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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
-Period: 1\1
+[[Comma list]]: 28/27
-Optimal ([[POTE]]) generator: ~3/2 = 721.5586
+{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
-EDO generators: [[5edo|3\5]], [[8edo|5\8]], [[13edo|8\13]], [[18edo|11\18]], [[23edo|14\23]]
+{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
+: mapping generators: ~2, ~3
-Scales (Scala files):
+[[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 }}
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Subgroup: 2.3..7
+[[Badness]] (Sintel): 0.235
-Comma list: 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: [<1 0 0 -2|, <0 1 0 3|]
+/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.
-{{Val list|legend=1| 2d, 3d, 5, 78bb }}
+Members of the clan discussed elsewhere are:
-</div></div>
+* [[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) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
+* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
+* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
-= Father =
+Considered below are uncle, octokaidecal, and parakangaroo.
-{{see also| Father family #Father }}
-Period: 1\1
+== Uncle ==
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''
-Optimal ([[POTE]]) generator: ~3/2 = 742.002
+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.
-EDO generators: [[5edo|3\5]], [[8edo|5\8]], [[13edo|8\13]]
+[[Subgroup]]: 2.3.5.7
-Scales (Scala files):
+[[Comma list]]: 28/27, 256/245
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 16/15, 28/27
+[[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 }}
-Mapping: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]
+[[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
-Wedgie: {{wedgie| 1 -1 3 -4 2 10 }}
+{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
-Minimax tuning:
+[[Badness]] (Sintel): 1.84
-* 7-odd-limit
-: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 0 -1/4 1/4 }}, {{monzo| 5/2 0 1/4 -1/4 }}, {{monzo| 5/2 0 -3/4 3/4 }}]
-: [[Eigenmonzo subgroup]]: 2.7/5
-* 9-odd-limit
-: Eigenmonzo subgroup: 2.9/5
-{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
+== 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]].
-Badness: 0.0213
+[[Subgroup]]: 2.3.5.7
-</div></div>
+[[Comma list]]: 28/27, 50/49
-= Uncle =
+{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
-Period: 1\1
+[[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 }}
-Optimal ([[POTE]]) generator: ~3/2 = 731.1774
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-EDO generators: [[18edo|11\18]], [[23edo|14\23]]
+{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
-Scales (Scala files):
+[[Badness]] (Sintel): 0.930
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+=== 11-limit ===
-<div style="line-height:1.6;">Technical data</div>
+Subgroup: 2.3.5.7.11
-<div class="mw-collapsible-content">
-Comma list: 28/27, 256/245
-Mapping: [{{val| 1 0 12 -2 }}, {{val| 0 1 -6 3 }}]
-Wedgie: {{wedgie| 1 -6 3 -12 2 24 }}
-Minimax tuning:
-* 7-odd-limit eigenmonzo subgroup: 2.5/3
-* 9-odd-limit eigenmonzo subgroup: 2.9/5
-{{Val list|legend=1| 5, 13d, 18, 23bc, 41bcd }}
-Badness: 0.0727
-</div></div>
-= Octokaidecal =
-Period: 1\2
-Optimal ([[POTE]]) generator: ~3/2 = 728.874
-EDO generators: [[8edo|5\8]], [[10edo|6\10]], [[18edo|11\18]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 28/27, 50/49
-Mapping: [{{val| 2 0 -5 -4 }}, {{val| 0 1 3 3 }}]
-Wedgie: {{wedgie| 2 6 6 5 4 -2 }}
-Minimax tuning:
-* 7-odd-limit eigenmonzo subgroup: 2.5
-* 9-odd-limit eigenmonzo subgroup: 2.5
-{{Val list|legend=1| 8d, 10, 18, 28b }}
-Badness: 0.0367
-</div></div>
-== 11-limit ==
-Period: 1\2
-Optimal ([[POTE]]) generator: ~3/2 = 732.330
-EDO generators: [[8edo|5\8]], [[10edo|6\10]], [[18edo|11\18]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
 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 }}
-{{Val list|legend=1| 8d, 10, 18e }}
-Badness: 0.0302
-</div></div>
-= Opossum =
-Period: 1\1
-Optimal ([[POTE]]) generator: ~10/9 = 159.691
-EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 28/27, 126/125
-Mapping: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]
-Wedgie: {{wedgie| 3 5 9 1 6 7 }}
-Minimax tuning:
-* 7-odd-limit eigenmonzo subgroup: 2.7
-* 9-odd-limit eigenmonzo subgroup: 2.7
-{{Val list|legend=1| 7d, 8d, 15 }}
-Badness: 0.0407
-</div></div>
-== 11-limit ==
-Period: 1\1
-Optimal ([[POTE]]) generator: ~10/9 = 159.807
-EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 28/27, 55/54, 77/75
-Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
-Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7
-{{Val list|legend=1| 7d, 8d, 15 }}
-Badness: 0.0223
-</div></div>
-== 13-limit ==
-Period: 1\1
-Optimal ([[POTE]]) generator: ~10/9 = 159.805
-EDO generators: [[7edo|1\7]], [[8edo|1\8]], [[15edo|2\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-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 }}]
-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
-</div></div>
-= Wallaby =
-Period: 1\1
-Optimal ([[POTE]]) generator: ~3/2 = 691.351
-EDO generators: [[5edo|3\5]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 28/27, 35/32
-Mapping: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]
-Wedgie: {{wedgie| 1 -3 3 -7 2 15 }}
-{{Val list|legend=1| 2d, 5c, 19ccdd }}
-Badness: 0.0585
-</div></div>
-= Kangaroo =
-== 5-limit ==
-Period: 1\1
+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}})
-Optimal ([[POTE]]) generator: ~27/20 = 561.195
+{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
-EDO generators: [[15edo|9\15]]
+Badness (Sintel): 1.00
-Scales (Scala files):
+== Parakangaroo ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+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.
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 64000/59049
+[[Subgroup]]: 2.3.5.7
-Mapping: [{{val| 1 0 -3 }}, {{val| 0 3 10 }}]
+[[Comma list]]: 28/27, 1029/1000
-{{Val list|legend=1| 15, 47b, 62b }}
+{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
-Badness: 0.2958
+: mapping generators: ~2, ~10/7
-</div></div>
+[[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 }}
-== 7-limit ==
+{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
-Period: 1\1
+[[Badness]] (Sintel): 1.97
-Optimal ([[POTE]]) generator: ~7/5 = 560.328
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-EDO generators: [[15edo|7\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 28/27, 1029/1000
-Mapping: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]
-Wedgie: {{wedgie| 3 10 9 9 6 -7 }}
-{{Val list|legend=1| 15 }}
-Badness: 0.0779
-</div></div>
-== 11-limit ==
-Period: 1\1
-Optimal ([[POTE]]) generator: ~7/5 = 560.155
-EDO generators: [[15edo|7\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
 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 }}
-{{Val list|legend=1| 15 }}
-Badness: 0.0432
-</div></div>
-== 13-limit ==
-Period: 1\1
-Optimal ([[POTE]]) generator: ~7/5 = 559.770
-EDO generators: [[15edo|7\15]]
-Scales (Scala files):
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-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 }}]
-{{Val list|legend=1| 15 }}
-Badness: 0.0327
-</div></div>
-= Quindecic =
-Period: 1\15
-Optimal ([[POTE]]) generator: ~13/8 = 852.924
+Optimal tunings:
+* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
-EDO generators: [[30edo|21\30]], [[45edo|32\45]]
+{{Optimal ET sequence|legend=0| 15 }}
-Scales (Scala files):
+Badness (Sintel): 1.43
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+=== 13-limit ===
-<div style="line-height:1.6;">Technical data</div>
+Subgroup: 2.3.5.7.11.13
-<div class="mw-collapsible-content">
-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 }}
-{{Val list|legend=1| 15, 30, 45bc }}
+Optimal tunings:
+* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
-Badness: 0.0289
+{{Optimal ET sequence|legend=0| 15 }}
-</div></div>
+Badness (Sintel): 1.35
-[[Category:Theory]]
+[[Category:Temperament clans]]
-[[Category:Temperament clan]]
+[[Category:Trienstonic clan| ]] <!-- main article -->
-[[Category:Trienstonic]]
 [[Category:Rank 2]]

Trienstonic clan: Difference between revisions