Ploidacot/Diploid monocot: Difference between revisions
Created page with "'''Diploid monocot''' is a temperament archetype where the generator is a 3/2 perfect fifth and the period is half a 2/1 octave, or 600 cents. The generator can also be characterized as a perfect fourth 4/3, or as a "perfect semitone" 3/(2*sqrt2). Diploid monocot temperaments usually generate the 2L 8s MOS structure and either 2L 10s (and thus 12L 2s) or 10L 2s as children. == Notation == Diploid monocot notation is complicated as it conventi..." |
Expansion and slight rewriting following Monocot (Also rewrote Kalismic because Kalismic doesn't belong in ploidacot - it has a 7-limit basis!) |
||
| (13 intermediate revisions by 7 users not shown) | |||
| Line 1: | Line 1: | ||
'''Diploid monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth and the period is half a [[2/1]] octave, or 600 | {{Breadcrumb}}{{Infobox ploidacot|Ploids=2|Shears=0|Cots=1|Pergen=[P8/2, P5]|Forms=10, 12, 22|Title=Diploid monocot|Wedgie=2}} | ||
'''Diploid monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth and the period is half a [[2/1]] octave, or 600{{c}}. The generator can also be characterized as a perfect fourth [[4/3]], or as a "perfect semitone" <math>\frac{3}{2\sqrt{2}}</math>. Diploid monocot temperaments usually generate the [[2L 8s]] MOS structure and either [[2L 10s]] (and thus [[12L 2s]]) or [[10L 2s]] as children. | |||
== Intervals and notation == | |||
Diploid monocot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" nominals, the use of scales other than the standard diatonic scale, or the use of halved pythagorean commas. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Diploid monocot intervals (assuming pure fifth and octave) | |+ style="font-size: 105%;" | Diploid monocot intervals (assuming pure fifth and octave) | ||
|- | |- | ||
!# | ! rowspan="2" | # | ||
! | ! colspan="3" | Ploid 1 | ||
! | ! colspan="3" | Ploid 2 | ||
|- | |- | ||
! Cents | |||
! Notation | |||
! Name | |||
! Cents | |||
! Notation | |||
! Name | |||
|- | |- | ||
| | | −6 | ||
| | | 588.27 | ||
| | | Gb | ||
| | | diminished fifth | ||
| | | 1188.27 | ||
| — | |||
| | | — | ||
| | |||
|- | |- | ||
| | | −5 | ||
| | | 90.22 | ||
| | | Db | ||
| | | minor second | ||
| 690.22 | |||
| | | — | ||
| | | — | ||
| | |||
|- | |- | ||
| | | −4 | ||
| | | 192.18 | ||
| | | — | ||
| | | — | ||
| | | 792.18 | ||
| | | Ab | ||
|minor | | minor sixth | ||
|- | |- | ||
| | | −3 | ||
| | | 294.13 | ||
| | | Eb | ||
| | | minor third | ||
| | | 894.13 | ||
| — | |||
| | | — | ||
| | |||
|- | |- | ||
| | | −2 | ||
| | | 396.09 | ||
| | | — | ||
| | | — | ||
| | | 996.09 | ||
| | | Bb | ||
| | | minor seventh | ||
|- | |- | ||
| | | −1 | ||
| | | 498.04 | ||
| | | F | ||
| | | perfect fourth | ||
| 1098.04 | |||
| | | — | ||
| | | — | ||
| | |||
|- | |- | ||
| | | 0 | ||
| | | 0 | ||
| | | C | ||
| | | unison | ||
| | | 600 | ||
| | | — | ||
| | | — | ||
|- | |- | ||
| | | 1 | ||
| | | 101.96 | ||
| | | — | ||
| | | — | ||
| | | 701.96 | ||
| G | |||
| | | perfect fifth | ||
| | |||
|- | |- | ||
| | | 2 | ||
| | | 203.91 | ||
| | | D | ||
| | | major second | ||
| 803.91 | |||
| | | — | ||
| | | — | ||
| | |||
|- | |- | ||
| | | 3 | ||
| | | 305.87 | ||
| - | | — | ||
| - | | — | ||
|5 | | 905.87 | ||
| | | A | ||
|major seventh | | major sixth | ||
| | |- | ||
| 4 | |||
| 407.82 | |||
| E | |||
| major third | |||
| 1007.82 | |||
| — | |||
| — | |||
|- | |||
| 5 | |||
| 509.78 | |||
| — | |||
| — | |||
| 1109.78 | |||
| B | |||
| major seventh | |||
|- | |||
| 6 | |||
| 11.73 | |||
| — | |||
| — | |||
| 611.73 | |||
| F# | |||
| augmented fourth | |||
|} | |} | ||
== Temperament interpretations == | == Temperament interpretations == | ||
By definition, diploid monocot temperaments equate | By definition, diploid monocot temperaments equate a pair of intervals, mapping them both to the semioctave. | ||
=== Medium accuracy === | |||
==== Diaschismic ==== | |||
Diaschismic equates [[45/32]] with [[64/45]], setting [[3/2]] equal to [[16/15]] plus a semioctave and mapping [[5/4]] to 2 generators down - a minor seventh minus a semioctave, or equivalently half of a minor sixth. Diaschismic naturally extends to prime 17 by equating the flat 16/15 generator with [[17/16]]. Diaschismic is tuned best with fifths slightly sharp of just. | |||
To extend to the 7-limit, [[pajara]] equates 7/5 with 10/7 (as per [[jubilismic]]), meaning 7/4 simply maps to a semioctave above 5/4. This suggests tuning the generator to about 709{{c}} (or equivalently 109{{c}}). | |||
[[Septimal diaschismic]] instead maps 7/4 to a semioctave 8 fifths down with more accuracy, and the [[pythagorean comma]] is tuned sharp enough so that its half can be used as a [[septimal comma]] or [[syntonic comma]] accidental. [[Srutal]] is the most accurate of diaschismic extensions, though the most complex, at +15 fifths. | |||
==== Injera ==== | |||
Injera extends [[meantone]] by equating 7/5 with 10/7 (as per jubilismic), so that both maps of [[5/4]] and [[7/4]] are found at 4 generators up, offset by a semioctave. Either sharper fifths (as in 12edo) or flatter fifths (as in 26edo) can work. | |||
=== | === High accuracy === | ||
==== Gariwizmic ==== | |||
[[Gariwizmic]] equates [[99/70]] with [[140/99]] (as per [[kalismic]]), so that maps of [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]; The tone is split into two [[35/33]] semitones, and the Pythagorean comma into two [[2835/2816|fwiwismas]]. Gariwizmic combines this with combining the haploid chain of fifths of [[gary]] for primes 7 and 11, with new diploid mappings for primes 5, 13, 17 and 19. | |||
It is incredibly accurate, qualifying as a [[microtemperament]]. Despite its great accuracy, it finds other primes deep in the diploid chain, with 5/4 at +39 fifths minus a semioctave, 13/8 at -27 fifths plus a semioctave, and 17/16 as +48 fifths (4 gary commas), and 19/16 as +44 fifths minus a semioctave. | |||
Latest revision as of 23:13, 22 July 2026
| Pergen | [P8/2, P5] |
| Numeral form | 2-ploid 1-cot |
| Pure generator size | 101.96 ¢ |
| Pure period size | 600 ¢ |
| Forms | 10, 12, 22 |
| Characteristic multival entry | 2 |
Diploid monocot is a temperament archetype where the generator is a 3/2 perfect fifth and the period is half a 2/1 octave, or 600 ¢. The generator can also be characterized as a perfect fourth 4/3, or as a "perfect semitone" [math]\displaystyle{ \frac{3}{2\sqrt{2}} }[/math]. Diploid monocot temperaments usually generate the 2L 8s MOS structure and either 2L 10s (and thus 12L 2s) or 10L 2s as children.
Intervals and notation
Diploid monocot notation is complicated as it conventionally requires either the introduction of new "hemipythagorean" nominals, the use of scales other than the standard diatonic scale, or the use of halved pythagorean commas. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals.
| # | Ploid 1 | Ploid 2 | ||||
|---|---|---|---|---|---|---|
| Cents | Notation | Name | Cents | Notation | Name | |
| −6 | 588.27 | Gb | diminished fifth | 1188.27 | — | — |
| −5 | 90.22 | Db | minor second | 690.22 | — | — |
| −4 | 192.18 | — | — | 792.18 | Ab | minor sixth |
| −3 | 294.13 | Eb | minor third | 894.13 | — | — |
| −2 | 396.09 | — | — | 996.09 | Bb | minor seventh |
| −1 | 498.04 | F | perfect fourth | 1098.04 | — | — |
| 0 | 0 | C | unison | 600 | — | — |
| 1 | 101.96 | — | — | 701.96 | G | perfect fifth |
| 2 | 203.91 | D | major second | 803.91 | — | — |
| 3 | 305.87 | — | — | 905.87 | A | major sixth |
| 4 | 407.82 | E | major third | 1007.82 | — | — |
| 5 | 509.78 | — | — | 1109.78 | B | major seventh |
| 6 | 11.73 | — | — | 611.73 | F# | augmented fourth |
Temperament interpretations
By definition, diploid monocot temperaments equate a pair of intervals, mapping them both to the semioctave.
Medium accuracy
Diaschismic
Diaschismic equates 45/32 with 64/45, setting 3/2 equal to 16/15 plus a semioctave and mapping 5/4 to 2 generators down - a minor seventh minus a semioctave, or equivalently half of a minor sixth. Diaschismic naturally extends to prime 17 by equating the flat 16/15 generator with 17/16. Diaschismic is tuned best with fifths slightly sharp of just.
To extend to the 7-limit, pajara equates 7/5 with 10/7 (as per jubilismic), meaning 7/4 simply maps to a semioctave above 5/4. This suggests tuning the generator to about 709 ¢ (or equivalently 109 ¢).
Septimal diaschismic instead maps 7/4 to a semioctave 8 fifths down with more accuracy, and the pythagorean comma is tuned sharp enough so that its half can be used as a septimal comma or syntonic comma accidental. Srutal is the most accurate of diaschismic extensions, though the most complex, at +15 fifths.
Injera
Injera extends meantone by equating 7/5 with 10/7 (as per jubilismic), so that both maps of 5/4 and 7/4 are found at 4 generators up, offset by a semioctave. Either sharper fifths (as in 12edo) or flatter fifths (as in 26edo) can work.
High accuracy
Gariwizmic
Gariwizmic equates 99/70 with 140/99 (as per kalismic), so that maps of 10/9 and 11/7 are a semioctave apart, as well as are 11/10 and 14/9; The tone is split into two 35/33 semitones, and the Pythagorean comma into two fwiwismas. Gariwizmic combines this with combining the haploid chain of fifths of gary for primes 7 and 11, with new diploid mappings for primes 5, 13, 17 and 19.
It is incredibly accurate, qualifying as a microtemperament. Despite its great accuracy, it finds other primes deep in the diploid chain, with 5/4 at +39 fifths minus a semioctave, 13/8 at -27 fifths plus a semioctave, and 17/16 as +48 fifths (4 gary commas), and 19/16 as +44 fifths minus a semioctave.