Tetracot (temperament): Difference between revisions

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'''Tetracot''' is a 2.3.5 temperament that splits 3/2 into four flattened 10/9's. Its 5-limit edo join is 27 & 34.
{{Infobox regtemp
| Title = Tetracot
| Subgroups = 2.3.5
| Comma basis = [[20000/19683]] (2.3.5)
| Edo join 1 = 27 | Edo join 2 = 34
| Mapping = 1; 4 9
| Generators = 10/9
| Generators tuning = 176.1
| Optimization method = CWE
| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]]
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13
}}
 
'''Tetracot''', 2.3.5[27 & 34] or 2.3.5[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'''.
{| class="wikitable right-1 right-2"
|-
! #
! 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'''
|}
<nowiki/>* in exact-5/2 tuning


== Extensions ==
== 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.
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.
* 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)^2 with 11/9, but this is questionable because it produces either a very sharp 11/8 or a flat 5/4.
* 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; there are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41), Modus (27e & 34d), and Wollemia (27e & 34). The weak extension [[Octacot (temperament)|Octacot]] (27 & 41) splits the Tetracot generator into two semitones representing 21/20, thus equating three Octacot generators with 7/6.
* 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).
** {{adv|In terms of commas:}}
*** {{adv|Bunya tempers out 225/224, the [[Marvel]] comma.}}
*** {{adv|Monkey is the extension tempering out [[5120/5103]], the aberschisma.}}
*** {{adv|Modus tempers out 64/63.}}
*** {{adv|Wollemia tempers out 126/125.}}
** The weak extension [[Octacot (temperament)|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.
{| class="wikitable sortable"
!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
|}
{{Navbox regtemp}}
{{Navbox regtemp}}
{{Cat|temperaments}}
{{Cat|temperaments}}

Latest revision as of 06:21, 1 July 2026

Tetracot
Subgroups 2.3.5
Reduced mapping ⟨1; 4 9]
ET join 27 & 34
Generators (CWE) ~10/9 = 176.1¢
MOS scales 6L 1s, 7L 6s, 7L 13s
Ploidacot tetracot
Comma basis 20000/19683 (2.3.5)
Minimax error 5-odd-limit: 3.07¢
Target scale size 5-odd-limit: 13 notes

Tetracot, 2.3.5[27 & 34] or 2.3.5[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).
    • In terms of commas:
      • Bunya tempers out 225/224, the Marvel comma.
      • Monkey is the extension tempering out 5120/5103, the aberschisma.
      • Modus tempers out 64/63.
      • Wollemia tempers out 126/125.
    • 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 • Gentle-fifth temperaments • 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) • Hedgehog (diploid alpha-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) • Paddle (alter-dicot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph) • Sidewalk (epsilon-heptaseph)
No-2 Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Aberschismic • Marvel • Parapyth • Pentagoth • Intergan
Straddle Rank-3 Hibernal • Monarch
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