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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 [[pergen #Pergen squares|pergen squares]] in that every pergen square that covers both 2/1 and 3/2 has a unique name in the ploidacot system.
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 ploidacot system was developed by [[Praveen Venkataramana]].
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.


== 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.
Any rank-2 temperament must split the octave into a number of '''ploids''', for instance pajara divides the octave into two so it's diploid. Temperaments that don't divide the octave are called haploid.


== Cots ==
== Specification ==
If [[3/2]] is represented by a linearly independent element to the ploid, there is a number of ploids which when added to 3/2 gives the interval which splits into the largest number of parts by the temperament. Each of these parts is called a '''cot''' or '''cotyledon''' and the ploidacot system uses one or more Greek letters to describe the smallest number of ploids should be added to 3/2 to form a whole number of cots, and this number of cots is indicated by a Greek numerical prefix.
=== 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.
 
=== Cots ===
If [[3/2]] is represented by a linearly independent element to the ploid, there is a number of ploids which when added to 3/2 gives the interval which is split into the largest number of parts, namely [[generator]]s, by the temperament. Each of these parts is called a '''cot''' or '''cotyledon'''. The ploidacot system uses Greek letters (alpha-, beta-, etc.) to describe the smallest nonnegative number of ploids that should be added to 3/2 to form a whole number of cots. If the number is zero, it is left empty. The number of cots is then indicated by a Greek numeral prefix. Temperaments that do not divide the fifth are called ''monocot'' (''not'' *haplocot). The full specification of cots is thus a (possibly empty) Greek letter prefix, followed by a Greek numeral prefix, and -cot.  


Temperaments where the image of 3/2 is a whole number of ploids are called '''acot'''.
Temperaments where the image of 3/2 is a whole number of ploids are called '''acot'''.


=== Greek letter prefixes ===
==== Greek letter prefixes ====
The Greek letter prefixes follow the ancient gematria/isopsephic system, detailed below:
The Greek letter prefixes follow the ancient gematria/isopsephic system, detailed below:
{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Greek letter prefixes in ploidacot
|+ style="font-size: 105%;" | Greek letter prefixes in ploidacot
Line 34: Line 38:
| delta
| delta
| epsilon
| epsilon
| digamma
| digamma/wau
| zeta
| zeta
| eta
| eta
Line 50: Line 54:
| qoppa
| qoppa
|-
|-
! style="white-space: nowrap;" | ''n'' + 10
! {{nowrap|''n'' + 10}}
| iota-alpha
| iota-alpha
| iota-beta
| iota-beta
Line 56: Line 60:
| iota-delta
| iota-delta
| iota-epsilon
| iota-epsilon
| iota-digamma
| iota-digamma/iota-wau
| iota-zeta
| iota-zeta
| iota-eta
| iota-eta
Line 66: Line 70:
Alternatively, Arabic numerals may be used in place of the Greek alphabetical and numeric prefixes, with the word "sheared" or its equivalent in other languages used in place of the alphabetic prefixes, so a diploid epsilon-heptacot system may be referred to as a 2-ploid 5-sheared 7-cot system.
Alternatively, Arabic numerals may be used in place of the Greek alphabetical and numeric prefixes, with the word "sheared" or its equivalent in other languages used in place of the alphabetic prefixes, so a diploid epsilon-heptacot system may be referred to as a 2-ploid 5-sheared 7-cot system.


== Examples ==
== Properties ==
* [[Meantone]] and [[schismic]] are haploid monocot
* For ''n''-cot systems there are exactly ''n'' settings of shear, or number of ploids to add to the step that represents the interval class of 3. The possible values of shear are 0, 1, 2, …, {{nowrap|(''n'' − 1)}}. For example, the tricot systems are tricot (0-sheared), alpha-tricot (1-sheared), and beta-tricot (2-sheared). There is not a *gamma-tricot since that would be equivalent to tricot.
* [[Mohajira]] and [[Dicot family|dicot]] are dicot
 
* [[Bug]] and [[semaphore]] are alpha-dicot
== Extensions ==
* [[Shrutar]] is diploid alpha-dicot
=== Omega extension ===
* [[Ennealimmal]] is enneaploid dicot
The Greek letter omega, proposed by [[User:Godtone|Godtone]], is used for −1. ("Contra" has also been used in place of omega.) This simplifies the classification of certain temperaments, e.g. porcupine, which instead of beta-tricot can be omega-tricot, as splitting the interval 4/3 into three is arguably more intuitive than splitting the interval 6. This effectively shifts the possible values of shear to -1, 0, 1, …, {{nowrap|(''n'' − 2)}} if ''n'' ≥ 3.
* [[Hemiennealimmal]] is octodecaploid (18-ploid) dicot
 
* [[Slendric]], [[Gamelismic clan|mothra]], and [[rodan]] are tricot
Note that omega should only be used with {{nowrap| ''n'' ≥ 3 }}. When {{nowrap| ''n'' {{=}} 1 }}, there is only monocot. When {{nowrap| ''n'' {{=}} 2 }}, alpha-dicot is preferred over omega-dicot. Omega-based names are also not preferred when dealing with temperaments that split the octave, as they may be confusing - for instance, diploid alpha-tricot splits 4/3 in three while diploid beta-tricot splits 3/1 in three.
* [[Tricot]] is alpha-tricot
 
* [[Porcupine]] is beta-tricot
=== No-twos or no-threes temperaments ===
* [[Hedgehog]] is diploid alpha-tricot
The ploidacot system, similarly to [[pergen]]s, relies on the presence of a [[3-limit]], i.e. 2.3 subgroup, spine, but its defining principles can be easily applied to a 2.5, 3.5, 3.7, etc. spine instead, and in the case of ploidacot, the "cot" suffix is simply replaced with a different suffix indicating the family of intervals being cloven. The existing extensions are "seph" for [[5/4]] with octave equivalence, and "gem" for [[7/3]] with tritave equivalence (note that 3.7 is preferred over 3.5 since [[9/7]] and 7/3 generate a much more commonly used structure in tritave systems, i.e. [[4L 5s (3/1-equivalent)|Lambda]], than [[5/3]] and [[9/5]]).
* [[Tetracot]] is tetracot
 
* [[Squares]] is beta-tetracot
For instance, in the 2.5.7 subgroup, [[didacus]] can be labeled as "diseph", because its generator divides 5/4 in two, and [[llywelyn]] can be labeled as "alpha-heptaseph" because seven generators make up [[5/2]]. In the tritave world, [[BPS]] (3.5.7) is "monogem" as its generator is 9/7, while [[mintaka]] (3.7.11) is alpha-trigem as its generator (of ~[[21/11]]) splits [[7/1]] in three.
* [[Bleu]] is pentacot
 
* [[Magic]] is alpha-pentacot
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).
* [[Amity]] is gamma-pentacot
 
* [[Miracle]] is hexacot
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]].
* [[Hanson]] is alpha-hexacot
 
* [[Harry]] is diploid delta-hexacot
== Origin ==
* [[Orwell]] is alpha-heptacot
The ploidacot system was developed by [[Praveen Venkataramana]], based on the pattern of certain individual temperament names: ''dicot'', ''tricot'' (now ''alphatricot''), and ''tetracot''.
* [[Sensi]] is beta-heptacot
 
* [[Vishnu]] is diploid epsilon-heptacot
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.
* [[Tetracot family|Octacot]] is octacot
 
* [[Würschmidt family|Würschmidt]] is beta-octacot
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.
* [[Valentine]] is enneacot
 
* [[Sycamore family|Sycamore]] is hendecacot
=== Relationship to pergens ===
* [[High badness temperaments#Chromo|Chromo]] is tridecacot
Each ploidacot has one pergen. The numbers of ploid (''p''), shear (''s''), and cot (''c'') are given, its pergen form has following features:
* [[Pajara]] and [[Meantone family#Injera|injera]] are diploid
* Every ''p''-ploid has a form of (P8/''p'', X).
* [[Very low accuracy temperaments#Antitonic|Antitonic]] is diploid acot
* Haploids (''p'' = 1) are of the form (P8, X/''c'') since the octave is unsplit.
* [[Augene]] is triploid
* Monocots (''c'' = 1) are of the form (P8/''p'', P5) since the fifth and its compounds are unsplit.
* [[Diminished]] is tetraploid
* If ''p'' and ''c'' are coprime, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* [[Blackwood]] is pentaploid acot
* If ''s'' mod GCD(''p'', ''c'') = 0, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* [[Whitewood family|Whitewood]] is heptaploid acot
* If ''s'' mod GCD(''p'', ''c'') is not 0, the ploidacot has an imperfect pergen.
* [[Ennealimmal]] is enneaploid
 
* [[Compton]] is dodecaploid acot
{| 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)
|}
 
== Relationship to 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]] or [[Stein–Zimmermann–Gould notation]]. 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 ⧵ .
 
== List of ploidacots ==
=== Acot ===
* Pentaploid acot ([[blackwood]], [[5edo]])
* Heptaploid acot ([[whitewood]], [[7edo]])
* Dodecaploid acot ([[compton]], [[12edo]])
 
=== Monocot ===
* [[Ploidacot/Monocot|Monocot]]
* [[Ploidacot/Diploid monocot|Diploid monocot]]
* [[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 ===
* [[Ploidacot/Dicot|Dicot]]
* [[Ploidacot/Alpha-dicot|Alpha-dicot]]
* [[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 ===
* [[Ploidacot/Tricot|Tricot]]
* [[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 ===
* [[Ploidacot/Tetracot|Tetracot]]
* [[Ploidacot/Alpha-tetracot|Alpha-tetracot]]
* [[Ploidacot/Beta-tetracot|Beta-tetracot]]
* [[Ploidacot/Omega-tetracot|Omega-tetracot]]
 
=== Pentacot ===
* [[Ploidacot/Pentacot|Pentacot]]
* [[Ploidacot/Alpha-pentacot|Alpha-pentacot]]
* [[Ploidacot/Beta-pentacot|Beta-pentacot]]
* [[Ploidacot/Gamma-pentacot|Gamma-pentacot]]
* [[Ploidacot/Omega-pentacot|Omega-pentacot]]
 
=== Hexacot ===
* [[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 ===
* [[Ploidacot/Heptacot|Heptacot]]
* [[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]]
 
=== Octacot ===
* [[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]]


== Notation ==
=== >10 cots ===
: ''TODO: Come up with canonical ups and downs notation systems for pergen squares''
* [[Ploidacot/Hendecacot|Hendecacot]]
* [[Ploidacot/Icosacot|Icosacot]]


[[Category:Temperament naming]]
[[Category:Temperament naming]]

Latest revision as of 08:44, 8 August 2026

The ploidacot system is a classification of rank-2 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 "ploid" number of a temperament refers to how many equal parts, or periods 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 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 ¢ each). Pajara is diploid monocot, since it is generated by the fifth and splits the octave in two 600 ¢ halves. Shrutar is diploid alpha-dicot, since it splits the octave in half, and splits the interval 600 ¢ above 3/2 (~1300 ¢) into two ~650 ¢ halves. Note that in shrutar the interval one ploid above 3/2 is ~1300 ¢ and not 3/1, since the octave is split into two 600 ¢ ploids.

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 periods, 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

If 3/2 is represented by a linearly independent element to the ploid, there is a number of ploids which when added to 3/2 gives the interval which is split into the largest number of parts, namely generators, by the temperament. Each of these parts is called a cot or cotyledon. The ploidacot system uses Greek letters (alpha-, beta-, etc.) to describe the smallest nonnegative number of ploids that should be added to 3/2 to form a whole number of cots. If the number is zero, it is left empty. The number of cots is then indicated by a Greek numeral prefix. Temperaments that do not divide the fifth are called monocot (not *haplocot). The full specification of cots is thus a (possibly empty) Greek letter prefix, followed by a Greek numeral prefix, and -cot.

Temperaments where the image of 3/2 is a whole number of ploids are called acot.

Greek letter prefixes

The Greek letter prefixes follow the ancient gematria/isopsephic system, detailed below:

Greek letter prefixes in ploidacot
Number n 1 2 3 4 5 6 7 8 9
Prefix n alpha beta gamma delta epsilon digamma/wau zeta eta theta
10n iota kappa lambda mu nu xi omicron pi qoppa
n + 10 iota-alpha iota-beta iota-gamma iota-delta iota-epsilon iota-digamma/iota-wau iota-zeta iota-eta iota-theta

Prefixes for numbers between 21 and 99 are constructed the same way as number words in English, for instance 21 is kappa-alpha and 99 is qoppa-theta.

Alternatively, Arabic numerals may be used in place of the Greek alphabetical and numeric prefixes, with the word "sheared" or its equivalent in other languages used in place of the alphabetic prefixes, so a diploid epsilon-heptacot system may be referred to as a 2-ploid 5-sheared 7-cot system.

Properties

  • For n-cot systems there are exactly n settings of shear, or number of ploids to add to the step that represents the interval class of 3. The possible values of shear are 0, 1, 2, …, (n − 1). For example, the tricot systems are tricot (0-sheared), alpha-tricot (1-sheared), and beta-tricot (2-sheared). There is not a *gamma-tricot since that would be equivalent to tricot.

Extensions

Omega extension

The Greek letter omega, proposed by Godtone, is used for −1. ("Contra" has also been used in place of omega.) This simplifies the classification of certain temperaments, e.g. porcupine, which instead of beta-tricot can be omega-tricot, as splitting the interval 4/3 into three is arguably more intuitive than splitting the interval 6. This effectively shifts the possible values of shear to -1, 0, 1, …, (n − 2) if n ≥ 3.

Note that omega should only be used with n ≥ 3. When n = 1, there is only monocot. When n = 2, alpha-dicot is preferred over omega-dicot. Omega-based names are also not preferred when dealing with temperaments that split the octave, as they may be confusing - for instance, diploid alpha-tricot splits 4/3 in three while diploid beta-tricot splits 3/1 in three.

No-twos or no-threes temperaments

The ploidacot system, similarly to pergens, relies on the presence of a 3-limit, i.e. 2.3 subgroup, spine, but its defining principles can be easily applied to a 2.5, 3.5, 3.7, etc. spine instead, and in the case of ploidacot, the "cot" suffix is simply replaced with a different suffix indicating the family of intervals being cloven. The existing extensions are "seph" for 5/4 with octave equivalence, and "gem" for 7/3 with tritave equivalence (note that 3.7 is preferred over 3.5 since 9/7 and 7/3 generate a much more commonly used structure in tritave systems, i.e. Lambda, than 5/3 and 9/5).

For instance, in the 2.5.7 subgroup, didacus can be labeled as "diseph", because its generator divides 5/4 in two, and llywelyn can be labeled as "alpha-heptaseph" because seven generators make up 5/2. In the tritave world, BPS (3.5.7) is "monogem" as its generator is 9/7, while mintaka (3.7.11) is alpha-trigem as its generator (of ~21/11) splits 7/1 in three.

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.

Origin

The ploidacot system was developed by Praveen Venkataramana, based on the pattern of certain individual temperament names: dicot, tricot (now alphatricot), and tetracot.

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.(…) subgroups in that every such pergen corresponds to a unique name in the ploidacot system.

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.
Pergen forms of ploidacot
Ploids Haploid Diploid Triploid Tetraploid Pentaploid Hexaploid Heptaploid Octaploid
Monocot (P8, P5) (P8/2, P5) (P8/3, P5) (P8/4, P5) (P8/5, P5) (P8/6, P5) (P8/7, P5) (P8/8, P5)
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)
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)
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)
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)
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)
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, c3P4/7) (P8/6, P4/7) (P8/7, cd8/49) (P8/8, P12/7)
beta-heptacot (P8, ccP5/7) (P8/2, P12/7) (P8/3, c3P4/7) (P8/4, ccP4/7) (P8/5, P4/7) (P8/6, P11/7) (P8/7, d8/49) (P8/8, ccP5/7)
gamma-heptacot (P8, c3P4/7) (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, c3P4/7)
delta-heptacot (P8, ccP4/7) (P8/2, ccP5/7) (P8/3, P4/7) (P8/4, P12/7) (P8/5, P11/7) (P8/6, c3P4/7) (P8/7, A8/49) (P8/8, ccP4/7)
epsilon-heptacot (P8, P11/7) (P8/2, P4/7) (P8/3, ccP4/7) (P8/4, c3P4/7) (P8/5, P12/7) (P8/6, ccP5/7) (P8/7, ccA1/49) (P8/8, P11/7)
wau-heptacot (P8, P4/7) (P8/2, c3P4/7) (P8/3, ccP5/7) (P8/4, P11/7) (P8/5, ccP4/7) (P8/6, P12/7) (P8/7, c3A1/49) (P8/8, P4/7)
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, c3P5/8) (P8/4, c3M3/32) (P8/5, ccP4/8) (P8/6, c3m7/16) (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, c3P5/8) (P8/2, c3m7/16) (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, c3P4/8) (P8/2, P11/8) (P8/3, c3P4/8) (P8/4, P4/8) (P8/5, c3P4/8) (P8/6, P11/8) (P8/7, c3P4/8) (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, c3P5/8) (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, c3P5/8) (P8/6, cm7/16) (P8/7, P12/8) (P8/8, c3A5/64)

Relationship to 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 or Stein–Zimmermann–Gould notation. For example, 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 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 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 diploid dicot) require another additional pair, such as lifts and drops, written / and ⧵ .

List of ploidacots

Acot

Monocot

Dicot

Tricot

Tetracot

Pentacot

Hexacot

Heptacot

Octacot

Enneacot

Decacot

>10 cots

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