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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]]. | ||
Revision as of 16:03, 9 September 2025
| Pergen | [P8, P5/4] |
| Numeral form | 4-cot |
| Pure generator size | 175.49 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 6, 7, 13 |
| Characteristic multival entry | TBD |
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 dicot intervals.
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, so Eb^^ = Evv = Ed.
| # | Cents | Notation | Name |
|---|---|---|---|
| −9 | 820.60 | ^Ab | supraminor sixth |
| −8 | 996.09 | Bb | minor seventh |
| −7 | 1,171.58 | vC | suboctave |
| −6 | 147.07 | Dd | neutral second |
| −5 | 322.56 | ^Eb | supraminor third |
| −4 | 498.05 | F | perfect fourth |
| −3 | 673.53 | vG | subfifth |
| −2 | 849.02 | Ad | neutral sixth |
| −1 | 1,024.51 | ^Bb | supraminor seventh |
| 0 | 0.00 | C | perfect unison / perfect octave |
| 1 | 175.49 | vD | submajor second |
| 2 | 350.98 | Ed | neutral third |
| 3 | 526.47 | ^F | superfourth |
| 4 | 701.96 | G | perfect fifth |
| 5 | 877.44 | vA | submajor sixth |
| 6 | 1,052.93 | Bd | neutral seventh |
| 7 | 28.42 | ^C | superunison |
| 8 | 203.91 | D | major second |
| 9 | 379.40 | vE | submajor third |
A notable feature of tetracot is the small diesis encountered after 7 steps. This makes tetracot scales cluster around 7edo.
Temperament interpretations
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 ¢.
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.