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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.  


== No-five 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 }}]
: mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 721.5586
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}


{{Val list|legend=1| 2d, 3d, 5, 78bb }}
[[Optimal tuning]]s:
* [[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 }}


== Father ==
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
{{see also| Father family #Father }}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 0.235


[[Comma list]]: 16/15, 28/27
=== 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.


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


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


[[POTE generator]]: ~3/2 = 742.002
Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.  


[[Minimax tuning]]:
== Father ==
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }}
{{Main| Father }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9
: Eigenmonzos (unchanged intervals): 2, 9/7


{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
See [[Father family #Septimal father]].


[[Badness]]: 0.021312
== Sharptone ==
See [[Meantone family #Sharptone]].  


== Uncle ==
== Uncle ==
Subgroup: 2.3.5.7
: ''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
[[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 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1190.224{{c}}, ~3/2 = 725.221{{c}}
[[POTE generator]]: ~3/2 = 731.177
: [[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 }}
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}


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


== Wallaby ==
== Octokaidecal ==
Subgroup: 2.3.5.7
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 [[generator sequence #JI scales obtained from guided generator sequences|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.


[[Comma list]]: 28/27, 35/32
=== 5-limit (supersharp) ===
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]
[[Comma list]]: 800/729


{{Multival|legend=1| 1 -3 3 -7 2 15 }}
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}


[[POTE generator]]: ~3/2 = 691.351
: mapping generators: ~27/20, ~3


{{Val list|legend=1| 2d, 5c, 19ccdd }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.058468
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}


== Octokaidecal ==
[[Badness]] (Sintel): 2.88
Subgroup: 2.3.5.7
 
=== 7-limit ===
[[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 }}


[[POTE generator]]: ~3/2 = 728.874
[[Optimal tuning]]s:
* [[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 97: Line 120:
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
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}})


Vals: {{Val list| 8d, 10, 18e }}
{{Optimal ET sequence|legend=0| 8d, 10, 18e }}


Badness: 0.030235
Badness (Sintel): 1.00


== Opossum ==
== Parakangaroo ==
Subgroup: 2.3.5.7
: ''For the 5-limit version of this temperament, 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 }}
 
[[POTE generator]]: ~10/9 = 159.691
 
[[Minimax tuning]]:
* [[7-odd-limit]] eigenmonzo subgroup: 2.7
* [[9-odd-limit]] eigenmonzo subgroup: 2.7
 
{{Val list|legend=1| 7d, 8d, 15 }}
 
[[Badness]]: 0.040650
 
=== 11-limit ===
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
 
Vals: {{Val list| 7d, 8d, 15 }}
 
Badness: 0.022325
 
=== 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
 
Vals: {{Val list| 7d, 8d, 15, 38bcef }}
 
Badness: 0.019389
 
== Kangaroo ==
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]]: 0.295767
 
=== 7-limit ===
Subgroup: 2.3.5.7


[[Comma list]]: 28/27, 1029/1000
[[Comma list]]: 28/27, 1029/1000


[[Mapping]]: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}


{{Multival|legend=1| 3 10 9 9 6 -7 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~7/5 = 560.328
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 190: Line 158:
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
Optimal tunings:  
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}


Vals: {{Val list| 15 }}
{{Optimal ET sequence|legend=0| 15 }}


Badness: 0.043195
Badness (Sintel): 1.43


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


Comma list: 28/27, 40/39, 77/75, 147/143
Comma list: 28/27, 40/39, 66/65, 147/143
 
Mapping: [{{val| 1 0 -3 -2 -4 0 }}, {{val| 0 3 10 9 14 7 }}]
 
POTE generator: ~7/5 = 559.770
 
Vals: {{Val list| 15 }}
 
Badness: 0.032653
 
== Quindecic ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 49/48, 55/54, 77/75


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}}


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


Badness: 0.028944
Badness (Sintel): 1.35


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Pages with mostly numerical content]]
[[Category:Trienstonic clan| ]] <!-- main article -->
[[Category:Trienstonic clan| ]] <!-- Main article -->
[[Category:Trienstonic]]
[[Category:Trienstonic| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Trienstonic_clan"