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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. 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.
* ''[[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]]
* ''[[Blacksmith]]'' (+49/48) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]


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


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{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}


{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}


[[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| 0.000 +15.562 -16.274 }}
: [[error map]]: {{val| -3.746 +13.604 -14.655 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 721.559
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
: error map: {{val| 0.000 +19.604 -4.150 }}
: error map: {{val| 0.000 +17.651 -10.007 }}


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


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


== Father ==
* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]
{{Main| Father }}
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
* ''[[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]]


See [[Father family #Septimal father]].
Considered below are uncle, octokaidecal, and parakangaroo.


== Uncle ==
== Uncle ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].''
: ''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
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 0 12 -2 | 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:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 731.394
* [[WE]]: ~2 = 1190.224{{c}}, ~3/2 = 725.221{{c}}
: [[error map]]: {{val| 0.000 +30.268 +20.350 +27.842 }}
: [[error map]]: {{val| -9.776 +13.490 +3.707 -2.939 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 731.177
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}}
: error map: {{val| 0.000 +29.222 +26.622 +24.706 }}
: error map: {{val| 0.000 +29.439 +25.324 +25.355 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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]] (Smith): 0.072653
[[Badness]] (Sintel): 1.84


== Octokaidecal ==
== 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 [[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.  
: ''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]].


=== 5-limit (supersharp) ===
[[Subgroup]]: 2.3.5
[[Comma list]]: 800/729
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
: mapping generators: ~27/20, ~3
[[Optimal tuning]]s:
* [[CTE]]: ~27/20 = 600.000, ~3/2 = 723.608 (~10/9 = 123.608)
: [[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 }}
[[Badness]] (Smith): 0.122848
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 2 0 -5 -4 | 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:
[[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| 0.000 +21.416 -16.201 +1.287 }}
: [[error map]]: {{val| -6.031 +17.224 -13.699 +0.774 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 728.874 (~15/14 = 128.874)
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}})
: error map: {{val| 0.000 +26.919 +0.307 +17.795 }}
: error map: {{val| 0.000 +24.364 -7.358 +10.130 }}


[[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|unchanged-interval (eigenmonzo) basis]]: 2.5


{{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 114: Line 99:


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}})


{{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 ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kangaroo]].''
: ''For the 5-limit version, 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 133: Line 118:


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


[[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| 0.000 +14.633 +2.314 -19.061 }}
: [[error map]]: {{val| -2.883 +12.450 +3.685 -19.845 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 639.672
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
: error map: {{val| 0.000 +17.060 +10.404 -11.780 }}
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }}


{{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 153: Line 136:
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}}


{{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 168: Line 151:
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}}


{{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]]
[[Category:Trienstonic clan| ]] <!-- Main article -->
[[Category:Trienstonic clan| ]] <!-- main article -->
[[Category:Trienstonic| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Trienstonic_clan"