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|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. | |||
== 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^^ {{=}} Evv {{=}} 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 | |||
|- | |- | ||
| | | −9 | ||
|820.60 | | 820.60 | ||
|^Ab | | ^Ab | ||
|supraminor sixth | | supraminor sixth | ||
|- | |- | ||
| | | −8 | ||
|996.09 | | 996.09 | ||
|Bb | | Bb | ||
|minor seventh | | minor seventh | ||
|- | |- | ||
| | | −7 | ||
|1,171.58 | | 1,171.58 | ||
|vC | | vC | ||
|suboctave | | suboctave | ||
|- | |- | ||
| | | −6 | ||
|147.07 | | 147.07 | ||
|Dd | | Dd | ||
|neutral second | | neutral second | ||
|- | |- | ||
| | | −5 | ||
|322.56 | | 322.56 | ||
|^Eb | | ^Eb | ||
|supraminor third | | supraminor third | ||
|- | |- | ||
| | | −4 | ||
|498.05 | | 498.05 | ||
|F | | F | ||
|perfect fourth | | perfect fourth | ||
|- | |- | ||
| | | −3 | ||
|673.53 | | 673.53 | ||
|vG | | vG | ||
|subfifth | | subfifth | ||
|- | |- | ||
| | | −2 | ||
|849.02 | | 849.02 | ||
|Ad | | Ad | ||
|neutral sixth | | neutral sixth | ||
|- | |- | ||
| | | −1 | ||
|1,024.51 | | 1,024.51 | ||
|^Bb | | ^Bb | ||
|supraminor seventh | | supraminor seventh | ||
|- | |- | ||
|0 | | 0 | ||
|0.00 | | 0.00 | ||
|C | | C | ||
|perfect unison / perfect octave | | perfect unison / perfect octave | ||
|- | |- | ||
|1 | | 1 | ||
|175.49 | | 175.49 | ||
|vD | | vD | ||
|submajor second | | submajor second | ||
|- | |- | ||
|2 | | 2 | ||
|350.98 | | 350.98 | ||
|Ed | | Ed | ||
|neutral third | | neutral third | ||
|- | |- | ||
|3 | | 3 | ||
|526.47 | | 526.47 | ||
|^F | | ^F | ||
|superfourth | | superfourth | ||
|- | |- | ||
|4 | | 4 | ||
|701.96 | | 701.96 | ||
|G | | G | ||
|perfect fifth | | perfect fifth | ||
|- | |- | ||
|5 | | 5 | ||
|877.44 | | 877.44 | ||
|vA | | vA | ||
|submajor sixth | | submajor sixth | ||
|- | |- | ||
|6 | | 6 | ||
|1,052.93 | | 1,052.93 | ||
|Bd | | Bd | ||
|neutral seventh | | neutral seventh | ||
|- | |- | ||
|7 | | 7 | ||
|28.42 | | 28.42 | ||
|^C | | ^C | ||
|superunison | | superunison | ||
|- | |- | ||
|8 | | 8 | ||
|203.91 | | 203.91 | ||
|D | | D | ||
|major second | | major second | ||
|- | |- | ||
|9 | | 9 | ||
|379.40 | | 379.40 | ||
|vE | | vE | ||
|submajor third | | submajor third | ||
|} | |} | ||
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]], 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 | 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. In monkey, the 7-step diesis represents both [[64/63]] and [[81/80]]. | 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]]. | ||