Trienstonic clan: Difference between revisions

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Unify to 3 decimal places. Misc cleanup
+Error maps, spell out 1\1 in decimal, and note the badness metric being used
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 717.517
* [[CTE]]: ~2 = 1200.000, ~3/2 = 717.517
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.559
: [[error map]]: {{val| 0.000 +15.562 -16.274 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 721.559
: 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


== Father ==
== Father ==
Line 40: Line 44:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 731.394
* [[CTE]]: ~2 = 1200.000, ~3/2 = 731.394
* [[POTE]]: ~2 = 1\1, ~3/2 = 731.177
: [[error map]]: {{val| 0.000 +30.268 +20.350 +27.842 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 731.177
: error map: {{val| 0.000 +29.222 +26.622 +24.706 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
* [[7-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
* [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
* [[9-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5


{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}


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


== Wallaby ==
== Wallaby ==
Line 63: Line 69:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 691.757
* [[CTE]]: ~2 = 1200.000, ~3/2 = 691.757
* [[POTE]]: ~2 = 1\1, ~3/2 = 691.351
: [[error map]]: {{val| 0.000 -8.515 -66.635 -88.505 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 691.351
: error map: {{val| 0.000 -10.604 -60.368 -94.772 }}


{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}
{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}


[[Badness]]: 0.058468
[[Badness]] (Smith): 0.058468


== Octokaidecal ==
== Octokaidecal ==
Line 83: Line 91:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~27/20 = 1\2, ~3/2 = 726.548
* [[CTE]]: ~27/20 = 600.000, ~3/2 = 723.608 (~10/9 = 123.608)
* [[POTE]]: ~27/20 = 1\2, ~3/2 = 729.097
: [[error map]]: {{val| 0.000 +21.653 -15.490 }}
* [[POTE]]: ~27/20 = 600.000, ~3/2 = 729.097 (~10/9 = 129.097)
: 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]]: 0.122848
[[Badness]] (Smith): 0.122848


=== 7-limit ===
=== 7-limit ===
Line 100: Line 110:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 1\2, ~3/2 = 723.371
* [[CTE]]: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
* [[POTE]]: ~7/5 = 1\2, ~3/2 = 728.874
: [[error map]]: {{val| 0.000 +21.416 -16.201 +1.287 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 728.874 (~15/14 = 128.874)
: error map: {{val| 0.000 +26.919 +0.307 +17.795 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


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


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


=== 11-limit ===
=== 11-limit ===
Line 118: Line 130:


Optimal tunings:
Optimal tunings:
* CTE: ~7/5 = 1\2, ~3/2 = 723.371
* CTE: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
* POTE: ~7/5 = 1\2, ~3/2 = 732.330
* POTE: ~7/5 = 600.000, ~3/2 = 732.330 (~15/14 = 132.330)


Optimal ET sequence: {{Optimal ET sequence| 8d, 10, 18e }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 10, 18e }}


Badness: 0.030235
Badness (Smith): 0.030235


== Parakangaroo ==
== Parakangaroo ==
Line 141: Line 153:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/7 = 638.863
* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.863
* [[POTE]]: ~2 = 1\1, ~10/7 = 639.672
: [[error map]]: {{val| 0.000 +14.633 +2.314 -19.061 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 639.672
: 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]]: 0.077857
[[Badness]] (Smith): 0.077857


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 639.036
* CTE: ~2 = 1200.000, ~10/7 = 639.036
* POTE: ~2 = 1\1, ~10/7 = 639.845
* POTE: ~2 = 1200.000, ~10/7 = 639.845


Optimal ET sequence: {{Optimal ET sequence| 15 }}
Optimal ET sequence: {{Optimal ET sequence| 15 }}


Badness: 0.043195
Badness (Smith): 0.043195


=== 13-limit ===
=== 13-limit ===
Line 171: Line 185:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 638.717
* CTE: ~2 = 1200.000, ~10/7 = 638.717
* POTE: ~2 = 1\1, ~10/7 = 640.230
* POTE: ~2 = 1200.000, ~10/7 = 640.230


Optimal ET sequence: {{Optimal ET sequence| 15 }}
Optimal ET sequence: {{Optimal ET sequence| 15 }}


Badness: 0.032653
Badness (Smith): 0.032653


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

Revision as of 09:24, 21 December 2024

The trienstonic clan of rank-2 temperaments tempers 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 sensibly be called a temperament at all; in other words, it is an exotemperament.

Adding 16/15 to 28/27 leads to father, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 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:

  • CTE: ~2 = 1200.000, ~3/2 = 717.517
error map: ⟨0.000 +15.562 -16.274]
  • POTE: ~2 = 1200.000, ~3/2 = 721.559
error map: ⟨0.000 +19.604 -4.150]

Optimal ET sequence: 2d, 3d, 5

Badness (Smith): 0.00685

Father

Main article: Father

See Father family #Septimal father.

Uncle

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Wedgie: ⟨⟨ 1 -6 3 -12 2 24 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 731.394
error map: ⟨0.000 +30.268 +20.350 +27.842]
  • POTE: ~2 = 1200.000, ~3/2 = 731.177
error map: ⟨0.000 +29.222 +26.622 +24.706]

Minimax tuning:

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

Badness (Smith): 0.072653

Wallaby

See also: Pelogic family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Wedgie: ⟨⟨ 1 -3 3 -7 2 15 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 691.757
error map: ⟨0.000 -8.515 -66.635 -88.505]
  • POTE: ~2 = 1200.000, ~3/2 = 691.351
error map: ⟨0.000 -10.604 -60.368 -94.772]

Optimal ET sequence: 2d, 5c, 7d, 19ccdd

Badness (Smith): 0.058468

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:

  • CTE: ~27/20 = 600.000, ~3/2 = 723.608 (~10/9 = 123.608)
error map: ⟨0.000 +21.653 -15.490]
  • POTE: ~27/20 = 600.000, ~3/2 = 729.097 (~10/9 = 129.097)
error map: ⟨0.000 +27.142 +0.976]

Optimal ET sequence: 8, 10, 18, 28b

Badness (Smith): 0.122848

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Wedgie: ⟨⟨ 2 6 6 5 4 -2 ]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
error map: ⟨0.000 +21.416 -16.201 +1.287]
  • POTE: ~7/5 = 600.000, ~3/2 = 728.874 (~15/14 = 128.874)
error map: ⟨0.000 +26.919 +0.307 +17.795]

Minimax tuning:

Optimal ET sequence: 8d, 10, 18, 28b

Badness (Smith): 0.036747

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:

  • CTE: ~7/5 = 600.000, ~3/2 = 723.371 (~15/14 = 123.371)
  • POTE: ~7/5 = 600.000, ~3/2 = 732.330 (~15/14 = 132.330)

Optimal ET sequence: 8d, 10, 18e

Badness (Smith): 0.030235

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

Wedgie: ⟨⟨ 3 10 9 9 6 -7 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/7 = 638.863
error map: ⟨0.000 +14.633 +2.314 -19.061]
  • POTE: ~2 = 1200.000, ~10/7 = 639.672
error map: ⟨0.000 +17.060 +10.404 -11.780]

Optimal ET sequence: 2cd, …, 13cd, 15

Badness (Smith): 0.077857

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:

  • CTE: ~2 = 1200.000, ~10/7 = 639.036
  • POTE: ~2 = 1200.000, ~10/7 = 639.845

Optimal ET sequence: 15

Badness (Smith): 0.043195

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:

  • CTE: ~2 = 1200.000, ~10/7 = 638.717
  • POTE: ~2 = 1200.000, ~10/7 = 640.230

Optimal ET sequence: 15

Badness (Smith): 0.032653

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