Tetracot (temperament)
From Xenharmonic Reference
| Tetracot |
Tetracot, [27 & 34] or [34 & 41], is a temperament that splits 3/2 into four flattened 10/9's.
Interval chain
In the following table, odd harmonics and subharmonics 1–15 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 176.3 | 10/9 |
| 2 | 352.5 | |
| 3 | 528.8 | 27/20 |
| 4 | 705.0 | 3/2 |
| 5 | 881.3 | 5/3 |
| 6 | 1057.5 | |
| 7 | 33.8 | 81/80 |
| 8 | 210.1 | 9/8 |
| 9 | 386.3 | 5/4 |
| 10 | 562.6 | |
| 11 | 738.8 | |
| 12 | 915.1 | 27/16 |
| 13 | 1091.3 | 15/8 |
* in exact-5/2 tuning
Extensions
Tetracot has a number of strong extensions, but most of them are problematic in some way. This is because the Tetracot generator is, optimally, approximately 31/28 — not easily interpretable as LCJI.
- Prime 13 can be added by equating (10/9)^2 (the neutral third) with 16/13. Note that this favors a sharp 3/2 (optimally around 3.2c sharp) and a sharp 13/8 (optimally around 6.9c sharp).
- Prime 11 is often added by equating 10/9 with 11/10 (thus placing 11/8 at +10 generators), but this is questionable because it produces either a very sharp 11/8 (as in 27edo and 34edo) or a flat 5/4 (as in 41edo and 48edo). An alternate extension (27p & 34), associated with 7-limit Wollemia, places 11/8 at -24 generators.
- There isn't a canonical way to add prime 7. This is because 27edo and 41edo have good 7 approximations but 34edo does not. There are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41 or 41 & 48), Modus (27 & 34d), and Wollemia (27 & 34).
- Monkey is notable because it's the extension tempering out 5120/5103, the aberschisma.
- The weak extension Octacot (27 & 41) is more elegant; it splits the Tetracot generator into two semitones (about 88.1c) representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4). Octacot can be extended to have prime 19 (at 17 generators) by equating 21/20 to 20/19 (equivalently, 10/9 to 21/19 or 27/20 to 19/14).
List of patent vals
The following patent vals support 2.3.5 Tetracot. Contorted vals are not included.
| Edo | Generator tuning | Fifth tuning |
|---|---|---|
| 7 | 171.429 | 685.714 |
| 48 | 175.000 | 700.000 |
| 41 | 175.610 | 702.439 |
| 116 | 175.862 | 703.448 |
| 191 | 175.916 | 703.665 |
| 75 | 176.000 | 704.000 |
| 259 | 176.062 | 704.247 |
| 184 | 176.087 | 704.348 |
| 109 | 176.147 | 704.587 |
| 143 | 176.224 | 704.895 |
| 177 | 176.271 | 705.085 |
| 34 | 176.471 | 705.882 |
| 95 | 176.842 | 707.368 |
| 61 | 177.049 | 708.197 |
| 27 | 177.778 | 711.111 |
| View • Talk • EditRegular temperaments | |
|---|---|
| Rank-2 | |
| Acot | Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave) |
| Monocot | Meantone • Schismic • Leapday • Archy |
| Complexity 2 | Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot) |
| Complexity 3 | Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) |
| Complexity 4 | Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot) |
| Complexity 5-6 | Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot) |
| Higher complexity | Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot) |
| Straddle-3 | A-Team (alter-tricot) • Machine (alter-monocot) |
| No-3 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
