Ploidacot/Alpha-dicot: Difference between revisions
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{{Breadcrumb}} | {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=2|Pergen=[P8, P4/2]|Forms=5, 9, 14|Title=Alpha-dicot; omega-dicot}}'''Alpha-dicot''' is a temperament archetype where the generator is a [[Interseptimal interval|semitwelfth]], two of which make a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Equivalently, the generator could be a semifourth, two of which make a [[4/3]], so '''omega-dicot''' means the same thing and is unused. | ||
Alpha-dicot temperaments usually generate the [[5L 4s]] MOS structure, named "semiquartal" after the semifourth generator, as well as the child scale [[5L 9s]]. Alpha-dicot temperaments tend to involve interseptimal intervals, which are in between conventional diatonic intervals. | |||
Alpha-dicot temperaments usually generate the [[5L 4s]] MOS structure, named "semiquartal" after the semifourth generator, | |||
== Intervals and notation == | == Intervals and notation == | ||
Alpha-dicot | Alpha-dicot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" ordinals or the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. Note and interval names are provided where alpha-dicot intervals align with standard monocot intervals (which use [[Chain-of-fifths notation]]). | ||
{| class="wikitable" | |||
|+Alpha-dicot intervals (assuming pure octave and fifth) | |||
|+ | |||
!# | !# | ||
!Cents | !Cents | ||
!Notation | !Notation | ||
!Name | !Name | ||
|- | |- | ||
| -9 | | -9 | ||
| | |1041.20 | ||
| | | | ||
| | | | ||
|- | |- | ||
| -8 | | -8 | ||
| Line 31: | Line 23: | ||
|- | |- | ||
| -7 | | -7 | ||
| | |543.16 | ||
| | | | ||
| | | | ||
|- | |- | ||
| -6 | | -6 | ||
| Line 41: | Line 33: | ||
|- | |- | ||
| -5 | | -5 | ||
| | |45.11 | ||
| | | | ||
| | | | ||
|- | |- | ||
| -4 | | -4 | ||
| Line 51: | Line 43: | ||
|- | |- | ||
| -3 | | -3 | ||
| | |747.07 | ||
| | | | ||
| | | | ||
|- | |- | ||
| -2 | | -2 | ||
| Line 61: | Line 53: | ||
|- | |- | ||
| -1 | | -1 | ||
| | |249.02 | ||
| | | | ||
| | | | ||
|- | |- | ||
|0 | |0 | ||
|0 | |0 | ||
|C | |C | ||
|perfect unison | |perfect unison | ||
|- | |- | ||
|1 | |1 | ||
| | |950.98 | ||
| | | | ||
| | | | ||
|- | |- | ||
|2 | |2 | ||
| Line 81: | Line 73: | ||
|- | |- | ||
|3 | |3 | ||
| | |452.93 | ||
| | | | ||
| | | | ||
|- | |- | ||
|4 | |4 | ||
| Line 91: | Line 83: | ||
|- | |- | ||
|5 | |5 | ||
| | |1154.89 | ||
| | | | ||
| | | | ||
|- | |- | ||
|6 | |6 | ||
| Line 101: | Line 93: | ||
|- | |- | ||
|7 | |7 | ||
| | |656.84 | ||
| | | | ||
| | | | ||
|- | |- | ||
|8 | |8 | ||
| Line 111: | Line 103: | ||
|- | |- | ||
|9 | |9 | ||
| | |158.8 | ||
| | | | ||
| | | | ||
|} | |} | ||
== Temperament interpretations == | == Temperament interpretations == | ||
| Line 128: | Line 114: | ||
=== Bug === | === Bug === | ||
[[Bug]] is an exotemperament, equating the | [[Bug]] is an exotemperament, equating the semitwelfth generator to 5/3. This means that 9/5 is the same interval (tempering out [[27/25]]), and the semifourth represents both 6/5 and 10/9. This is clearly badly inaccurate, but is probably the best 5-limit interpretation of this ploidacot. | ||
The best tunings tend to be around 940{{c}} for the semitwelfth, with a somewhat flat twelfth. This sets the semifourth to 260{{c}}, which is close to [[7/6]]. | The best tunings tend to be around 940{{c}} for the semitwelfth, with a somewhat flat twelfth. This sets the semifourth to 260{{c}}, which is close to [[7/6]]. | ||
=== Semaphore === | === Semaphore === | ||
Given that bug sets the | Given that bug sets the semifourth close to 7/6, what happens if it is set equal to 7/6 in the 2.3.7 subgroup? Then, it is equated to [[8/7]], and [[49/48]] is tempered out. The semitwelfth is equated to [[12/7]] and [[7/4]]. This is still an inaccurate temperament, but is on the edge as to whether it counts as exo. | ||
The best tunings tend to be around 950{{c}} here, with a far more accurate twelfth. But, just like how bug's semifourth generator was close to 7/6, this is close to [[26/15]]. | The best tunings tend to be around 950{{c}} here, with a far more accurate twelfth. But, just like how bug's semifourth generator was close to 7/6, this is close to [[26/15]]. | ||
=== Barbados === | === Barbados === | ||
Here, the generator actually is 26/15, equated with [[45/26]]. This is | Here, the generator actually is 26/15, equated with [[45/26]]. This is an accurate temperament, tempering out the unnoticeable comma of [[676/675]], but it is defined in the awkward 2.3.13/5 subgroup. The semifourth here is [[15/13]][[~]][[52/45]]. | ||
As the comma is so small, the best tunings are close to just. The semitwelfth is around 951{{c}}, leading to a near-just twelfth. | As the comma is so small, the best tunings are close to just. The semitwelfth is around 951{{c}}, leading to a near-just twelfth. | ||
{{Todo| unify precision }} | |||
Latest revision as of 00:20, 2 September 2025
| Pergen | [P8, P4/2] |
| Numeral form | 1-sheared 2-cot |
| Pure generator size | 249.03 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 5, 9, 14 |
Alpha-dicot is a temperament archetype where the generator is a semitwelfth, two of which make a perfect twelfth of 3/1, and the period is a 2/1 octave. Equivalently, the generator could be a semifourth, two of which make a 4/3, so omega-dicot means the same thing and is unused.
Alpha-dicot temperaments usually generate the 5L 4s MOS structure, named "semiquartal" after the semifourth generator, as well as the child scale 5L 9s. Alpha-dicot temperaments tend to involve interseptimal intervals, which are in between conventional diatonic intervals.
Intervals and notation
Alpha-dicot notation is complicated as it conventionally requires either the introduction of new "hemipythagorean" ordinals or the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. Note and interval names are provided where alpha-dicot intervals align with standard monocot intervals (which use Chain-of-fifths notation).
| # | Cents | Notation | Name |
|---|---|---|---|
| -9 | 1041.20 | ||
| -8 | 792.18 | Ab | minor sixth |
| -7 | 543.16 | ||
| -6 | 294.14 | Eb | minor third |
| -5 | 45.11 | ||
| -4 | 996.09 | Bb | minor seventh |
| -3 | 747.07 | ||
| -2 | 498.05 | F | perfect fourth |
| -1 | 249.02 | ||
| 0 | 0 | C | perfect unison |
| 1 | 950.98 | ||
| 2 | 701.96 | G | perfect fifth |
| 3 | 452.93 | ||
| 4 | 203.91 | D | major second |
| 5 | 1154.89 | ||
| 6 | 905.87 | A | major sixth |
| 7 | 656.84 | ||
| 8 | 407.82 | E | major third |
| 9 | 158.8 |
Temperament interpretations
By definition, alpha-dicot temperaments equate some interval to its twelfth complement, and so some other interval to its fourth complement.
In order of accuracy, we have bug (a clear exotemperament), semaphore (borderline) and barbados (awkward subgroup, but accurate).
Bug
Bug is an exotemperament, equating the semitwelfth generator to 5/3. This means that 9/5 is the same interval (tempering out 27/25), and the semifourth represents both 6/5 and 10/9. This is clearly badly inaccurate, but is probably the best 5-limit interpretation of this ploidacot.
The best tunings tend to be around 940 ¢ for the semitwelfth, with a somewhat flat twelfth. This sets the semifourth to 260 ¢, which is close to 7/6.
Semaphore
Given that bug sets the semifourth close to 7/6, what happens if it is set equal to 7/6 in the 2.3.7 subgroup? Then, it is equated to 8/7, and 49/48 is tempered out. The semitwelfth is equated to 12/7 and 7/4. This is still an inaccurate temperament, but is on the edge as to whether it counts as exo.
The best tunings tend to be around 950 ¢ here, with a far more accurate twelfth. But, just like how bug's semifourth generator was close to 7/6, this is close to 26/15.
Barbados
Here, the generator actually is 26/15, equated with 45/26. This is an accurate temperament, tempering out the unnoticeable comma of 676/675, but it is defined in the awkward 2.3.13/5 subgroup. The semifourth here is 15/13~52/45.
As the comma is so small, the best tunings are close to just. The semitwelfth is around 951 ¢, leading to a near-just twelfth.