Xenharmonic Wiki

Ploidacot: Difference between revisions

← Older edit
VisualWikitext
Better integration of the special case
→Relationship to pergens: cut the table down to 8, as that's where it occupies full width in the Vector skin
 
(One intermediate revision by one other user not shown)
Line 91: Line 91:


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 131: 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 ==
Retrieved from "https://en.xen.wiki/w/Ploidacot"