Ploidacot: Difference between revisions

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== Specification ==

=== Ploids ===

Any rank-2 temperament of the 2.3.… subgroup has an octave, and it ~~must~~ 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, 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 ===

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. The number of cots is indicated by a Greek numeral prefix. The full specification of cots is thus a Greek letter prefix ~~(alpha-, beta-, etc.)~~, followed by a Greek numeral prefix ~~(mono-, di-, etc.)~~, and -cot.

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'''.

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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, …, {{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.

== Extensions ==

=== Omega extension ===

The Greek letter omega, proposed by [[User:Godtone|Godtone]], is used for −1. 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.

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.

=== Non-octave temperaments ===

== Examples ==

* [[Meantone]] and [[~~schismic~~]] are haploid monocot

The ploidacots of most common temperaments can be intuitively derived from a basic understanding of its mapping. [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are monocot since they have a period of a whole octave and are generated by the perfect fifth. Dicot is dicot since it has a period of a whole octave and splits the perfect fifth in two. Semaphore has a period of a whole octave and splits the perfect twelfth in two. It requires one period to add to the fifth to make it a twelfth, and one is alpha. So it is alpha-dicot.

* [[Mohajira]] and [[~~Dicot family|~~dicot]] are dicot

For a more complex example, let us consider sensi and its weak extension bison. Sensi splits 6/1 in seven. It requires two periods to the fifth to reach 6/1, and two is beta. So it is beta-heptacot. Bison splits the period of sensi in two. As a result, it now requires four periods to the fifth to reach 6/1, and four is delta. So it is diploid delta-heptacot.

Below is a list of ploidacots for common temperaments

* [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are haploid monocot

* [[Mohajira]] and [[dicot]] are dicot

* [[Bug]] and [[semaphore]] are alpha-dicot

* [[Shrutar]] is diploid alpha-dicot

* [[Ennealimmal]] is enneaploid dicot

* [[Hemiennealimmal]] is octodecaploid (18-ploid) dicot

* [[Slendric]], [[~~Gamelismic clan|~~mothra]], and [[rodan]] are tricot

* [[Slendric]], [[mothra]], and [[rodan]] are tricot

* [[~~Tricot~~]] is alpha-tricot

* [[Alphatricot]] is alpha-tricot

* [[Porcupine]] is beta-tricot

* [[Hedgehog]] is diploid alpha-tricot

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* [[Sensi]] is beta-heptacot

* [[Vishnu]] is diploid epsilon-heptacot

* [[~~Tetracot family|~~Octacot]] is octacot

* [[Octacot]] is octacot

* [[~~Würschmidt family|~~Würschmidt]] is beta-octacot

* [[Würschmidt]] is beta-octacot

* [[Valentine]] is enneacot

* [[~~Sycamore family|~~Sycamore]] is hendecacot

* [[Sycamore]] is hendecacot

* [[~~High badness temperaments#Chromo|~~Chromo]] is tridecacot

* [[Chromo]] is tridecacot

* [[Pajara]] and [[~~Meantone family#Injera|~~injera]] are diploid

* [[Pajara]] and [[injera]] are diploid

* [[~~Very low accuracy temperaments#Antitonic|~~Antitonic]] is diploid acot

* [[Antitonic]] is diploid acot

* [[Augene]] is triploid

* [[Diminished]] is tetraploid

* [[Blackwood]] is pentaploid acot

* [[~~Whitewood family|~~Whitewood]] is heptaploid acot

* [[Whitewood]] is heptaploid acot

* [[Ennealimmal]] is enneaploid

* [[Compton]] is dodecaploid acot

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== Notation ==

: ''TODO: Come up with canonical ups and downs notation systems for pergen squares''

== 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]]

=== Dicot ===

* [[Ploidacot/Dicot|Dicot]]

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

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

=== Tricot ===

* [[Ploidacot/Tricot|Tricot]]

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

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

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

=== Hexacot ===

* [[Ploidacot/Hexacot|Hexacot]]

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

=== Heptacot ===

* [[Ploidacot/Heptacot|Heptacot]]

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

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

=== Octacot ===

* [[Ploidacot/Octacot|Octacot]]

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

=== Enneacot ===

* [[Ploidacot/Enneacot|Enneacot]]

=== >10 cots ===

* [[Ploidacot/Icosacot|Icosacot]]

== See also ==

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

[[Category:Temperament naming]]

@@ Line 5: / Line 5: @@
 == Specification ==
 === Ploids ===
-Any rank-2 temperament of the 2.3.… subgroup has an octave, and it must 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, 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 ===
-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. The number of cots is indicated by a Greek numeral prefix. The full specification of cots is thus a Greek letter prefix (alpha-, beta-, etc.), followed by a Greek numeral prefix (mono-, di-, etc.), and -cot.
+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'''.
@@ Line 67: / Line 67: @@
 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, …, {{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.
+== Extensions ==
+=== Omega extension ===
+The Greek letter omega, proposed by [[User:Godtone|Godtone]], is used for −1. 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.
+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.
+=== Non-octave temperaments ===
+{{Todo|inline=1| expand }}
 == Examples ==
-* [[Meantone]] and [[schismic]] are haploid monocot
+The ploidacots of most common temperaments can be intuitively derived from a basic understanding of its mapping. [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are monocot since they have a period of a whole octave and are generated by the perfect fifth. Dicot is dicot since it has a period of a whole octave and splits the perfect fifth in two. Semaphore has a period of a whole octave and splits the perfect twelfth in two. It requires one period to add to the fifth to make it a twelfth, and one is alpha. So it is alpha-dicot.
-* [[Mohajira]] and [[Dicot family|dicot]] are dicot
+For a more complex example, let us consider sensi and its weak extension bison. Sensi splits 6/1 in seven. It requires two periods to the fifth to reach 6/1, and two is beta. So it is beta-heptacot. Bison splits the period of sensi in two. As a result, it now requires four periods to the fifth to reach 6/1, and four is delta. So it is diploid delta-heptacot.
+Below is a list of ploidacots for common temperaments
+* [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are haploid monocot
+* [[Mohajira]] and [[dicot]] are dicot
 * [[Bug]] and [[semaphore]] are alpha-dicot
 * [[Shrutar]] is diploid alpha-dicot
 * [[Ennealimmal]] is enneaploid dicot
 * [[Hemiennealimmal]] is octodecaploid (18-ploid) dicot
-* [[Slendric]], [[Gamelismic clan|mothra]], and [[rodan]] are tricot
+* [[Slendric]], [[mothra]], and [[rodan]] are tricot
-* [[Tricot]] is alpha-tricot
+* [[Alphatricot]] is alpha-tricot
 * [[Porcupine]] is beta-tricot
 * [[Hedgehog]] is diploid alpha-tricot
@@ Line 90: / Line 107: @@
 * [[Sensi]] is beta-heptacot
 * [[Vishnu]] is diploid epsilon-heptacot
-* [[Tetracot family|Octacot]] is octacot
+* [[Octacot]] is octacot
-* [[Würschmidt family|Würschmidt]] is beta-octacot
+* [[Würschmidt]] is beta-octacot
 * [[Valentine]] is enneacot
-* [[Sycamore family|Sycamore]] is hendecacot
+* [[Sycamore]] is hendecacot
-* [[High badness temperaments#Chromo|Chromo]] is tridecacot
+* [[Chromo]] is tridecacot
-* [[Pajara]] and [[Meantone family#Injera|injera]] are diploid
+* [[Pajara]] and [[injera]] are diploid
-* [[Very low accuracy temperaments#Antitonic|Antitonic]] is diploid acot
+* [[Antitonic]] is diploid acot
 * [[Augene]] is triploid
 * [[Diminished]] is tetraploid
 * [[Blackwood]] is pentaploid acot
-* [[Whitewood family|Whitewood]] is heptaploid acot
+* [[Whitewood]] is heptaploid acot
 * [[Ennealimmal]] is enneaploid
 * [[Compton]] is dodecaploid acot
@@ Line 106: / Line 123: @@
 == Notation ==
 : ''TODO: Come up with canonical ups and downs notation systems for pergen squares''
+== 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]]
+=== Dicot ===
+* [[Ploidacot/Dicot|Dicot]]
+* [[Ploidacot/Alpha-dicot|Alpha-dicot]]
+* [[Ploidacot/Diploid dicot|Diploid dicot]]
+=== Tricot ===
+* [[Ploidacot/Tricot|Tricot]]
+* [[Ploidacot/Alpha-tricot|Alpha-tricot]]
+* [[Ploidacot/Diploid alpha-tricot|Diploid alpha-tricot]]
+* [[Ploidacot/Omega-tricot|Omega-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/Gamma-pentacot|Gamma-pentacot]]
+=== Hexacot ===
+* [[Ploidacot/Hexacot|Hexacot]]
+* [[Ploidacot/Alpha-hexacot|Alpha-hexacot]]
+=== Heptacot ===
+* [[Ploidacot/Heptacot|Heptacot]]
+* [[Ploidacot/Alpha-heptacot|Alpha-heptacot]]
+* [[Ploidacot/Beta-heptacot|Beta-heptacot]]
+=== Octacot ===
+* [[Ploidacot/Octacot|Octacot]]
+* [[Ploidacot/Beta-octacot|Beta-octacot]]
+=== Enneacot ===
+* [[Ploidacot/Enneacot|Enneacot]]
+=== >10 cots ===
+* [[Ploidacot/Icosacot|Icosacot]]
+== See also ==
+* [[Wedgie]] - a mathematical generalization of the concept of ploidacots that uniquely characterizes a temperament
 [[Category:Temperament naming]]