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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=2|Pergen=[P8, P5/2]|Forms=7, 10|Title=Dicot}} | {{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 tend to involve "neutral" intervals, which are in-between conventional diatonic intervals. | '''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 tend to involve "neutral" intervals, which are in-between conventional diatonic intervals. | ||
== Intervals and notation == | == Intervals and notation == | ||
Dicot temperaments can be notated using [[Neutral chain-of-fifths notation|neutral chain-of-fifths notation | Dicot temperaments can be notated using [[Neutral chain-of-fifths notation|neutral chain-of-fifths notation]]. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Dicot intervals (assuming pure fifth and octave) | |+ style="font-size: 105%;" | Dicot intervals (assuming pure fifth and octave) | ||
!# | |- | ||
!Cents | ! # | ||
!Notation | ! Cents | ||
!Name | ! Notation | ||
! Name | |||
|- | |- | ||
| | | −10 | ||
|90.22 | | 90.22 | ||
|Db | | Db | ||
|minor second | | minor second | ||
|- | |- | ||
| | | −9 | ||
|441.20 | | 441.20 | ||
|Fd | | Fd | ||
|semidiminished fourth | | semidiminished fourth | ||
|- | |- | ||
| | | −8 | ||
|792.18 | | 792.18 | ||
|Ab | | Ab | ||
|minor sixth | | minor sixth | ||
|- | |- | ||
| | | −7 | ||
| | | 1143.16 | ||
|Cd | | Cd | ||
|semidiminished octave | | semidiminished octave | ||
|- | |- | ||
| | | −6 | ||
|294. | | 294.13 | ||
|Eb | | Eb | ||
|minor third | | minor third | ||
|- | |- | ||
| | | −5 | ||
|645.11 | | 645.11 | ||
|Gd | | Gd | ||
|semidiminished fifth | | semidiminished fifth | ||
|- | |- | ||
| | | −4 | ||
|996.09 | | 996.09 | ||
|Bb | | Bb | ||
|minor seventh | | minor seventh | ||
|- | |- | ||
| | | −3 | ||
|147.07 | | 147.07 | ||
|Dd | | Dd | ||
|neutral second | | neutral second | ||
|- | |- | ||
| | | −2 | ||
|498. | | 498.04 | ||
|F | | F | ||
|perfect fourth | | perfect fourth | ||
|- | |- | ||
| | | −1 | ||
|849.02 | | 849.02 | ||
|Ad | | Ad | ||
|neutral sixth | | neutral sixth | ||
|- | |- | ||
|0 | | 0 | ||
|0 | | 0 | ||
|C | | C | ||
|perfect unison | | perfect unison | ||
|- | |- | ||
|1 | | 1 | ||
|350.98 | | 350.98 | ||
|Ed | | Ed | ||
|neutral third | | neutral third | ||
|- | |- | ||
|2 | | 2 | ||
|701.96 | | 701.96 | ||
|G | | G | ||
|perfect fifth | | perfect fifth | ||
|- | |- | ||
|3 | | 3 | ||
| | | 1052.93 | ||
|Bd | | Bd | ||
|neutral seventh | | neutral seventh | ||
|- | |- | ||
|4 | | 4 | ||
|203.91 | | 203.91 | ||
|D | | D | ||
|major second | | major second | ||
|- | |- | ||
|5 | | 5 | ||
|554.89 | | 554.89 | ||
|Ft | | Ft | ||
|semiaugmented fourth | | semiaugmented fourth | ||
|- | |- | ||
|6 | | 6 | ||
|905.87 | | 905.87 | ||
|A | | A | ||
|major sixth | | major sixth | ||
|- | |- | ||
|7 | | 7 | ||
|56.84 | | 56.84 | ||
|Ct | | Ct | ||
|semiaugmented unison | | semiaugmented unison | ||
|- | |- | ||
|8 | | 8 | ||
|407.82 | | 407.82 | ||
|E | | E | ||
|major third | | major third | ||
|- | |- | ||
|9 | | 9 | ||
|758.80 | | 758.80 | ||
|Gt | | Gt | ||
|semiaugmented fifth | | semiaugmented fifth | ||
|- | |- | ||
|10 | | 10 | ||
| | | 1109.78 | ||
|B | | B | ||
|major seventh | | major seventh | ||
|} | |} | ||
| Line 122: | Line 124: | ||
=== Dicot === | === Dicot === | ||
The temperament named "[[dicot]]" is an exotemperament, equating the neutral third to 5/4. This means that 6/5 is the same interval, and the neutral sixth represents both 5/3 and 8/5. It is best tuned with either a sharpened generator of around 360 | The temperament named "[[dicot]]" is an exotemperament, equating the neutral third to 5/4. This means that 6/5 is the same interval, and the neutral sixth represents both 5/3 and 8/5. It is best tuned with either 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). | ||
=== Neutral === | === Neutral === | ||
[[Neutral]] is the temperament equating [[11/9]] with [[27/22]]. This makes 11/9 the neutral third and [[11/8]] the semiaugmented fourth. | [[Neutral (temperament)|Neutral]] is the temperament equating [[11/9]] with [[27/22]]. This makes 11/9 the neutral third and [[11/8]] the semiaugmented fourth. [[Namo]] extends neutral so that [[16/13]] is found at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments. | ||
=== Mohajira === | === Mohajira === | ||
When neutral is combined with [[meantone]] (which sets the major third equal to [[5/4]]), the result is [[mohajira]], which tunes the generator of ~11/9 to about 348 | When neutral is combined with [[meantone]] (which sets the major third equal 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 === | === Hemififths === | ||
Fittingly to its name, [[hemififths]] divides the fifth evenly into two [[49/40]]~[[60/49]]<nowiki/>s. 7/4 is the semiaugmented sixth, and consequently 5/4 is the sesqui-augmented ( | Fittingly to its name, [[hemififths]] divides the fifth evenly into two {{nowrap|[[49/40]]~[[60/49]]<nowiki/>s}}. 7/4 is the semiaugmented sixth, and consequently 5/4 is the sesqui-augmented (one-and-a-half augmented) second. | ||
[[Category:Ploidacots|Dicot]] | |||
Latest revision as of 20:19, 30 April 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 tend to involve "neutral" intervals, which are in-between conventional diatonic intervals.
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 equate some interval to its fifth complement.
Dicot
The temperament named "dicot" is an exotemperament, equating the neutral third to 5/4. This means that 6/5 is the same interval, and the neutral sixth represents both 5/3 and 8/5. It is best tuned with either 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).
Neutral
Neutral is the temperament equating 11/9 with 27/22. This makes 11/9 the neutral third and 11/8 the semiaugmented fourth. Namo extends neutral so that 16/13 is found 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 sets the major third equal 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
Fittingly to its name, hemififths divides the fifth evenly into two 49/40~60/49s. 7/4 is the semiaugmented sixth, and consequently 5/4 is the sesqui-augmented (one-and-a-half augmented) second.