Trienstonic clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 717.517
* [[WE]]: ~2 = 1196.254{{c}}, ~3/2 = 719.306{{c}}
: [[error map]]: {{val| -3.746 +13.604 -14.655 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
: error map: {{val| 0.000 +17.651 -10.007 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 717.517{{c}}
: [[error map]]: {{val| 0.000 +15.562 -16.274 }}
: [[error map]]: {{val| 0.000 +15.562 -16.274 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 721.559
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 721.559{{c}}
: error map: {{val| 0.000 +19.604 -4.150 }}
: error map: {{val| 0.000 +19.604 -4.150 }} -->


{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}


[[Badness]] (Smith): 0.00685
[[Badness]] (Sintel): 0.235


== Father ==
== Father ==
Line 47: Line 51:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 731.394
* [[WE]]: ~2 = 1190.224{{c}}, ~3/2 = 725.221{{c}}
: [[error map]]: {{val| -9.776 +13.490 +3.707 -2.939 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}}
: error map: {{val| 0.000 +29.439 +25.324 +25.355 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 732.223{{c}}
: [[error map]]: {{val| 0.000 +30.268 +20.350 +27.842 }}
: [[error map]]: {{val| 0.000 +30.268 +20.350 +27.842 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 731.177
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.177{{c}}
: error map: {{val| 0.000 +29.222 +26.622 +24.706 }}
: error map: {{val| 0.000 +29.222 +26.622 +24.706 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 58: Line 66:
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}


[[Badness]] (Smith): 0.072653
[[Badness]] (Sintel): 1.84


== Octokaidecal ==
== Octokaidecal ==
Line 73: Line 81:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~27/20 = 600.000, ~3/2 = 723.608 (~10/9 = 123.608)
* [[WE]]: ~27/20 = 596.986{{c}}, ~3/2 = 725.434{{c}} (~10/9 = 128.448{{c}})
: [[error map]]: {{val| -6.029 +17.450 -13.027 }}
* [[CWE]]: ~27/20 = 600.000{{c}}, ~3/2 = 726.548{{c}} (~10/9 = 126.548{{c}})
: error map: {{val| 0.000 +24.593 -6.670 }}
<!-- * [[CTE]]: ~27/20 = 600.000{{c}}, ~3/2 = 723.608{{c}} (~10/9 = 123.608{{c}})
: [[error map]]: {{val| 0.000 +21.653 -15.490 }}
: [[error map]]: {{val| 0.000 +21.653 -15.490 }}
* [[POTE]]: ~27/20 = 600.000, ~3/2 = 729.097 (~10/9 = 129.097)
* [[POTE]]: ~27/20 = 600.000{{c}}, ~3/2 = 729.097{{c}} (~10/9 = 129.097{{c}})
: error map: {{val| 0.000 +27.142 +0.976 }}
: error map: {{val| 0.000 +27.142 +0.976 }} -->


{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}


[[Badness]] (Smith): 0.122848
[[Badness]] (Sintel): 2.88


=== 7-limit ===
=== 7-limit ===
Line 90: Line 102:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
* [[WE]]: ~7/5 = 596.984{{c}}, ~3/2 = 725.210{{c}} (~15/14 = 128.226{{c}})
: [[error map]]: {{val| -6.031 +17.224 -13.699 +0.774 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}})
: error map: {{val| 0.000 +24.364 -7.358 +10.130 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 723.371{{c}} (~15/14 = 123.371{{c}})
: [[error map]]: {{val| 0.000 +21.416 -16.201 +1.287 }}
: [[error map]]: {{val| 0.000 +21.416 -16.201 +1.287 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 728.874 (~15/14 = 128.874)
* [[POTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 728.874{{c}} (~15/14 = 128.874{{c}})
: error map: {{val| 0.000 +26.919 +0.307 +17.795 }}
: error map: {{val| 0.000 +26.919 +0.307 +17.795 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 100: Line 116:
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}


[[Badness]] (Smith): 0.036747
[[Badness]] (Sintel): 0.930


=== 11-limit ===
=== 11-limit ===
Line 110: Line 126:


Optimal tunings:
Optimal tunings:
* CTE: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
* WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 732.330 (~15/14 = 132.330)
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 729.485{{c}} (~15/14 = 129.485{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~3/2 = 723.371{{c}} (~15/14 = 123.371{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~3/2 = 732.330 (~15/14 = 132.330{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
{{Optimal ET sequence|legend=0| 8d, 10, 18e }}


Badness (Smith): 0.030235
Badness (Sintel): 1.00


== Parakangaroo ==
== Parakangaroo ==
Line 131: Line 149:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.863
* [[WE]]: ~2 = 596.984{{c}}, ~10/7 = 638.135{{c}}
: [[error map]]: {{val| -2.883 +12.450 +3.685 -19.845 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/7 = 638.863{{c}}
: [[error map]]: {{val| 0.000 +14.633 +2.314 -19.061 }}
: [[error map]]: {{val| 0.000 +14.633 +2.314 -19.061 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 639.672
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.672{{c}}
: error map: {{val| 0.000 +17.060 +10.404 -11.780 }}
: error map: {{val| 0.000 +17.060 +10.404 -11.780 }} -->


{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}


[[Badness]] (Smith): 0.077857
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Line 147: Line 169:
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 639.036
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 639.845
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 639.036{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 639.845{{c}} -->


{{Optimal ET sequence|legend=0| 15 }}
{{Optimal ET sequence|legend=0| 15 }}


Badness (Smith): 0.043195
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 162: Line 186:
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 638.717
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 640.230
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 638.717{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 640.230{{c}} -->


{{Optimal ET sequence|legend=0| 15 }}
{{Optimal ET sequence|legend=0| 15 }}


Badness (Smith): 0.032653
Badness (Sintel): 1.35


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 17:43, 15 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The trienstonic clan of rank-2 temperaments are low-complexity, high-error temperaments that temper out 28/27, the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, 9/8 with 7/6, 8/7 with 32/27, and 4/3 with 9/7. Trienstonian is close to the edge of what can be sensibly called a temperament at all; in other words, it is an exotemperament.

Adding 16/15 to 28/27 leads to father, 256/245 gives uncle, 50/49 gives octokaidecal, and 35/32 gives wallaby. Other members of the clan discussed elsewhere are:

Trienstonian

Subgroup: 2.3.7

Comma list: 28/27

Sval mapping: [⟨1 0 -2], ⟨0 1 3]]

mapping generators: ~2, ~3

Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]

Optimal tunings:

  • WE: ~2 = 1196.254 ¢, ~3/2 = 719.306 ¢
error map: ⟨-3.746 +13.604 -14.655]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 719.606 ¢
error map: ⟨0.000 +17.651 -10.007]

Optimal ET sequence: 2d, 3d, 5

Badness (Sintel): 0.235

Father

Main article: Father

See Father family #Septimal father.

Uncle

For the 5-limit version, see Syntonic–diatonic equivalence continuum.

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Optimal tunings:

  • WE: ~2 = 1190.224 ¢, ~3/2 = 725.221 ¢
error map: ⟨-9.776 +13.490 +3.707 -2.939]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 731.394 ¢
error map: ⟨0.000 +29.439 +25.324 +25.355]

Minimax tuning:

Optimal ET sequence: 5, 13d, 18, 23bc, 41bbcd

Badness (Sintel): 1.84

Octokaidecal

The 5-limit restriction of octokaidecal is supersharp, which tempers out 800/729, the difference between the 27/20 wolf fourth and the 40/27 wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5, as a generator. This means that five steps from the Zarlino generator sequence starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.

5-limit (supersharp)

Subgroup: 2.3.5

Comma list: 800/729

Mapping: [⟨2 0 -5], ⟨0 1 3]]

mapping generators: ~27/20, ~3

Optimal tunings:

  • WE: ~27/20 = 596.986 ¢, ~3/2 = 725.434 ¢ (~10/9 = 128.448 ¢)
error map: ⟨-6.029 +17.450 -13.027]
  • CWE: ~27/20 = 600.000 ¢, ~3/2 = 726.548 ¢ (~10/9 = 126.548 ¢)
error map: ⟨0.000 +24.593 -6.670]

Optimal ET sequence: 8, 10, 18, 28b

Badness (Sintel): 2.88

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Optimal tunings:

  • WE: ~7/5 = 596.984 ¢, ~3/2 = 725.210 ¢ (~15/14 = 128.226 ¢)
error map: ⟨-6.031 +17.224 -13.699 +0.774]
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 726.319 ¢ (~15/14 = 126.319 ¢)
error map: ⟨0.000 +24.364 -7.358 +10.130]

Minimax tuning:

Optimal ET sequence: 8d, 10, 18, 28b

Badness (Sintel): 0.930

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 50/49, 55/54

Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]

Optimal tunings:

  • WE: ~7/5 = 595.139 ¢, ~3/2 = 726.397 ¢ (~15/14 = 131.258 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 729.485 ¢ (~15/14 = 129.485 ¢)

Optimal ET sequence: 8d, 10, 18e

Badness (Sintel): 1.00

Parakangaroo

For the 5-limit version of this temperament, see High badness temperaments #Kangaroo.

This temperament used to be known as kangaroo.

Subgroup: 2.3.5.7

Comma list: 28/27, 1029/1000

Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 596.984 ¢, ~10/7 = 638.135 ¢
error map: ⟨-2.883 +12.450 +3.685 -19.845]
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.302 ¢
error map: ⟨0.000 +15.952 +6.710 -15.104]

Optimal ET sequence: 2cd, …, 13cd, 15

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 77/75, 245/242

Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]

Optimal tunings:

  • WE: ~2 = 1196.971 ¢, ~10/7 = 638.230 ¢
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.480 ¢

Optimal ET sequence: 15

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 66/65, 147/143

Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]

Optimal tunings:

  • WE: ~2 = 1194.720 ¢, ~10/7 = 637.413 ¢
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.609 ¢

Optimal ET sequence: 15

Badness (Sintel): 1.35

Retrieved from "https://en.xen.wiki/index.php?title=Trienstonic_clan&oldid=204562"