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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 sharptone, sharp, and blacksmith.
+== 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.
-= Father =
+[[Subgroup]]: 2.3.7
-{{see also| Father family #Father }}
-Commas: 16/15, 28/27
+[[Comma list]]: 28/27
--limit minimax
+{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
-[&lt;1 0 0 0|, &lt;3/2 0 -1/4 1/4|, &lt;5/2 0 1/4 -1/4|, &lt;5/2 0 -3/4 3/4|]
+: mapping generators: ~2, ~3
-[[Eigenmonzo subgroup]]: 2.7/5
+{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
--limit eigenmonzo subgroup: 2.9/5
+[[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 }}
-[[POTE generator]]: 742.002
+{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
-Map: [&lt;1 0 4 -2|, &lt;0 1 -1 3|]
+[[Badness]] (Sintel): 0.235
-Wedgie: &lt;&lt;1 -1 3 -4 2 10||
+=== 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.
-EDOs: 5, 8d, 13cd, 21bcd
+/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.
-Badness: 0.0213
+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]]
-= Uncle =
+Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.
-Commas: 28/27, 256/245
--limit eigenmonzo subgroup: 2.5/3
+== Father ==
+{{Main| Father }}
--limit eigenmonzo subgroup: 2.9/5
+See [[Father family #Septimal father]].
-Map: [&lt;1 0 12 -2|, &lt;0 1 -6 3|]
+== Sharptone ==
+See [[Meantone family #Sharptone]].
-Wedgie: &lt;&lt;1 -6 3 -12 2 24||
+== Uncle ==
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].''
-EDOs: 13d, 18, 23bc, 41bcd
+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.
-Badness: 0.0727
+[[Subgroup]]: 2.3.5.7
-= Octokaidecal =
+[[Comma list]]: 28/27, 256/245
-Commas: 28/27, 50/49
--limit eigenmonzo subgroup: 2.5
+{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
--limit eigenmonzo subgroup: 2.5
+[[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 }}
-[[POTE generator]]: 728.874
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
+* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
-Map: [&lt;2 0 -5 -4|, &lt;0 1 3 3|]
+{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
-Wedgie: &lt;&lt;2 6 6 5 4 -2||
+[[Badness]] (Sintel): 1.84
-EDOs: 10, 18, 28b
+== 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.
-Badness: 0.0367
+=== 5-limit (supersharp) ===
+[[Subgroup]]: 2.3.5
-== 11-limit ==
+[[Comma list]]: 800/729
-Commas: 28/27, 50/49, 55/54
-POTE generator: ~3/2 = 732.330
+{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
-Map: [&lt;2 0 -5 -4 7|, &lt;0 1 3 3 0|]
+: mapping generators: ~27/20, ~3
-EDOs: 10, 18e
+[[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: 0.0302
+{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
-= Opossum =
+[[Badness]] (Sintel): 2.88
-Commas: 28/27, 126/125
--limit eigenmonzo subgroup: 2.7
+=== 7-limit ===
+[[Subgroup]]: 2.3.5.7
--limit eigenmonzo subgroup: 2.7
+[[Comma list]]: 28/27, 50/49
-[[POTE generator]]: ~10/9 = 159.691
+{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
-Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -9|]
+[[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 }}
-Wedgie: &lt;&lt;3 5 9 1 6 7||
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-EDOs: 7d, 8d, 15
+{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
-Badness: 0.0407
+[[Badness]] (Sintel): 0.930
-== 11-limit ==
+=== 11-limit ===
-Commas: 28/27, 55/54, 77/75
+Subgroup: 2.3.5.7.11
--limit eigenmonzo subgroup: 2.7
+Comma list: 28/27, 50/49, 55/54
-[[POTE generator]]: ~10/9 = 159.807
+Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
-Map: [&lt;1 2 3 4 4|, &lt;0 -3 -5 -9 -4|]
+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}})
-EDOs: 7d, 8d, 15
+{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
-Badness: 0.0223
+Badness (Sintel): 1.00
-== 13-limit ==
+== Parakangaroo ==
-Commas: 28/27, 40/39, 55/54, 66/65
+: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
--limit eigenmonzo subgroup: 2.7
+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.
--limit eigenmonzo subgroup: 2.7
+[[Subgroup]]: 2.3.5.7
-[[POTE generator]]: ~10/9 = 158.805
+[[Comma list]]: 28/27, 1029/1000
-Map: [&lt;1 2 3 4 4 4|, &lt;0 -3 -5 -9 -4 -2|]
+{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
-EDOs: 7d, 8d, 15, 38bcef
+: mapping generators: ~2, ~10/7
-Badness: 0.0194
+[[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 }}
-= Wallaby =
+{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
-Commas: 28/27, 35/32
-POTE generator: ~3/2 = 691.351
+[[Badness]] (Sintel): 1.97
-Map: [&lt;1 0 7 -2|, &lt;0 1 -3 3|]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Wedgie: &lt;&lt;1 -3 3 -7 2 15||
+Comma list: 28/27, 77/75, 245/242
-EDOs: 2d, 5c, 19ccdd
+Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
-Badness: 0.0585
+Optimal tunings:
+* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
-= Kangaroo =
+{{Optimal ET sequence|legend=0| 15 }}
-== 5-limit ==
-Comma: 64000/59049
-POTE generator: ~27/20 = 561.195
+Badness (Sintel): 1.43
-Map: [&lt;1 0 -3|, &lt;0 3 10|]
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-EDOs: 15, 47b, 62b
+Comma list: 28/27, 40/39, 66/65, 147/143
-Badness: 0.2958
+Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
-== 7-limit ==
+Optimal tunings:
-Commas: 28/27, 1029/1000
+* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
-POTE generator: ~7/5 = 560.328
+{{Optimal ET sequence|legend=0| 15 }}
-Map: [&lt;1 0 -3 -2|, &lt;0 3 10 9|]
+Badness (Sintel): 1.35
-Wedgie: &lt;&lt;3 10 9 9 6 -7||
+[[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
-EDOs: 15
+[[Category:Trienstonic clan| ]] <!-- Main article -->
+[[Category:Trienstonic| ]] <!-- Key article -->
-Badness: 0.0779
-== 11-limit ==
-Commas: 28/27, 77/75, 245/242
-POTE generator: ~7/5 = 560.155
-Map: [&lt;1 0 -3 -2 -4|, &lt;0 3 10 9 14|]
-EDOs: 15
-Badness: 0.0432
-== 13-limit ==
-Commas: 28/27, 40/39, 77/75, 147/143
-POTE generator: ~7/5 = 559.770
-Map: [&lt;1 0 -3 -2 -4 0|, &lt;0 3 10 9 14 7|]
-EDOs: 15
-Badness: 0.0327
-= Quindecic =
-Commas: 28/27, 49/48, 55/54, 77/75
-POTE generator: ~13/8 = 852.924
-Map: [&lt;15 24 35 42 52 0|, &lt;0 0 0 0 0 1|]
-EDOs: 15, 30, 45bc
-Badness: 0.0289
-[[Category:Theory]]
-[[Category:Temperament clan]]
-[[Category:Trienstonic]]
 [[Category:Rank 2]]

Trienstonic clan: Difference between revisions