tricot
| (P8, P4/3)
| (P8/2, P11/3)
| (P8/3, M6/9)
| (P8/4, P4/3)
| (P8/5, P11/3)
| (P8/6, m3/9)
| (P8/7, P4/3)
| (P8/8, P11/3)
|-
! rowspan="4" | Tetracots
! tetracot
| (P8, P5/4)
| (P8/2, P5/4)
| (P8/3, P5/4)
| (P8/4, P4/4)
| (P8/5, P5/4)
| (P8/6, P5/4)
| (P8/7, P5/4)
| (P8/8, P4/4)
|-
! alpha-tetracot
| (P8, P12/4)
| (P8/2, cm7/8)
| (P8/3, P4/4)
| (P8/4, m6/16)
| (P8/5, P12/4)
| (P8/6, M2/8)
| (P8/7, P4/4)
| (P8/8, A12/32)
|-
! beta-tetracot
| (P8, P11/4)
| (P8/2, P4/4)
| (P8/3, P11/4)
| (P8/4, M2/8)
| (P8/5, P11/4)
| (P8/6, P4/4)
| (P8/7, P11/4)
| (P8/8, m6/16)
|-
! gamma-tetracot
| (P8, P4/4)
| (P8/2, M2/8)
| (P8/3, P12/4)
| (P8/4, M10/16)
| (P8/5, P4/4)
| (P8/6, cm7/8)
| (P8/7, P12/4)
| (P8/8, d4/32)
|-
! rowspan="5" | Pentacots
! pentacot
| (P8, P5/5)
| (P8/2, P5/5)
| (P8/3, P5/5)
| (P8/4, P5/5)
| (P8/5, P4/5)
| (P8/6, P5/5)
| (P8/7, P5/5)
| (P8/8, P5/5)
|-
! alpha-pentacot
| (P8, P12/5)
| (P8/2, P11/5)
| (P8/3, ccP4/5)
| (P8/4, P4/5)
| (P8/5, m9/25)
| (P8/6, P12/5)
| (P8/7, P11/5)
| (P8/8, ccP4/5)
|-
! beta-pentacot
| (P8, ccP4/5)
| (P8/2, P12/5)
| (P8/3, P4/5)
| (P8/4, P11/5)
| (P8/5, m2/25)
| (P8/6, ccP4/5)
| (P8/7, P12/5)
| (P8/8, P4/5)
|-
! gamma-pentacot
| (P8, P11/5)
| (P8/2, P4/5)
| (P8/3, P12/5)
| (P8/4, ccP4/5)
| (P8/5, M7/25)
| (P8/6, P11/5)
| (P8/7, P4/5)
| (P8/8, P12/5)
|-
! delta-pentacot
| (P8, P4/5)
| (P8/2, ccP4/5)
| (P8/3, P11/5)
| (P8/4, P12/5)
| (P8/5, cM7/25)
| (P8/6, P4/5)
| (P8/7, ccP4/5)
| (P8/8, P11/5)
|-
! rowspan="6" | Hexacots
! hexacot
| (P8, P5/6)
| (P8/2, P5/6)
| (P8/3, P5/6)
| (P8/4, P5/6)
| (P8/5, P5/6)
| (P8/6, P4/6)
| (P8/7, P5/6)
| (P8/8, P5/6)
|-
! alpha-hexacot
| (P8, P12/6)
| (P8/2, ccM2/12)
| (P8/3, ccM6/18)
| (P8/4, ccm6/24)
| (P8/5, P4/6)
| (P8/6, d12/36)
| (P8/7, P12/6)
| (P8/8, d4/48)
|-
! beta-hexacot
| (P8, ccP5/6)
| (P8/2, P11/6)
| (P8/3, ccm3/18)
| (P8/4, P4/6)
| (P8/5, P11/6)
| (P8/6, m3/18)
| (P8/7, ccP5/6)
| (P8/8, P11/6)
|-
! gamma-hexacot
| (P8, ccP4/6)
| (P8/2, cm7/12)
| (P8/3, P4/6)
| (P8/4, m6/24)
| (P8/5, ccP4/6)
| (P8/6, M2/12)
| (P8/7, ccP4/6)
| (P8/8, A12/48)
|-
! delta-hexacot
| (P8, P11/6)
| (P8/2, P4/6)
| (P8/3, m3/18)
| (P8/4, P11/6)
| (P8/5, ccP5/6)
| (P8/6, M6/18)
| (P8/7, P11/6)
| (P8/8, P4/6)
|-
! epsilon-hexacot
| (P8, P4/6)
| (P8/2, M2/12)
| (P8/3, M6/18)
| (P8/4, M10/24)
| (P8/5, P12/6)
| (P8/6, ccA4/36)
| (P8/7, P4/6)
| (P8/8, ccd4/48)
|-
! rowspan="7" | Heptacots
! heptacot
| (P8, P5/7)
| (P8/2, P5/7)
| (P8/3, P5/7)
| (P8/4, P5/7)
| (P8/5, P5/7)
| (P8/6, P5/7)
| (P8/7, P4/7)
| (P8/8, P5/7)
|-
! alpha-heptacot
| (P8, P12/7)
| (P8/2, ccP4/7)
| (P8/3, P11/7)
| (P8/4, ccP5/7)
| (P8/5, c<sup>3</sup>P4/7)
| (P8/6, P4/7)
| (P8/7, cd8/49)
| (P8/8, P12/7)
|-
! beta-heptacot
| (P8, ccP5/7)
| (P8/2, P12/7)
| (P8/3, c<sup>3</sup>P4/7)
| (P8/4, ccP4/7)
| (P8/5, P4/7)
| (P8/6, P11/7)
| (P8/7, d8/49)
| (P8/8, ccP5/7)
|-
! gamma-heptacot
| (P8, c<sup>3</sup>P4/7)
| (P8/2, P11/7)
| (P8/3, P12/7)
| (P8/4, P4/7)
| (P8/5, ccP5/7)
| (P8/6, ccP4/7)
| (P8/7, A1/49)
| (P8/8, c<sup>3</sup>P4/7)
|-
! delta-heptacot
| (P8, ccP4/7)
| (P8/2, ccP5/7)
| (P8/3, P4/7)
| (P8/4, P12/7)
| (P8/5, P11/7)
| (P8/6, c<sup>3</sup>P4/7)
| (P8/7, A8/49)
| (P8/8, ccP4/7)
|-
! epsilon-heptacot
| (P8, P11/7)
| (P8/2, P4/7)
| (P8/3, ccP4/7)
| (P8/4, c<sup>3</sup>P4/7)
| (P8/5, P12/7)
| (P8/6, ccP5/7)
| (P8/7, ccA1/49)
| (P8/8, P11/7)
|-
! wau-heptacot
| (P8, P4/7)
| (P8/2, c<sup>3</sup>P4/7)
| (P8/3, ccP5/7)
| (P8/4, P11/7)
| (P8/5, ccP4/7)
| (P8/6, P12/7)
| (P8/7, c<sup>3</sup>A1/49)
| (P8/8, P4/7)
|-
! rowspan="8" | Octacots
! octacot
| (P8, P5/8)
| (P8/2, P5/8)
| (P8/3, P5/8)
| (P8/4, P5/8)
| (P8/5, P5/8)
| (P8/6, P5/8)
| (P8/7, P5/8)
| (P8/8, P4/8)
|-
! alpha-octacot
| (P8, P12/8)
| (P8/2, ccM2/16)
| (P8/3, c<sup>3</sup>P5/8)
| (P8/4, c<sup>3</sup>M3/32)
| (P8/5, ccP4/8)
| (P8/6, c<sup>3</sup>m7/16)
| (P8/7, P4/8)
| (P8/8, ccd4/64)
|-
! beta-octacot
| (P8, ccP5/8)
| (P8/2, P12/8)
| (P8/3, P11/8)
| (P8/4, cm7/16)
| (P8/5, ccP5/8)
| (P8/6, P4/8)
| (P8/7, P11/8)
| (P8/8, m6/32)
|-
! gamma-octacot
| (P8, c<sup>3</sup>P5/8)
| (P8/2, c<sup>3</sup>m7/16)
| (P8/3, P12/8)
| (P8/4, ccm6/32)
| (P8/5, P4/8)
| (P8/6, ccM2/16)
| (P8/7, ccP4/8)
| (P8/8, d4/64)
|-
! delta-octacot
| (P8, c<sup>3</sup>P4/8)
| (P8/2, P11/8)
| (P8/3, c<sup>3</sup>P4/8)
| (P8/4, P4/8)
| (P8/5, c<sup>3</sup>P4/8)
| (P8/6, P11/8)
| (P8/7, c<sup>3</sup>P4/8)
| (P8/8, M2/16)
|-
! epsilon-octacot
| (P8, ccP4/8)
| (P8/2, cm7/16)
| (P8/3, P4/8)
| (P8/4, m6/32)
| (P8/5, P12/8)
| (P8/6, M2/16)
| (P8/7, c<sup>3</sup>P5/8)
| (P8/8, A12/64)
|-
! wau-octacot
| (P8, P11/8)
| (P8/2, P4/8)
| (P8/3, ccP5/8)
| (P8/4, M2/16)
| (P8/5, P11/8)
| (P8/6, P12/8)
| (P8/7, ccP5/8)
| (P8/8, M10/32)
|-
! zeta-octacot
| (P8, P4/8)
| (P8/2, M2/16)
| (P8/3, ccP4/8)
| (P8/4, M10/16)
| (P8/5, c<sup>3</sup>P5/8)
| (P8/6, cm7/16)
| (P8/7, P12/8)
| (P8/8, c<sup>3</sup>A5/64)
|}
== Notation ==
While there are no agreed-upon notation system for many ploidacots, some of which can be notated by [[Kite's ups and downs notation|ups and downs notation system]]. For example, [[Ploidacot/Tricot|tricot]] is based on interpreting the generator as a supermajor second, allowing for an ^ or v to stand for 1/3 of a diatonic semitone, and [[Ploidacot/Tetracot|tetracot]] is based on interpreting the generator as a submajor second, allowing for an ^ or v to stand for 1/4 of a chromatic semitone, and [[Ploidacot/Triploid monocot|triploid monocot]] is based on interpreting the period as a submajor third, allowing for an ^ or v to stand for 1/3 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]). Certain ploidacots (such as [[Ploidacot/Diploid dicot|diploid dicot]]) require another additional pair, such as lifts and drops, written / and ⧵ .


== Examples ==
== Examples ==
Line 125: Line 524:
* [[Whitewood]] is heptaploid acot
* [[Whitewood]] is heptaploid acot
* [[Compton]] is dodecaploid acot
* [[Compton]] is dodecaploid acot
== Notation ==
: ''TODO: Come up with canonical ups and downs notation systems for pergen squares''


== List of ploidacots ==
== List of ploidacots ==
Line 188: Line 584:
* [[Ploidacot/Gamma-heptacot|Gamma-heptacot]]
* [[Ploidacot/Gamma-heptacot|Gamma-heptacot]]
* [[Ploidacot/Delta-heptacot|Delta-heptacot]]
* [[Ploidacot/Delta-heptacot|Delta-heptacot]]
* [[Ploidacot/Epsilon-heptacot|Epsilon-heptacot]]
* [[Ploidacot/Omega-heptacot|Omega-heptacot]]


=== Octacot ===
=== Octacot ===
Line 201: Line 599:


=== Decacot ===
=== Decacot ===
* [[Ploidacot/Decacot|Decacot]]
* [[Ploidacot/Beta-decacot|Beta-decacot]]
* [[Ploidacot/Beta-decacot|Beta-decacot]]
* [[Ploidacot/Epsilon-decacot|Epsilon-decacot]]
* [[Ploidacot/Epsilon-decacot|Epsilon-decacot]]
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