Ploidacot/Tetracot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=4|Pergen=[P8, P5/4]|Forms=6, 7, 13|Title=Tetracot}}'''Tetracot''' is a temperament archetype where the generator is a submajor second, four of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Tetracot temperaments typically generate the [[1L 5s]] and [[6L 1s]] MOS scales, and they split the chromatic semitone into four equal parts, creating "supraminor", "neutral", and "submajor" intervals and containing all [[Ploidacot/Dicot|dicot]] intervals. | {{Breadcrumb}} | ||
{{Infobox ploidacot|Ploids=1|Shears=0|Cots=4|Pergen=[P8, P5/4]|Forms=6, 7, 13, 20|Title=Tetracot|Wedgie=4}} | |||
'''Tetracot''' is a temperament archetype where the generator is a submajor second, four of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Tetracot temperaments typically generate the [[1L 5s]] and [[6L 1s]] MOS scales, and they split the chromatic semitone into four equal parts, creating "supraminor", "neutral", and "submajor" intervals and containing all [[Ploidacot/Dicot|dicot]] intervals. | |||
Tetracot temperaments often generate [[7L 6s]] and [[7L 13s]] as chromatic scales, and for particularly sharp tunings [[13L 7s]]. More info may be found at [[Ploidacot/Tetracot/Generator tunings]]. | |||
== Intervals and notation == | == Intervals and notation == | ||
While there is no agreed-upon notation system for tetracot, the notation provided here is based on interpreting the generator as a submajor second, and produced by extending dicot notation, allowing for an ^ or v to stand for a quarter of a chromatic semitone | While there is no agreed-upon notation system for tetracot, the notation provided here is based on interpreting the generator as a submajor second, and produced by extending dicot notation, allowing for an ^ or v to stand for a quarter of a chromatic semitone, so {{nowrap|^^Eb {{=}} vvE {{=}} Ed}}. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Tetracot intervals (assuming pure fifth and octave) | |+ style="font-size: 105%;" | Tetracot intervals (assuming pure fifth and octave) | ||
!# | |- | ||
!Cents | ! # | ||
!Notation | ! Cents | ||
!Name | ! Notation | ||
! Name | |||
|- | |- | ||
| | | −13 | ||
| | | 118.65 | ||
|^ | | ^Db | ||
|supraminor | | supraminor second | ||
|- | |- | ||
| | | −12 | ||
| | | 294.13 | ||
| | | Eb | ||
|minor | | minor third | ||
|- | |- | ||
| | | −11 | ||
| | | 469.62 | ||
| | | vF | ||
| | | subfourth | ||
|- | |- | ||
| | | −10 | ||
| | | 645.11 | ||
| | | Gd | ||
| | | semidiminished fifth | ||
|- | |- | ||
| | | −9 | ||
| | | 820.60 | ||
|^ | | ^Ab | ||
|supraminor | | supraminor sixth | ||
|- | |- | ||
| | | −8 | ||
| | | 996.09 | ||
| | | Bb | ||
| | | minor seventh | ||
|- | |- | ||
| | | −7 | ||
| | | 1171.58 | ||
| | | vC | ||
| | | suboctave | ||
|- | |- | ||
| | | −6 | ||
| | | 147.07 | ||
| | | Dd | ||
|neutral | | neutral second | ||
|- | |- | ||
| | | −5 | ||
| | | 322.56 | ||
|^ | | ^Eb | ||
|supraminor | | supraminor third | ||
|- | |- | ||
| | | −4 | ||
| | | 498.04 | ||
| | | F | ||
|perfect | | perfect fourth | ||
|- | |- | ||
| | | −3 | ||
| | | 673.53 | ||
| | | vG | ||
| | | subfifth | ||
|- | |- | ||
| | | −2 | ||
| | | 849.02 | ||
| | | Ad | ||
|neutral | | neutral sixth | ||
|- | |- | ||
| | | −1 | ||
| | | 1024.51 | ||
|^ | | ^Bb | ||
| | | supraminor seventh | ||
|- | |- | ||
| | | 0 | ||
| | | 0.00 | ||
| | | C | ||
|perfect | | perfect unison | ||
|- | |- | ||
| | | 1 | ||
| | | 175.49 | ||
| | | vD | ||
|submajor | | submajor second | ||
|- | |- | ||
| | | 2 | ||
| | | 350.98 | ||
| | | Ed | ||
|neutral | | neutral third | ||
|- | |- | ||
| | | 3 | ||
| | | 526.47 | ||
|^ | | ^F | ||
| | | superfourth | ||
|- | |- | ||
| | | 4 | ||
| | | 701.96 | ||
| | | G | ||
| | | perfect fifth | ||
|- | |- | ||
|9 | | 5 | ||
|379.40 | | 877.44 | ||
|vE | | vA | ||
|submajor third | | submajor sixth | ||
|- | |||
| 6 | |||
| 1052.93 | |||
| Bd | |||
| neutral seventh | |||
|- | |||
| 7 | |||
| 28.42 | |||
| ^C | |||
| superunison | |||
|- | |||
| 8 | |||
| 203.91 | |||
| D | |||
| major second | |||
|- | |||
| 9 | |||
| 379.40 | |||
| vE | |||
| submajor third | |||
|- | |||
| 10 | |||
| 554.89 | |||
| Ft | |||
| semiaugmented fourth | |||
|- | |||
| 11 | |||
| 730.38 | |||
| ^G | |||
| superfifth | |||
|- | |||
| 12 | |||
| 905.87 | |||
| A | |||
| major sixth | |||
|- | |||
| 13 | |||
| 1081.35 | |||
| vB | |||
| submajor seventh | |||
|} | |} | ||
A notable feature of tetracot is the small diesis encountered after 7 steps. This makes tetracot scales cluster around 7edo. | A notable feature of tetracot is the small diesis encountered after 7 steps. This makes tetracot scales cluster around 7edo. | ||
== Temperament interpretations == | == Temperament interpretations == | ||
=== Tetracot === | === Tetracot === | ||
The temperament named "tetracot" is one of the simplest 5-limit interpretations. The step is interpreted as [[10/9]], meaning the submajor third is interpreted as [[5/4]]. It is best tuned with a sharpened generator of around 176 | The temperament named "tetracot" is one of the simplest 5-limit interpretations. The step is interpreted as [[10/9]], meaning the submajor third is interpreted as [[5/4]], and extends to include prime 11 by mapping the neutral third to [[11/9]] (so that the step stands for both 10/9 and [[11/10]]). It is best tuned with a sharpened generator of around 176{{c}}. | ||
=== Monkey === | ==== Monkey ==== | ||
Usually framed as an extension of tetracot due to the availability of tetracot's 5-limit mapping, monkey maps 7/4 to 15 steps down, the subminor seventh. | Usually framed as an extension of tetracot due to the availability of tetracot's 5-limit mapping, monkey maps 7/4 to 15 steps down, the subminor seventh. In monkey, the 7-step diesis represents both [[64/63]] and [[81/80]]. | ||