Trienstonic clan: Difference between revisions

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The '''trienstonic clan''' of [[rank-2 temperament|rank-2]] [[temperament]]s are low-complexity, high-error temperaments that [[tempering out|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]].
The '''trienstonic clan''' of [[rank-2 temperament|rank-2]] [[temperament]]s are low-complexity, high-error temperaments that [[tempering out|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 ==
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps.
* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]


== Trienstonian ==
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


Line 27: Line 20:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
: error map: {{val| 0.000 +17.651 -10.007 }}
: 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 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 721.559{{c}}
: 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]] (Sintel): 0.235
[[Badness]] (Sintel): 0.235
=== Overview to extensions ===
Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian.
50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period.
Members of the clan discussed elsewhere are:
* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.


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


See [[Father family #Septimal father]].
See [[Father family #Septimal father]].
== Sharptone ==
See [[Meantone family #Sharptone]].


== Uncle ==
== Uncle ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].''
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].''
Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 55: Line 64:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}}
: error map: {{val| 0.000 +29.439 +25.324 +25.355 }}
: 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 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.177{{c}}
: error map: {{val| 0.000 +29.222 +26.622 +24.706 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 85: Line 90:
* [[CWE]]: ~27/20 = 600.000{{c}}, ~3/2 = 726.548{{c}} (~10/9 = 126.548{{c}})
* [[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 }}
: 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 }}
* [[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 }} -->


{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
Line 106: Line 107:
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}})
* [[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 }}
: 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 }}
* [[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 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 127: Line 124:
Optimal tunings:
Optimal tunings:
* WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}})
* WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 729.485{{c}} (~15/14 = 129.485{{c}})  
* 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 }}
Line 136: Line 131:


== Parakangaroo ==
== Parakangaroo ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kangaroo]].''
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''


This temperament used to be known as '''kangaroo'''.  
This temperament used to be known as ''kangaroo'', but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. [[15edo]] shows us an obvious tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 153: Line 148:
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }}
: 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 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.672{{c}}
: 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 }}
Line 172: Line 163:
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
* 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 }}
Line 189: Line 178:
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
* 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 }}

Latest revision as of 11:45, 16 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.

Trienstonian

This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. 5edo is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the 7-odd-limit, leading to an oneirotonic scale or otherwise a diatonic scale with negative small steps.

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

Overview to extensions

Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian.

50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period.

Members of the clan discussed elsewhere are:

Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.

Father

Main article: Father

See Father family #Septimal father.

Sharptone

See Meantone family #Sharptone.

Uncle

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

Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.

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 Miscellaneous 5-limit temperaments #Kangaroo.

This temperament used to be known as kangaroo, but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 15edo shows us an obvious tuning.

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

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