@@ Line 1: / Line 1: @@
-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. 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.
-= 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|legend=3| 1 0 0 -2 | 0 1 0 3 }}
+: mapping generators: ~2, ~3
-[[Eigenmonzo subgroup]]: 2.7/5
+[[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 }}
--limit eigenmonzo subgroup: 2.9/5
+{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
-[[POTE generator]]: 742.002
+[[Badness]] (Sintel): 0.235
-Map: [&lt;1 0 4 -2|, &lt;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.
-Wedgie: &lt;&lt;1 -1 3 -4 2 10||
+/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.
-EDOs: 5, 8d, 13cd, 21bcd
+Members of the clan discussed elsewhere are:
-Badness: 0.0213
+* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]
+* [[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) → [[Blackwood family #Blackwood|Blackwood family]]
+* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
+* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
-= Uncle =
+Considered below are uncle, octokaidecal, and parakangaroo.
-Commas: 28/27, 256/245
--limit eigenmonzo subgroup: 2.5/3
+== Uncle ==
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''
--limit eigenmonzo subgroup: 2.9/5
+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.
-Map: [&lt;1 0 12 -2|, &lt;0 1 -6 3|]
+[[Subgroup]]: 2.3.5.7
-Wedgie: &lt;&lt;1 -6 3 -12 2 24||
+[[Comma list]]: 28/27, 256/245
-EDOs: 13d, 18, 23bc, 41bcd
+{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
-Badness: 0.0727
+[[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 }}
-= Octokaidecal =
+[[Minimax tuning]]:
-Commas: 28/27, 50/49
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
+* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
--limit eigenmonzo subgroup: 2.5
+{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
--limit eigenmonzo subgroup: 2.5
+[[Badness]] (Sintel): 1.84
-[[POTE generator]]: 728.874
+== 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]].
-Map: [&lt;2 0 -5 -4|, &lt;0 1 3 3|]
+[[Subgroup]]: 2.3.5.7
-Wedgie: &lt;&lt;2 6 6 5 4 -2||
+[[Comma list]]: 28/27, 50/49
-EDOs: 10, 18, 28b
+{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
-Badness: 0.0367
+[[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 }}
-== 11-limit ==
+[[Minimax tuning]]:
-Commas: 28/27, 50/49, 55/54
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-POTE generator: ~3/2 = 732.330
+{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
-Map: [&lt;2 0 -5 -4 7|, &lt;0 1 3 3 0|]
+[[Badness]] (Sintel): 0.930
-EDOs: 10, 18e
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.0302
+Comma list: 28/27, 50/49, 55/54
-= Opossum =
+Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
-Commas: 28/27, 126/125
--limit eigenmonzo subgroup: 2.7
+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}})
--limit eigenmonzo subgroup: 2.7
+{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
-[[POTE generator]]: ~10/9 = 159.691
+Badness (Sintel): 1.00
-Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -9|]
+== Parakangaroo ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
-Wedgie: &lt;&lt;3 5 9 1 6 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.
-EDOs: 7d, 8d, 15
+[[Subgroup]]: 2.3.5.7
-Badness: 0.0407
+[[Comma list]]: 28/27, 1029/1000
-== 11-limit ==
+{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
-Commas: 28/27, 55/54, 77/75
--limit eigenmonzo subgroup: 2.7
+: mapping generators: ~2, ~10/7
-[[POTE generator]]: ~10/9 = 159.807
+[[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 }}
-Map: [&lt;1 2 3 4 4|, &lt;0 -3 -5 -9 -4|]
+{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
-EDOs: 7d, 8d, 15
+[[Badness]] (Sintel): 1.97
-Badness: 0.0223
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 13-limit ==
+Comma list: 28/27, 77/75, 245/242
-Commas: 28/27, 40/39, 55/54, 66/65
--limit eigenmonzo subgroup: 2.7
+Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
--limit eigenmonzo subgroup: 2.7
+Optimal tunings:
+* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
-[[POTE generator]]: ~10/9 = 158.805
+{{Optimal ET sequence|legend=0| 15 }}
-Map: [&lt;1 2 3 4 4 4|, &lt;0 -3 -5 -9 -4 -2|]
+Badness (Sintel): 1.43
-EDOs: 7d, 8d, 15, 38bcef
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Badness: 0.0194
+Comma list: 28/27, 40/39, 66/65, 147/143
-= Wallaby =
+Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
-Commas: 28/27, 35/32
-POTE generator: ~3/2 = 691.351
+Optimal tunings:
+* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
-Map: [&lt;1 0 7 -2|, &lt;0 1 -3 3|]
+{{Optimal ET sequence|legend=0| 15 }}
-Wedgie: &lt;&lt;1 -3 3 -7 2 15||
+Badness (Sintel): 1.35
-EDOs: 2d, 5c, 19ccdd
+[[Category:Trienstonic clan| ]] <!-- main article -->
+[[Category:Temperament clans]]
-Badness: 0.0585
+[[Category:Catalogs of rank-2 temperaments]]
-= Kangaroo =
-== 5-limit ==
-Comma: 64000/59049
-POTE generator: ~27/20 = 561.195
-Map: [&lt;1 0 -3|, &lt;0 3 10|]
-EDOs: 15, 47b, 62b
-Badness: 0.2958
-== 7-limit ==
-Commas: 28/27, 1029/1000
-POTE generator: ~7/5 = 560.328
-Map: [&lt;1 0 -3 -2|, &lt;0 3 10 9|]
-Wedgie: &lt;&lt;3 10 9 9 6 -7||
-EDOs: 15
-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