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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=2|Cots=3|Pergen=[P8, P4/3]|Forms=7, 8, 15|Title=Omega-tricot}}'''Omega-tricot''' is a temperament archetype where the generator is a submajor second, three of which stack to form a perfect fourth of [[4/3]], and the period is a [[2/1]] octave. Omega-tricot temperaments usually generate the [[1L 6s]] and [[7L 1s]] MOS structures. Omega-tricot temperaments produce "supraminor" and "submajor" intervals, splitting the chromatic semitone into three parts. | {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=2|Cots=3|Pergen=[P8, P4/3]|Forms=7, 8, 15, 22|Title=Omega-tricot (beta-tricot)|Wedgie=3}} | ||
'''Omega-tricot''' is a temperament archetype where the generator is a submajor second, three of which stack to form a perfect fourth of [[4/3]], and the period is a [[2/1]] octave. Omega-tricot temperaments usually generate the [[1L 6s]] and [[7L 1s]] MOS structures. Omega-tricot temperaments produce "supraminor" and "submajor" intervals, splitting the chromatic semitone into three parts. | |||
== Intervals and notation == | |||
While there is no agreed-upon notation system for omega-tricot, the following is based on interpreting the generator as a submajor second, allowing for an ^ or v to stand for 1/3 of a chromatic semitone, so ^^C and vC# are enharmonic. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ style="font-size: 105%;" | Omega-tricot intervals (assuming pure fifth and octave) | ||
!# | |- | ||
!Cents | ! # | ||
!Notation | ! Cents | ||
!Name | ! Notation | ||
! Name | |||
|- | |- | ||
| | | −9 | ||
|905.865 | | 905.865 | ||
|A | | A | ||
|major sixth | | major sixth | ||
|- | |- | ||
| | | −8 | ||
|1071.88 | | 1071.88 | ||
|vB | | vB | ||
|submajor seventh | | submajor seventh | ||
|- | |- | ||
| | | −7 | ||
|37.895 | | 37.895 | ||
|^C | | ^C | ||
|superunison | | superunison | ||
|- | |- | ||
| | | −6 | ||
|203.91 | | 203.91 | ||
|D | | D | ||
|major second | | major second | ||
|- | |- | ||
| | | −5 | ||
|369.925 | | 369.925 | ||
|vE | | vE | ||
|submajor third | | submajor third | ||
|- | |- | ||
| | | −4 | ||
|535.94 | | 535.94 | ||
|^F | | ^F | ||
|superfourth | | superfourth | ||
|- | |- | ||
| | | −3 | ||
|701.955 | | 701.955 | ||
|G | | G | ||
|perfect fifth | | perfect fifth | ||
|- | |- | ||
| | | −2 | ||
|867.97 | | 867.97 | ||
|vA | | vA | ||
|submajor sixth | | submajor sixth | ||
|- | |- | ||
| | | −1 | ||
|1033.985 | | 1033.985 | ||
|^Bb | | ^Bb | ||
|supraminor seventh | | supraminor seventh | ||
|- | |- | ||
|0 | | 0 | ||
|0 | | 0 | ||
|C | | C | ||
|perfect unison | | perfect unison | ||
|- | |- | ||
|1 | | 1 | ||
|166.015 | | 166.015 | ||
|vD | | vD | ||
|submajor second | | submajor second | ||
|- | |- | ||
|2 | | 2 | ||
|332.03 | | 332.03 | ||
|^Eb | | ^Eb | ||
|supraminor third | | supraminor third | ||
|- | |- | ||
|3 | | 3 | ||
|498.045 | | 498.045 | ||
|F | | F | ||
|perfect fourth | | perfect fourth | ||
|- | |- | ||
|4 | | 4 | ||
|664.06 | | 664.06 | ||
|vG | | vG | ||
|subfifth | | subfifth | ||
|- | |- | ||
|5 | | 5 | ||
|830.075 | | 830.075 | ||
|^Ab | | ^Ab | ||
|supraminor sixth | | supraminor sixth | ||
|- | |- | ||
|6 | | 6 | ||
|996.09 | | 996.09 | ||
|Bb | | Bb | ||
|minor seventh | | minor seventh | ||
|- | |- | ||
|7 | | 7 | ||
|1162.105 | | 1162.105 | ||
|vC | | vC | ||
|suboctave | | suboctave | ||
|- | |- | ||
|8 | | 8 | ||
|128.12 | | 128.12 | ||
|^Db | | ^Db | ||
|supraminor second | | supraminor second | ||
|- | |- | ||
|9 | | 9 | ||
|294.135 | | 294.135 | ||
|Eb | | Eb | ||
|minor third | | minor third | ||
|} | |} | ||
== Temperament interpretations == | == Temperament interpretations == | ||
=== Porcupine === | === Porcupine === | ||
In [[porcupine]], the generator is [[11/10]], two generators make [[6/5]], and three make 4/3. This is tuned best with a considerably flat generator of about 162 | In [[porcupine]], the generator is [[11/10]], two generators make [[6/5]], and three make 4/3. This is tuned best with a considerably flat generator of about 162{{c}} or so, and naturally extends to the full 11-limit as in [[superpyth]], so the minor seventh is [[7/4]]. | ||
=== Superpine === | === Superpine === | ||
In [[superpine]], the | In [[superpine]], the generator is again mapped to 11/10, and the mapping for 5 is as in [[meantone]], so [[5/4]] is found 12 generators down. The interval of two generators no longer represents a minor third, but more of a neutral one, as it is mapped to [[39/32]] (so that 5 generators span [[13/8]]). The canonical mapping of 7 places [[7/2]] at 13 generators, equating the generator to [[35/32]]. It is best tuned with a slightly sharp generator of about 168{{c}}. | ||
{{ | |||
Latest revision as of 06:38, 16 July 2026
| Pergen | [P8, P4/3] |
| Numeral form | 2-sheared 3-cot |
| Pure generator size | 166.01 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 7, 8, 15, 22 |
| Characteristic multival entry | 3 |
Omega-tricot is a temperament archetype where the generator is a submajor second, three of which stack to form a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-tricot temperaments usually generate the 1L 6s and 7L 1s MOS structures. Omega-tricot temperaments produce "supraminor" and "submajor" intervals, splitting the chromatic semitone into three parts.
Intervals and notation
While there is no agreed-upon notation system for omega-tricot, the following is based on interpreting the generator as a submajor second, allowing for an ^ or v to stand for 1/3 of a chromatic semitone, so ^^C and vC# are enharmonic.
| # | Cents | Notation | Name |
|---|---|---|---|
| −9 | 905.865 | A | major sixth |
| −8 | 1071.88 | vB | submajor seventh |
| −7 | 37.895 | ^C | superunison |
| −6 | 203.91 | D | major second |
| −5 | 369.925 | vE | submajor third |
| −4 | 535.94 | ^F | superfourth |
| −3 | 701.955 | G | perfect fifth |
| −2 | 867.97 | vA | submajor sixth |
| −1 | 1033.985 | ^Bb | supraminor seventh |
| 0 | 0 | C | perfect unison |
| 1 | 166.015 | vD | submajor second |
| 2 | 332.03 | ^Eb | supraminor third |
| 3 | 498.045 | F | perfect fourth |
| 4 | 664.06 | vG | subfifth |
| 5 | 830.075 | ^Ab | supraminor sixth |
| 6 | 996.09 | Bb | minor seventh |
| 7 | 1162.105 | vC | suboctave |
| 8 | 128.12 | ^Db | supraminor second |
| 9 | 294.135 | Eb | minor third |
Temperament interpretations
Porcupine
In porcupine, the generator is 11/10, two generators make 6/5, and three make 4/3. This is tuned best with a considerably flat generator of about 162 ¢ or so, and naturally extends to the full 11-limit as in superpyth, so the minor seventh is 7/4.
Superpine
In superpine, the generator is again mapped to 11/10, and the mapping for 5 is as in meantone, so 5/4 is found 12 generators down. The interval of two generators no longer represents a minor third, but more of a neutral one, as it is mapped to 39/32 (so that 5 generators span 13/8). The canonical mapping of 7 places 7/2 at 13 generators, equating the generator to 35/32. It is best tuned with a slightly sharp generator of about 168 ¢.