Trienstonic clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 277116006 - Original comment: **
+ missing intro to each entry
 
(67 intermediate revisions by 13 users not shown)
Line 1: Line 1:
<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>2011-11-18 19:24:50 UTC</tt>.<br>
: The original revision id was <tt>277116006</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">[[toc|flat]]


Adding 16/15 to the trienstonic comma of 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
Commas: 16/15, 28/27


7-limit minimax
[[Comma list]]: 28/27
[&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|]
[[Eigenmonzo subgroup]]: 2.7/5


9-limit eigenmonzo subgroup: 2.9/5
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}


[[POTE tuning|POTE generator]]: 742.002
: mapping generators: ~2, ~3


Map: [&lt;1 0 4 -2|, &lt;0 1 -1 3|]
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
Wedgie: &lt;&lt;1 -1 3 -4 2 10||
EDOs: 5, 8d, 13cd, 21bcd
Badness: 0.0213


=Uncle=
[[Optimal tuning]]s:
Commas: 28/27, 256/245
* [[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 }}


7-limit eigenmonzo subgroup: 2.5/3
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
9-limit eigenmonzo subgroup: 2.9/5


Map: [&lt;1 0 12 -2|, &lt;0 1 -6 3|]
[[Badness]] (Sintel): 0.235
Wedgie: &lt;&lt;1 -6 3 -12 2 24||
EDOs: 13d, 18, 23bc, 41bcd
Badness: 0.0727


=Octokaidecal=
=== Overview to extensions ===
Commas: 28/27, 50/49
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.


7-limit eigenmonzo subgroup: 2.5
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.  
9-limit eigenmonzo subgroup: 2.5


[[POTE tuning|POTE generator]]: 728.874
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]]


Map: [&lt;2 0 -5 -4|, &lt;0 1 3 3|]
Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.  
Wedgie: &lt;&lt;2 6 6 5 4 -2||
EDOs: 10, 18, 28b
Badness: 0.0367


=Opossum=
== Father ==
Commas: 28/27, 126/125
{{Main| Father }}


7-limit eigenmonzo subgroup: 2.7
See [[Father family #Septimal father]].
9-limit eigenmonzo subgroup: 2.7


[[POTE tuning|POTE generator]]: ~10/9 = 159.691
== Sharptone ==
See [[Meantone family #Sharptone]].  


Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -9|]
== Uncle ==
Wedgie: &lt;&lt;3 5 9 1 6 7||
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].''
EDOs: 7d, 8d, 15
Badness: 0.0407


==11-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.
Commas: 28/27, 55/54, 77/75


11-limit eigenmonzo subgroup: 2.7
[[Subgroup]]: 2.3.5.7


[[POTE tuning|POTE generator]]: ~10/9 = 159.807
[[Comma list]]: 28/27, 256/245


Map: [&lt;1 2 3 4 4|, &lt;0 -3 -5 -9 -4|]
{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
EDOs: 7d, 8d, 15
Badness: 0.0223


==13-limit==
[[Optimal tuning]]s:
Commas: 28/27, 40/39, 55/54, 66/65
* [[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 }}


13-limit eigenmonzo subgroup: 2.7
[[Minimax tuning]]:
15-limit eigenmonzo subgroup: 2.7
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[POTE tuning|POTE generator]]: ~10/9 = 158.805
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}


Map: [&lt;1 2 3 4 4 4|, &lt;0 -3 -5 -9 -4 -2|]
[[Badness]] (Sintel): 1.84
EDOs: 7d, 8d, 15, 38bcef
 
Badness: 0.0194</pre></div>
== Octokaidecal ==
<h4>Original HTML content:</h4>
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.  
<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;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Father"&gt;Father&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Uncle"&gt;Uncle&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Octokaidecal"&gt;Octokaidecal&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Opossum"&gt;Opossum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
=== 5-limit (supersharp) ===
Adding 16/15 to the trienstonic comma of 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.&lt;br /&gt;
[[Subgroup]]: 2.3.5
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Father"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Father&lt;/h1&gt;
[[Comma list]]: 800/729
Commas: 16/15, 28/27&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
7-limit minimax&lt;br /&gt;
 
[&amp;lt;1 0 0 0|, &amp;lt;3/2 0 -1/4 1/4|, &amp;lt;5/2 0 1/4 -1/4|, &amp;lt;5/2 0 -3/4 3/4|]&lt;br /&gt;
: mapping generators: ~27/20, ~3
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.7/5&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
9-limit eigenmonzo subgroup: 2.9/5&lt;br /&gt;
* [[WE]]: ~27/20 = 596.986{{c}}, ~3/2 = 725.434{{c}} (~10/9 = 128.448{{c}})
&lt;br /&gt;
: [[error map]]: {{val| -6.029 +17.450 -13.027 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 742.002&lt;br /&gt;
* [[CWE]]: ~27/20 = 600.000{{c}}, ~3/2 = 726.548{{c}} (~10/9 = 126.548{{c}})
&lt;br /&gt;
: error map: {{val| 0.000 +24.593 -6.670 }}
Map: [&amp;lt;1 0 4 -2|, &amp;lt;0 1 -1 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -1 3 -4 2 10||&lt;br /&gt;
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
EDOs: 5, 8d, 13cd, 21bcd&lt;br /&gt;
 
Badness: 0.0213&lt;br /&gt;
[[Badness]] (Sintel): 2.88
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Uncle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Uncle&lt;/h1&gt;
=== 7-limit ===
Commas: 28/27, 256/245&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
7-limit eigenmonzo subgroup: 2.5/3&lt;br /&gt;
[[Comma list]]: 28/27, 50/49
9-limit eigenmonzo subgroup: 2.9/5&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
Map: [&amp;lt;1 0 12 -2|, &amp;lt;0 1 -6 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -6 3 -12 2 24||&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 13d, 18, 23bc, 41bcd&lt;br /&gt;
* [[WE]]: ~7/5 = 596.984{{c}}, ~3/2 = 725.210{{c}} (~15/14 = 128.226{{c}})
Badness: 0.0727&lt;br /&gt;
: [[error map]]: {{val| -6.031 +17.224 -13.699 +0.774 }}
&lt;br /&gt;
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Octokaidecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Octokaidecal&lt;/h1&gt;
: error map: {{val| 0.000 +24.364 -7.358 +10.130 }}
Commas: 28/27, 50/49&lt;br /&gt;
 
&lt;br /&gt;
[[Minimax tuning]]:
7-limit eigenmonzo subgroup: 2.5&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
9-limit eigenmonzo subgroup: 2.5&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 728.874&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.930
Map: [&amp;lt;2 0 -5 -4|, &amp;lt;0 1 3 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;2 6 6 5 4 -2||&lt;br /&gt;
=== 11-limit ===
EDOs: 10, 18, 28b&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0367&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 28/27, 50/49, 55/54
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Opossum"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Opossum&lt;/h1&gt;
 
Commas: 28/27, 126/125&lt;br /&gt;
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}
&lt;br /&gt;
 
7-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
Optimal tunings:
9-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
* WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}})
&lt;br /&gt;
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 729.485{{c}} (~15/14 = 129.485{{c}})
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 159.691&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -3 -5 -9|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 5 9 1 6 7||&lt;br /&gt;
Badness (Sintel): 1.00
EDOs: 7d, 8d, 15&lt;br /&gt;
 
Badness: 0.0407&lt;br /&gt;
== Parakangaroo ==
&lt;br /&gt;
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Opossum-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
 
Commas: 28/27, 55/54, 77/75&lt;br /&gt;
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.
&lt;br /&gt;
 
11-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 159.807&lt;br /&gt;
[[Comma list]]: 28/27, 1029/1000
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 4|, &amp;lt;0 -3 -5 -9 -4|]&lt;br /&gt;
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
EDOs: 7d, 8d, 15&lt;br /&gt;
 
Badness: 0.0223&lt;br /&gt;
: mapping generators: ~2, ~10/7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Opossum-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
[[Optimal tuning]]s:
Commas: 28/27, 40/39, 55/54, 66/65&lt;br /&gt;
* [[WE]]: ~2 = 596.984{{c}}, ~10/7 = 638.135{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -2.883 +12.450 +3.685 -19.845 }}
13-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
15-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 158.805&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 4 4|, &amp;lt;0 -3 -5 -9 -4 -2|]&lt;br /&gt;
[[Badness]] (Sintel): 1.97
EDOs: 7d, 8d, 15, 38bcef&lt;br /&gt;
 
Badness: 0.0194&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 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]]

Latest revision as of 11:45, 16 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The trienstonic clan of rank-2 temperaments are low-complexity, high-error temperaments that 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.

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 oneirotonic scale or otherwise a diatonic scale with negative small steps.

Subgroup: 2.3.7

Comma list: 28/27

Sval mapping: [⟨1 0 -2], ⟨0 1 3]]

mapping generators: ~2, ~3

Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]

Optimal tunings:

  • WE: ~2 = 1196.254 ¢, ~3/2 = 719.306 ¢
error map: ⟨-3.746 +13.604 -14.655]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 719.606 ¢
error map: ⟨0.000 +17.651 -10.007]

Optimal ET sequence: 2d, 3d, 5

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:

Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.

Father

Main article: 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: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Optimal tunings:

  • WE: ~2 = 1190.224 ¢, ~3/2 = 725.221 ¢
error map: ⟨-9.776 +13.490 +3.707 -2.939]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 731.394 ¢
error map: ⟨0.000 +29.439 +25.324 +25.355]

Minimax tuning:

Optimal ET sequence: 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 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: [⟨2 0 -5], ⟨0 1 3]]

mapping generators: ~27/20, ~3

Optimal tunings:

  • WE: ~27/20 = 596.986 ¢, ~3/2 = 725.434 ¢ (~10/9 = 128.448 ¢)
error map: ⟨-6.029 +17.450 -13.027]
  • CWE: ~27/20 = 600.000 ¢, ~3/2 = 726.548 ¢ (~10/9 = 126.548 ¢)
error map: ⟨0.000 +24.593 -6.670]

Optimal ET sequence: 8, 10, 18, 28b

Badness (Sintel): 2.88

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Optimal tunings:

  • WE: ~7/5 = 596.984 ¢, ~3/2 = 725.210 ¢ (~15/14 = 128.226 ¢)
error map: ⟨-6.031 +17.224 -13.699 +0.774]
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 726.319 ¢ (~15/14 = 126.319 ¢)
error map: ⟨0.000 +24.364 -7.358 +10.130]

Minimax tuning:

Optimal ET sequence: 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: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]

Optimal tunings:

  • WE: ~7/5 = 595.139 ¢, ~3/2 = 726.397 ¢ (~15/14 = 131.258 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 729.485 ¢ (~15/14 = 129.485 ¢)

Optimal ET sequence: 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: [⟨1 0 -3 -2], ⟨0 3 10 9]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 596.984 ¢, ~10/7 = 638.135 ¢
error map: ⟨-2.883 +12.450 +3.685 -19.845]
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.302 ¢
error map: ⟨0.000 +15.952 +6.710 -15.104]

Optimal ET sequence: 2cd, …, 13cd, 15

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 77/75, 245/242

Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]

Optimal tunings:

  • WE: ~2 = 1196.971 ¢, ~10/7 = 638.230 ¢
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.480 ¢

Optimal ET sequence: 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: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]

Optimal tunings:

  • WE: ~2 = 1194.720 ¢, ~10/7 = 637.413 ¢
  • CWE: ~2 = 1200.000 ¢, ~10/7 = 639.609 ¢

Optimal ET sequence: 15

Badness (Sintel): 1.35

Retrieved from "https://en.xen.wiki/index.php?title=Trienstonic_clan&oldid=204618"