Ploidacot/Triploid monocot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=3|Shears=0|Cots=1|Pergen=[P8/3, P5]|Forms=9, 12, 15|Title=Triploid monocot|Wedgie=3}} | {{Breadcrumb}}{{Infobox ploidacot|Ploids=3|Shears=0|Cots=1|Pergen=[P8/3, P5]|Forms=9, 12, 15, 27|Title=Triploid monocot|Wedgie=3}} | ||
'''Triploid monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth and the period is 1/3 of a [[2/1]] octave, or 400{{c}}. The generator can also be characterized as a perfect fourth [[4/3]], or as a "perfect semitone" <math>\frac{2\sqrt[3]{4}}{3}</math>. Triploid monocot temperaments usually generate the [[3L 6s]] MOS structure and either [[3L 9s]] (and thus [[12L 3s]]) or [[9L 3s]] as children. | '''Triploid monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth and the period is 1/3 of a [[2/1]] octave, or 400{{c}}. The generator can also be characterized as a perfect fourth [[4/3]], or as a "perfect semitone" <math>\frac{2\sqrt[3]{4}}{3}</math>. Triploid monocot temperaments usually generate the [[3L 6s]] MOS structure and either [[3L 9s]] (and thus [[12L 3s]]) or [[9L 3s]] as children. | ||
== | == Intervals and notation == | ||
While there is no agreed-upon notation system for triploid monocot, the following is based on interpreting the generator as a semitone (1/3 of a minor 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]]), so vvC# and ^Db are enharmonic. | |||
While there is no agreed-upon notation system for triploid monocot, the following is based on interpreting the generator as a semitone (1/3 of a minor third), allowing for an ^ or v to stand for | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 16: | Line 14: | ||
|- | |- | ||
! Cents | ! Cents | ||
! Notation | |||
! Name | ! Name | ||
! Cents | |||
! Notation | ! Notation | ||
! Name | |||
! Cents | ! Cents | ||
! Notation | ! Notation | ||
! Name | ! Name | ||
|- | |- | ||
| −6 | | −6 | ||
| 188.27 | | 188.27 | ||
| ^Ebb | |||
| — | | — | ||
| 588.27 | | 588.27 | ||
| Gb | |||
| diminished fifth | | diminished fifth | ||
| 988.27 | | 988.27 | ||
| vBb | |||
| — | | — | ||
|- | |- | ||
| −5 | | −5 | ||
| 90.22 | | 90.22 | ||
| Db | |||
| minor second | | minor second | ||
| 490.22 | | 490.22 | ||
| vF | |||
| — | | — | ||
| 890.22 | | 890.22 | ||
| ^Bbb | |||
| — | | — | ||
|- | |- | ||
| −4 | | −4 | ||
| 392.18 | | 392.18 | ||
| ^Fb | |||
| — | | — | ||
| 792.18 | | 792.18 | ||
| Ab | |||
| minor sixth | | minor sixth | ||
| 1192.18 | | 1192.18 | ||
| vC | |||
| — | | — | ||
|- | |- | ||
| −3 | | −3 | ||
| 294.13 | | 294.13 | ||
| Eb | |||
| minor third | | minor third | ||
| 694.13 | | 694.13 | ||
| vG | |||
| — | | — | ||
| 1094.13 | | 1094.13 | ||
| ^Cb | |||
| — | | — | ||
|- | |- | ||
| −2 | | −2 | ||
| 196.09 | | 196.09 | ||
| vD | |||
| — | | — | ||
| 596.09 | | 596.09 | ||
| ^Gb | |||
| — | | — | ||
| 996.09 | | 996.09 | ||
| Bb | |||
| minor seventh | | minor seventh | ||
|- | |- | ||
| −1 | | −1 | ||
| 98.04 | | 98.04 | ||
| ^Db | |||
| — | | — | ||
| 498.04 | | 498.04 | ||
| F | |||
| perfect fourth | | perfect fourth | ||
| 898.04 | | 898.04 | ||
| vA | |||
| — | | — | ||
|- | |- | ||
| 0 | | 0 | ||
| 0 | | 0 | ||
| C | |||
| unison | | unison | ||
| 400 | | 400 | ||
| vE | |||
| — | | — | ||
| 800 | | 800 | ||
| ^Ab | |||
| — | | — | ||
|- | |- | ||
| 1 | | 1 | ||
| 301.96 | | 301.96 | ||
| ^Eb | |||
| — | | — | ||
| 701.96 | | 701.96 | ||
| G | |||
| perfect fifth | | perfect fifth | ||
| 1101.96 | | 1101.96 | ||
| vB | |||
| — | | — | ||
|- | |- | ||
| 2 | | 2 | ||
| 203.91 | | 203.91 | ||
| D | |||
| major second | | major second | ||
| 603.91 | | 603.91 | ||
| vF# | |||
| — | | — | ||
| 1003.91 | | 1003.91 | ||
| ^Bb | |||
| — | | — | ||
|- | |- | ||
| 3 | | 3 | ||
| 105.87 | | 105.87 | ||
| vC# | |||
| — | | — | ||
| 505.87 | | 505.87 | ||
| ^F | |||
| — | | — | ||
| 905.87 | | 905.87 | ||
| A | |||
| major sixth | | major sixth | ||
|- | |- | ||
| 4 | | 4 | ||
| 7.82 | | 7.82 | ||
| ^C | |||
| — | | — | ||
| 407.82 | | 407.82 | ||
| E | |||
| major third | | major third | ||
| 807.82 | | 807.82 | ||
| vG# | |||
| — | | — | ||
|- | |- | ||
| 5 | | 5 | ||
| 309.78 | | 309.78 | ||
| vD# | |||
| — | | — | ||
| 709.78 | | 709.78 | ||
| ^G | |||
| — | | — | ||
| 1109.78 | | 1109.78 | ||
| B | |||
| major seventh | | major seventh | ||
|- | |- | ||
| 6 | | 6 | ||
| 211.73 | | 211.73 | ||
| ^D | |||
| — | | — | ||
| 611.73 | | 611.73 | ||
| F# | |||
| augmented fourth | | augmented fourth | ||
| 1011.73 | | 1011.73 | ||
| vA# | |||
| — | | — | ||
|} | |} | ||
| Line 175: | Line 173: | ||
=== Augmented === | === Augmented === | ||
[[Augmented family|Augmented]] sets [[5/4]] as a period, and uses a fifth as a free generator. There are some extensions for 7-limit | [[Augmented family|Augmented]] sets [[5/4]] as a period, and uses a fifth as a free generator. There are some extensions for 7-limit: augene (12 & 15), august (9 & 12), inflated (3d & 15), and deflated (3 & 9). In the 11-limit, these extensions equate 5/4 and [[14/11]] as a period. | ||
[[Category:Ploidacot]] | [[Category:Ploidacot]] | ||
Revision as of 13:06, 5 January 2026
| Pergen | [P8/3, P5] |
| Numeral form | 3-ploid 1-cot |
| Pure generator size | 98.04 ¢ |
| Pure period size | 400 ¢ |
| Forms | 9, 12, 15, 27 |
| Characteristic multival entry | 3 |
Triploid monocot is a temperament archetype where the generator is a 3/2 perfect fifth and the period is 1/3 of a 2/1 octave, or 400 ¢. The generator can also be characterized as a perfect fourth 4/3, or as a "perfect semitone" [math]\displaystyle{ \frac{2\sqrt[3]{4}}{3} }[/math]. Triploid monocot temperaments usually generate the 3L 6s MOS structure and either 3L 9s (and thus 12L 3s) or 9L 3s as children.
Intervals and notation
While there is no agreed-upon notation system for triploid monocot, the following is based on interpreting the generator as a semitone (1/3 of a minor 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), so vvC# and ^Db are enharmonic.
| # | Ploid 1 | Ploid 2 | Ploid 3 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cents | Notation | Name | Cents | Notation | Name | Cents | Notation | Name | |
| −6 | 188.27 | ^Ebb | — | 588.27 | Gb | diminished fifth | 988.27 | vBb | — |
| −5 | 90.22 | Db | minor second | 490.22 | vF | — | 890.22 | ^Bbb | — |
| −4 | 392.18 | ^Fb | — | 792.18 | Ab | minor sixth | 1192.18 | vC | — |
| −3 | 294.13 | Eb | minor third | 694.13 | vG | — | 1094.13 | ^Cb | — |
| −2 | 196.09 | vD | — | 596.09 | ^Gb | — | 996.09 | Bb | minor seventh |
| −1 | 98.04 | ^Db | — | 498.04 | F | perfect fourth | 898.04 | vA | — |
| 0 | 0 | C | unison | 400 | vE | — | 800 | ^Ab | — |
| 1 | 301.96 | ^Eb | — | 701.96 | G | perfect fifth | 1101.96 | vB | — |
| 2 | 203.91 | D | major second | 603.91 | vF# | — | 1003.91 | ^Bb | — |
| 3 | 105.87 | vC# | — | 505.87 | ^F | — | 905.87 | A | major sixth |
| 4 | 7.82 | ^C | — | 407.82 | E | major third | 807.82 | vG# | — |
| 5 | 309.78 | vD# | — | 709.78 | ^G | — | 1109.78 | B | major seventh |
| 6 | 211.73 | ^D | — | 611.73 | F# | augmented fourth | 1011.73 | vA# | — |
A notable feature of triploid monocot is the small comma, encountered after 4 steps, which represents 1/3 of a Pythagorean comma (or its equivalence, inversed diminished second). This makes triploid monocot scales cluster around 12edo.
Temperament interpretations
By definition, triploid monocot temperaments split the octave in three.
Augmented
Augmented sets 5/4 as a period, and uses a fifth as a free generator. There are some extensions for 7-limit: augene (12 & 15), august (9 & 12), inflated (3d & 15), and deflated (3 & 9). In the 11-limit, these extensions equate 5/4 and 14/11 as a period.