Xenharmonic Wiki

Diminished family: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 332260622 - Original comment: **
Post-move cleanup
 
(36 intermediate revisions by 10 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 '''diminished family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the major diesis a.k.a. diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-09 01:09:29 UTC</tt>.<br>
: The original revision id was <tt>332260622</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]]


The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.
== Diminished ==
{{Main| Diminished (temperament) }}


=Dimipent=
The [[generator]] of diminished can be taken as a fifth or a semitone, and 12edo, with its excellent fifth, is an obvious tuning, though a flatter fifth might be preferred to go with the flat minor third. Its [[ploidacot]] is tetraploid monocot.
Comma: 648/625


POTE generator: ~3/2 =  699.507
[[Subgroup]]: 2.3.5


Map: [&lt;4 0 3|, &lt;0 1 1|]
[[Comma list]]: 648/625
EDOs: 8, 12
Badness: 0.0472


=Hemidim=
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
Commas: 49/48, 648/625


POTE generator: ~8/7 = 252.555
[[Optimal tuning]]s:  
* [[WE]]: ~6/5 = 299.6476{{c}}, ~3/2 = 698.6854{{c}} (~25/24 = 99.3903{{c}})
: [[error map]]: {{val| -1.410 -4.679 +9.905 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 698.2660{{c}} (~25/24 = 98.2660{{c}})
: error map: {{val| 0.000 -3.689 +11.952 }}


Map: [&lt;4 0 3 8|, &lt;0 2 2 1|]
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
Wedgie: &lt;&lt;8 8 4 -6 -16 -13||
EDOs: 24, 52d, 76cd
Badness: 0.0864


==11-limit==
[[Badness]] (Sintel): 1.11
Commas: 49/48, 77/75, 243/242


POTE generator: ~8/7 = 251.658
== Septimal diminished ==
{{Main| Diminished (temperament) }}


Map: [&lt;4 0 3 8 -2|, &lt;0 2 2 1 5|]
[[Subgroup]]: 2.3.5.7
EDOs: 24, 76cde
Badness: 0.0566


==13-limit==
[[Comma list]]: 36/35, 50/49
Commas: 49/48, 66/65, 77/75, 648/625


POTE generator: ~8/7 = 252.225
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}


Map: [&lt;4 0 3 8 -2 -1|, &lt;0 2 2 1 5 5|]
[[Optimal tuning]]s:  
EDOs: 24, 52de, 76cde
* [[WE]]: ~6/5 = 299.0347{{c}}, ~3/2 = 697.2727{{c}} (~21/20 = 99.2032{{c}})
Badness: 0.039
: [[error map]]: {{val| -3.861 -8.543 +4.202 +19.759 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 695.9619{{c}} (~21/20 = 95.9619{{c}})
: error map: {{val| 0.000 -5.993 +9.648 +27.136 }}


=Semidim=
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
Commas: 245/243, 392/375


POTE generator4: ~3/2 = 707.014
[[Badness]] (Sintel): 0.567


Map: [&lt;8 0 6 -3|, &lt;0 1 1 2|]
=== 11-limit ===
Wedgie: &lt;&lt;8 8 16 -6 3 15||
Subgroup: 2.3.5.7.11
EDOs: 24, 32c, 56c
Badness: 0.1075


==11-limit==
Comma list: 36/35, 50/49, 56/55
Commas: 56/55, 77/75, 245/243


POTE generator: ~3/2 = 706.645
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}


Map: [&lt;8 0 6 -3 15|, &lt;0 1 1 2 1|]
Optimal tunings:  
EDOs: 24, 32c, 56c
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
Badness: 0.0476
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})


==13-limit==
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}
Commas: 56/55, 66/65, 77/75, 507/500


POTE generator: ~3/2 = 707.376
Badness (Sintel): 0.732


Map: [&lt;8 0 6 -3 15 17|, &lt;0 1 1 2 1 1|]
==== 13-limit ====
EDOs: 24, 32cf, 56cf
Subgroup: 2.3.5.7.11.13
Badness: 0.0306


Comma list: 36/35, 40/39, 50/49, 66/65


Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}


Optimal tunings:
* WE: ~6/5 = 297.2520{{c}}, ~3/2 = 707.2352{{c}} (~15/14 = 112.7312{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 708.4648{{c}} (~15/14 = 108.4648{{c}})


{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}


Badness (Sintel): 0.806


=== Demolished ===
Subgroup: 2.3.5.7.11


</pre></div>
Comma list: 36/35, 45/44, 50/49
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Dimipent family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Dimipent"&gt;Dimipent&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Hemidim"&gt;Hemidim&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;a href="#Semidim"&gt;Semidim&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
 
The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo. &lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~6/5 = 299.6308{{c}}, ~3/2 = 689.0322{{c}} (~21/20 = 89.7707{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Dimipent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Dimipent&lt;/h1&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 688.9304{{c}} (~21/20 = 88.9304{{c}})
Comma: 648/625&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}
POTE generator: ~3/2 = 699.507&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.879
Map: [&amp;lt;4 0 3|, &amp;lt;0 1 1|]&lt;br /&gt;
 
EDOs: 8, 12&lt;br /&gt;
=== Cohedim ===
Badness: 0.0472&lt;br /&gt;
This extension has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[#Hemidim|11-limit 4e & 24 and 13-limit 4ef & 24]]. For the 11-limit 8bce & 12 temperament, ''cohedim'' arguably makes more sense. Its ploidacot is tetraploid alpha-dicot.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Hemidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Hemidim&lt;/h1&gt;
Subgroup: 2.3.5.7.11
Commas: 49/48, 648/625&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 50/49, 125/121
POTE generator: ~8/7 = 252.555&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
Map: [&amp;lt;4 0 3 8|, &amp;lt;0 2 2 1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 8 4 -6 -16 -13||&lt;br /&gt;
: mapping generators: ~6/5, ~11/7
EDOs: 24, 52d, 76cd&lt;br /&gt;
 
Badness: 0.0864&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~6/5 = 298.7799{{c}}, ~11/7 = 795.0744{{c}} (~12/11 = 101.2653{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Hemidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 796.0102{{c}} (~12/11 = 103.9898{{c}})
Commas: 49/48, 77/75, 243/242&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8bce, 12 }}
POTE generator: ~8/7 = 251.658&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.82
Map: [&amp;lt;4 0 3 8 -2|, &amp;lt;0 2 2 1 5|]&lt;br /&gt;
 
EDOs: 24, 76cde&lt;br /&gt;
==== 13-limit ====
Badness: 0.0566&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Hemidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Comma list: 36/35, 50/49, 66/65, 125/121
Commas: 49/48, 66/65, 77/75, 648/625&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
POTE generator: ~8/7 = 252.225&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
Map: [&amp;lt;4 0 3 8 -2 -1|, &amp;lt;0 2 2 1 5 5|]&lt;br /&gt;
* WE: ~6/5 = 298.4646{{c}}, ~11/7 = 793.6185{{c}} (~12/11 = 101.7754{{c}})
EDOs: 24, 52de, 76cde&lt;br /&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 794.7323{{c}} (~12/11 = 105.2677{{c}})
Badness: 0.039&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8bcef, 12f }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Semidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Semidim&lt;/h1&gt;
 
Commas: 245/243, 392/375&lt;br /&gt;
Badness (Sintel): 1.72
&lt;br /&gt;
 
POTE generator4: ~3/2 = 707.014&lt;br /&gt;
== Hemidim ==
&lt;br /&gt;
Hemidim tempers out 49/48 and may be described as the {{nowrap| 4 & 24 }} temperament. Its ploidcot is tetraploid dicot.
Map: [&amp;lt;8 0 6 -3|, &amp;lt;0 1 1 2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 8 16 -6 3 15||&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 24, 32c, 56c &lt;br /&gt;
 
Badness: 0.1075&lt;br /&gt;
[[Comma list]]: 49/48, 648/625
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Semidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}
Commas: 56/55, 77/75, 245/243&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~6/5, ~7/4
POTE generator: ~3/2 = 706.645&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
Map: [&amp;lt;8 0 6 -3 15|, &amp;lt;0 1 1 2 1|]&lt;br /&gt;
* [[WE]]: ~6/5 = 300.5053{{c}}, ~7/4 = 949.0409{{c}} (~36/35 = 47.5250{{c}})
EDOs: 24, 32c, 56c&lt;br /&gt;
: [[error map]]: {{val| +2.021 -3.873 +13.284 -15.743 }}
Badness: 0.0476&lt;br /&gt;
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~7/4 = 948.2575{{c}} (~36/35 = 48.2575{{c}})
&lt;br /&gt;
: error map: {{val| 0.000 -5.440 +10.201 -20.568 }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Semidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
 
Commas: 56/55, 66/65, 77/75, 507/500&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 707.376&lt;br /&gt;
[[Badness]] (Sintel): 2.19
&lt;br /&gt;
 
Map: [&amp;lt;8 0 6 -3 15 17|, &amp;lt;0 1 1 2 1 1|]&lt;br /&gt;
=== 11-limit ===
EDOs: 24, 32cf, 56cf&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0306&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Comma list: 49/48, 77/75, 243/242
 
Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}
 
Optimal tunings:
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.6958{{c}} (~36/35 = 48.4112{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.9065{{c}} (~36/35 = 48.9065{{c}})
 
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}
 
Badness (Sintel): 1.87
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 66/65, 77/75, 243/242
 
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
 
Optimal tunings:
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.2440{{c}} (~36/35 = 47.8487{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.3581{{c}} (~36/35 = 48.3581{{c}})
 
{{Optimal ET sequence|legend=0| 4ef, 24 }}
 
Badness (Sintel): 1.61
 
== Octonion ==
Octonion tempers out 245/243, and may be described as the {{nowrap| 8d & 24 }} temperament. Its ploidacot is octoploid monocot.
 
It was formerly known as ''semidim'' but renamed to avoid confusion with another temperament of the same name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 392/375
 
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}
 
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 149.6673{{c}}, ~3/2 = 705.4455{{c}} (~36/35 = 42.8910{{c}})
: [[error map]]: {{val| -2.662 +0.828 +14.474 -12.260 }}
* [[CWE]]: ~15/14 = 150.0000{{c}}, ~3/2 = 704.9636{{c}} (~36/35 = 45.0364{{c}})
: error map: {{val| 0.000 +3.008 +18.650 -8.899 }}
 
{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}
 
[[Badness]] (Sintel): 2.72
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 77/75, 245/243
 
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}
 
Optimal tunings:
* WE: ~12/11 = 149.7102{{c}}, ~3/2 = 705.2799{{c}} (~36/35 = 43.2712{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 704.9285{{c}} (~36/35 = 45.0715{{c}})
 
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }}
 
Badness (Sintel): 1.57
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 66/65, 77/75, 507/500
 
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
 
Optimal tunings:
* WE: ~12/11 = 149.6311{{c}}, ~3/2 = 705.6367{{c}} (~36/35 = 42.5188{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 705.2777{{c}} (~36/35 = 44.7223{{c}})
 
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}
 
Badness (Sintel): 1.26
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Diminished family| ]] <!-- main article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Diminished_family"