Trienstonic clan: Difference between revisions

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Technical data page}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+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]].
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-20 21:59:24 UTC</tt>.<br>
-: The original revision id was <tt>189481425</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Adding 16/15 to the trienstonic comma of 28/27 leads to father temperament, and adding 50/49 gives octokaidecal. Other members of the clan discussed elsewhere are sharptone, sharp, blacksmith and opossum.
-===Father===
+== Trienstonian ==
-Commas: 16/15, 28/27
+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.
-[[POTE tuning|POTE generator]]: 742.002
+[[Subgroup]]: 2.3.7
-Map: [&lt;1 0 4 -2|, &lt;0 1 -1 3|]
+[[Comma list]]: 28/27
-Wedgie: &lt;&lt;1 -1 3 -4 2 10||
-EDOs: 5, 8, 13, 21, 55, 76
-===Octokaidecal===
+{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}
-Commas: 28/27, 50/49</pre></div>
-<h4>Original HTML content:</h4>
+: mapping generators: ~2, ~3
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Trienstonic clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Adding 16/15 to the trienstonic comma of 28/27 leads to father temperament, and adding 50/49 gives octokaidecal. Other members of the clan discussed elsewhere are sharptone, sharp, blacksmith and opossum.&lt;br /&gt;
-&lt;br /&gt;
+{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Father"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Father&lt;/h3&gt;
-Commas: 16/15, 28/27&lt;br /&gt;
+[[Optimal tuning]]s:
-&lt;br /&gt;
+* [[WE]]: ~2 = 1196.254{{c}}, ~3/2 = 719.306{{c}}
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 742.002&lt;br /&gt;
+: [[error map]]: {{val| -3.746 +13.604 -14.655 }}
-&lt;br /&gt;
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
-Map: [&amp;lt;1 0 4 -2|, &amp;lt;0 1 -1 3|]&lt;br /&gt;
+: error map: {{val| 0.000 +17.651 -10.007 }}
-Wedgie: &amp;lt;&amp;lt;1 -1 3 -4 2 10||&lt;br /&gt;
-EDOs: 5, 8, 13, 21, 55, 76&lt;br /&gt;
+{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Octokaidecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Octokaidecal&lt;/h3&gt;
+[[Badness]] (Sintel): 0.235
-Commas: 28/27, 50/49&lt;/body&gt;&lt;/html&gt;</pre></div>
+=== 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.
+/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:
+* ''[[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]]
+Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.
+== Father ==
+{{Main| Father }}
+See [[Father family #Septimal father]].
+== Sharptone ==
+See [[Meantone family #Sharptone]].
+== Uncle ==
+: ''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
+{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
+[[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 }}
+[[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
+{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
+[[Badness]] (Sintel): 1.84
+== 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.
+=== 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:
+* [[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 }}
+{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
+[[Badness]] (Sintel): 2.88
+=== 7-limit ===
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 28/27, 50/49
+{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
+[[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]]:
+* [[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 }}
+[[Badness]] (Sintel): 0.930
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 28/27, 50/49, 55/54
+Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
+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 ET sequence|legend=0| 8d, 10, 18e }}
+Badness (Sintel): 1.00
+== Parakangaroo ==
+: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #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
+[[Comma list]]: 28/27, 1029/1000
+{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
+: mapping generators: ~2, ~10/7
+[[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 }}
+{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
+[[Badness]] (Sintel): 1.97
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 28/27, 77/75, 245/242
+Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
+Optimal tunings:
+* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
+{{Optimal ET sequence|legend=0| 15 }}
+Badness (Sintel): 1.43
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 28/27, 40/39, 66/65, 147/143
+Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
+Optimal tunings:
+* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
+{{Optimal ET sequence|legend=0| 15 }}
+Badness (Sintel): 1.35
+[[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Trienstonic clan| ]] <!-- Main article -->
+[[Category:Trienstonic| ]] <!-- Key article -->
+[[Category:Rank 2]]