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The '''trienstonic clan''' of temperaments tempers out [[28/27]], the septimal third-tone or trienstonic comma.  
{{Technical data page}}
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 temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are [[Dicot family #Sharp|sharp]], [[Archytas clan #Blacksmith|blacksmith]], and [[Meantone family #Sharptone|sharptone]].  
== 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 [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones, being on the other side of 3\5 from [[archy]] fifths, just like how mavila fifths are on the other side of 4\7 from [[meantone]] fifths.


== Trienstonic  ==
[[Subgroup]]: 2.3.7
 
Subgroup: 2.3.7


[[Comma list]]: 28/27
[[Comma list]]: 28/27


[[Sval]] [[mapping]]: [{{val| 1 0 -2 }}, {{val| 0 1 3 }}]
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
 
[[Gencom]] mapping: [{{val| 1 0 0 -2 }}, {{val| 0 1 0 3 }}]
 
[[POTE generator]]: ~3/2 = 721.5586


{{Val list|legend=1| 2d, 3d, 5, 78bb }}
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
: mapping generators: ~2, ~3


== Father ==
[[Optimal tuning]]s:
{{see also| Father family #Father }}
* [[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 }}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}


[[Comma list]]: 16/15, 28/27
[[Badness]] (Sintel): 0.235


[[Mapping]]: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]
=== 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.


{{Multival|legend=1| 1 -1 3 -4 2 10 }}
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.


[[POTE generator]]: ~3/2 = 742.002
Members of the clan discussed elsewhere are:  


[[Minimax tuning]]:
* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }}
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
: [[Eigenmonzo]]s: 2, 7/5
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
: Eigenmonzos: 2, 9/7
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]


{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
Considered below are uncle, octokaidecal, and parakangaroo.


[[Badness]]: 0.021312
== Uncle ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''


== Uncle ==
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


[[Comma list]]: 28/27, 256/245
[[Comma list]]: 28/27, 256/245


[[Mapping]]: [{{val| 1 0 12 -2 }}, {{val| 0 1 -6 3 }}]
{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
 
{{Multival|legend=1| 1 -6 3 -12 2 24 }}


[[POTE generator]]: ~3/2 = 731.1774
[[Optimal tuning]]s:
* [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzo subgroup: 2.5/3
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]] eigenmonzo subgroup: 2.9/5
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 
{{Val list|legend=1| 5, 13d, 18, 23bc, 41bcd }}
 
[[Badness]]: 0.0727


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


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 1.84


[[Comma list]]: 28/27, 35/32
== Octokaidecal ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Supersharp]].''
Octokaidecal extends trienstonian by tempering out [[50/49]], thus splitting the octave in half. It generates the [[8L 2s]] (taric) mos scale, with tunings on the other side of [[10edo]] as [[2L 8s]] (jaric). Compared to [[pajara]], decatonic thirds ([[8/7]] and [[7/6]]) and fourths ([[6/5]] and [[5/4]]) have their mappings reversed, meaning octokaidecal can be considered what is to pajara as [[mavila]] is to [[meantone]].


[[Mapping]]: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]
[[Subgroup]]: 2.3.5.7
 
{{Multival|legend=1| 1 -3 3 -7 2 15 }}
 
[[POTE generator]]: ~3/2 = 691.351
 
{{Val list|legend=1| 2d, 5c, 19ccdd }}
 
[[Badness]]: 0.0585
 
== Octokaidecal  ==
 
Subgroup: 2.3.5.7


[[Comma list]]: 28/27, 50/49
[[Comma list]]: 28/27, 50/49


[[Mapping]]: [{{val| 2 0 -5 -4 }}, {{val| 0 1 3 3 }}]
{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}


{{Multival|legend=1| 2 6 6 5 4 -2 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~7/5 = 596.984{{c}}, ~3/2 = 725.210{{c}} (~15/14 = 128.226{{c}})
[[POTE generator]]: ~3/2 = 728.874
: [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzo subgroup: 2.5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]] eigenmonzo subgroup: 2.5
 
{{Val list|legend=1| 8d, 10, 18, 28b }}


[[Badness]]: 0.0367
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}


=== 11-limit  ===
[[Badness]] (Sintel): 0.930


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 2 0 -5 -4 7 }}, {{val| 0 1 3 3 0 }}]
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
 
POTE generator: ~3/2 = 732.330


{{Val list|legend=1| 8d, 10, 18e }}
Optimal tunings:
* 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}})


Badness: 0.0302
{{Optimal ET sequence|legend=0| 8d, 10, 18e }}


== Opossum  ==
Badness (Sintel): 1.00


Subgroup: 2.3.5.7
== Parakangaroo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''


[[Comma list]]: 28/27, 126/125
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.


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 3 5 9 1 6 7 }}
[[Comma list]]: 28/27, 1029/1000


[[POTE generator]]: ~10/9 = 159.691
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}


[[Minimax tuning]]:  
: mapping generators: ~2, ~10/7
* 7-odd-limit eigenmonzo subgroup: 2.7
* 9-odd-limit eigenmonzo subgroup: 2.7


{{Val list|legend=1| 7d, 8d, 15 }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.0407
{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}


=== 11-limit  ===
[[Badness]] (Sintel): 1.97
 
Subgroup: 2.3.5.7.11
 
Comma list: 28/27, 55/54, 77/75
 
Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
 
POTE generator: ~10/9 = 159.807
 
Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7
 
{{Val list|legend=1| 7d, 8d, 15 }}
 
Badness: 0.0223
 
=== 13-limit  ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 55/54, 66/65
 
Mapping: [{{val| 1 2 3 4 4 4 }}, {{val| 0 -3 -5 -9 -4 -2 }}]
 
POTE generator: ~10/9 = 159.805
 
Minimax tuning:
* 13-odd-limit eigenmonzo subgroup: 2.7
* 15-odd-limit eigenmonzo subgroup: 2.7
 
{{Val list|legend=1| 7d, 8d, 15, 38bcef }}
 
Badness: 0.0194
 
== Kangaroo  ==
=== 5-limit  ===
 
Subgroup: 2.3.5
 
Comma list: 64000/59049
 
Mapping: [{{val| 1 0 -3 }}, {{val| 0 3 10 }}]
 
POTE generator: ~27/20 = 561.195
 
{{Val list|legend=1| 15, 47b, 62b }}
 
Badness: 0.2958
 
=== 7-limit  ===
 
Subgroup: 2.3.5.7
 
Comma list: 28/27, 1029/1000
 
Mapping: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]
 
{{Multival|legend=1| 3 10 9 9 6 -7 }}
 
POTE generator: ~7/5 = 560.328
 
{{Val list|legend=1| 15 }}
 
Badness: 0.0779
 
=== 11-limit  ===


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 0 -3 -2 -4 }}, {{val| 0 3 10 9 14 }}]
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
 
POTE generator: ~7/5 = 560.155
 
{{Val list|legend=1| 15 }}
 
Badness: 0.0432
 
=== 13-limit  ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 77/75, 147/143
 
Mapping: [{{val| 1 0 -3 -2 -4 0 }}, {{val| 0 3 10 9 14 7 }}]
 
POTE generator: ~7/5 = 559.770


{{Val list|legend=1| 15 }}
Optimal tunings:
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}


Badness: 0.0327
{{Optimal ET sequence|legend=0| 15 }}


== Quindecic  ==
Badness (Sintel): 1.43


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 28/27, 49/48, 55/54, 77/75
Comma list: 28/27, 40/39, 66/65, 147/143


Mapping: [{{val| 15 24 35 42 52 0 }}, {{val| 0 0 0 0 0 1 }}]
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}


POTE generator: ~13/8 = 852.924
Optimal tunings:  
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}


{{Val list|legend=1| 15, 30, 45bc }}
{{Optimal ET sequence|legend=0| 15 }}


Badness: 0.0289
Badness (Sintel): 1.35


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Trienstonic clan| ]] <!-- main article -->
[[Category:Trienstonic clan| ]] <!-- main article -->
[[Category:Trienstonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 01:45, 16 May 2026

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. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the pentic range, instead generating an antipentic scale. Trienstonian can be considered the 2.3.7 analog of mavila temperament, with extremely sharp fifths rather than extremely flat ones, being on the other side of 3\5 from archy fifths, just like how mavila fifths are on the other side of 4\7 from meantone fifths.

Subgroup: 2.3.7

Comma list: 28/27

Subgroup-val mapping: [⟨1 0 -2], ⟨0 1 3]]

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

mapping generators: ~2, ~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 uncle, octokaidecal, and parakangaroo.

Uncle

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Uncle (5-limit).

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

For the 5-limit version, see Miscellaneous 5-limit temperaments #Supersharp.

Octokaidecal extends trienstonian by tempering out 50/49, thus splitting the octave in half. It generates the 8L 2s (taric) mos scale, with tunings on the other side of 10edo as 2L 8s (jaric). Compared to pajara, decatonic thirds (8/7 and 7/6) and fourths (6/5 and 5/4) have their mappings reversed, meaning octokaidecal can be considered what is to pajara as mavila is to meantone.

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, 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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