Ploidacot/Dicot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=2|Pergen=[P8, P5/2]|Forms=7, 10, 17|Title=Dicot|Wedgie=2}} | {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=2|Pergen=[P8, P5/2]|Forms=7, 10, 17|Title=Dicot|Wedgie=2}} | ||
'''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L 3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L 7s]] or [[7L 10s]]. Dicot temperaments | '''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L 3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L 7s]] or [[7L 10s]]. Dicot temperaments involve "neutral" intervals - right in the middle of typical diatonic intervals. More precisely, they are ''interchromatic'', as the neutrals are found between [[Apotome|apotomes]] (chromatic semitones). | ||
== Intervals and notation == | == Intervals and notation == | ||
| Line 121: | Line 121: | ||
== Temperament interpretations == | == Temperament interpretations == | ||
By definition, dicot temperaments | By definition, dicot temperaments map a pair of intervals to the neutral third, equating them. | ||
=== | === Low accuracy === | ||
=== | ==== Dicot ==== | ||
[[ | The temperament named "[[dicot]]" is an exotemperament that equates 5/4 with 6/5, as well as 5/3 with 8/5 with It is well tuned either way with a sharpened generator of around 360{{c}} (optimizing for the tuning of 5/4) or a flattened generator of around 340{{c}} (optimizing for the tuning of 5/3). | ||
Despite its noticeable innacuracy, it is the first "usable" dicot temperament, as the one before it would be the extreme temperament that "equates" 9/8 and 4/3. | |||
=== | === Medium accuracy === | ||
[[ | ==== Neutral ==== | ||
[[Neutral (temperament)|Neutral]] (AKA the 2.3.11 restriction of [[rastmic]]) is the temperament equating [[11/9]] with [[27/22]]. This maps 11/9 to the neutral third and [[11/8]] to the semiaugmented fourth. [[Namo]] extends neutral so that [[16/13]] is mapped at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments. | |||
==== Mohajira ==== | |||
When neutral is combined with [[meantone]] (which maps the major third to [[5/4]]), the result is [[mohajira]], which tunes the generator of ~11/9 to about 348{{c}} and extends to the full 11-limit by setting 7/4 equal to the semidiminished seventh. | |||
==== Hemififths ==== | |||
As its name implies, [[hemififths]] has hemififths as generators, equating [[49/40]] with [[60/49]]. 7/4 maps to a semiaugmented sixth, and consequently 5/4 maps to a sesqui-augmented (one-and-a-half augmented) second. | |||
=== High accuracy === | |||
==== Newt ==== | |||
Much like hemififths, [[newt]] equates 49/40 with 60/49, but primes are found much deeper in the dicot chain, albeit its accuracy is astounding, solidifying it as a [[microtemperament]]. It maps, with incredible exactitude: 7/4 to a subminor seventh (doubly diminished octave), 11/8 to a hyperfourth (3x-augmented second), 5/4 at -28.5 hemififths, 13/8 at 40.5 fifths, 19/16 at -23.5 fifths. | |||
Latest revision as of 22:16, 22 July 2026
| Pergen | [P8, P5/2] |
| Numeral form | 2-cot |
| Pure generator size | 350.98 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 7, 10, 17 |
| Characteristic multival entry | 2 |
Dicot is a temperament archetype where the generator is a neutral third, two of which make a perfect fifth of 3/2, and the period is a 2/1 octave. Dicot temperaments usually generate the 7L 3s MOS structure, fittingly named "dicoid", and one of its children 10L 7s or 7L 10s. Dicot temperaments involve "neutral" intervals - right in the middle of typical diatonic intervals. More precisely, they are interchromatic, as the neutrals are found between apotomes (chromatic semitones).
Intervals and notation
Dicot temperaments can be notated using neutral chain-of-fifths notation.
| # | Cents | Notation | Name |
|---|---|---|---|
| −10 | 90.22 | Db | minor second |
| −9 | 441.20 | Fd | semidiminished fourth |
| −8 | 792.18 | Ab | minor sixth |
| −7 | 1143.16 | Cd | semidiminished octave |
| −6 | 294.13 | Eb | minor third |
| −5 | 645.11 | Gd | semidiminished fifth |
| −4 | 996.09 | Bb | minor seventh |
| −3 | 147.07 | Dd | neutral second |
| −2 | 498.04 | F | perfect fourth |
| −1 | 849.02 | Ad | neutral sixth |
| 0 | 0 | C | perfect unison |
| 1 | 350.98 | Ed | neutral third |
| 2 | 701.96 | G | perfect fifth |
| 3 | 1052.93 | Bd | neutral seventh |
| 4 | 203.91 | D | major second |
| 5 | 554.89 | Ft | semiaugmented fourth |
| 6 | 905.87 | A | major sixth |
| 7 | 56.84 | Ct | semiaugmented unison |
| 8 | 407.82 | E | major third |
| 9 | 758.80 | Gt | semiaugmented fifth |
| 10 | 1109.78 | B | major seventh |
Temperament interpretations
By definition, dicot temperaments map a pair of intervals to the neutral third, equating them.
Low accuracy
Dicot
The temperament named "dicot" is an exotemperament that equates 5/4 with 6/5, as well as 5/3 with 8/5 with It is well tuned either way with a sharpened generator of around 360 ¢ (optimizing for the tuning of 5/4) or a flattened generator of around 340 ¢ (optimizing for the tuning of 5/3).
Despite its noticeable innacuracy, it is the first "usable" dicot temperament, as the one before it would be the extreme temperament that "equates" 9/8 and 4/3.
Medium accuracy
Neutral
Neutral (AKA the 2.3.11 restriction of rastmic) is the temperament equating 11/9 with 27/22. This maps 11/9 to the neutral third and 11/8 to the semiaugmented fourth. Namo extends neutral so that 16/13 is mapped at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments.
Mohajira
When neutral is combined with meantone (which maps the major third to 5/4), the result is mohajira, which tunes the generator of ~11/9 to about 348 ¢ and extends to the full 11-limit by setting 7/4 equal to the semidiminished seventh.
Hemififths
As its name implies, hemififths has hemififths as generators, equating 49/40 with 60/49. 7/4 maps to a semiaugmented sixth, and consequently 5/4 maps to a sesqui-augmented (one-and-a-half augmented) second.
High accuracy
Newt
Much like hemififths, newt equates 49/40 with 60/49, but primes are found much deeper in the dicot chain, albeit its accuracy is astounding, solidifying it as a microtemperament. It maps, with incredible exactitude: 7/4 to a subminor seventh (doubly diminished octave), 11/8 to a hyperfourth (3x-augmented second), 5/4 at -28.5 hemififths, 13/8 at 40.5 fifths, 19/16 at -23.5 fifths.