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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 '''minortone family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[minortone comma]] ({{monzo|legend=1| -16 35 -17 }}).  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-12 14:02:51 UTC</tt>.<br>
: The original revision id was <tt>209871482</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">This tempers out the minortone comma, |-16 35 -17&gt;. The head of the family is minortonic temperament, with generator a minor tone.


Comma: |-16 35 -17&gt;
== Minortone ==
The head of this family is the 5-limit minortone temperament, a super-accurate microtemperament with generator a minor tone of [[10/9]], seventeen of which make the [[6/1|6th harmonic]], and another eighteen generators make the interval class of the [[5/1|5th harmonic]]. Its [[ploidacot]] is beta-17-cot. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. [[171edo|171-]], [[559edo|559-]] and [[730edo]] are among the possible edo tunings.


[[POTE tuning|POTE generator]]: ~10/9 = 182.466
[[Subgroup]]: 2.3.5


Map: [&lt;1 16 32|, &lt;0 -17 -35|]
[[Comma list]]: {{monzo| -16 35 -17 }}
EDOs: 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091


[[toc|flat]]
{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
: mapping generators: ~2, ~10/9


=Mitonic=
[[Optimal tuning]]s:
As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, |-16 35 -17&gt;. Flipping that gives the 5-limit wedgie &lt;&lt;17 35 16||, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
* [[WE]]: ~2 = 1200.0036{{c}}, ~10/9 = 182.4670{{c}}
: [[error map]]: {{val| +0.004 -0.020 +0.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4666{{c}}
: error map: {{val| 0.000 -0.023 +0.017 }}


However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. The wedgie is now &lt;&lt;17 35 -21 16 -81 -147||, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.
{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091, 25859b }}


Commas: 4375/4374, 2100875/2097152
[[Badness]] (Sintel): 0.698


POTE generator: ~10/9 = 182.458
== Mitonic ==
Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo|559]] or [[730edo|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 171e). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. The latter further branches to goldmine (46 & 125f), which equates [[16/13]] with a stack of two [[10/9]]'s and tempers together [[13/12]] and [[14/13]], and orefield (46 & 125), which uses the same mapping for 13 as mineral. All have the same mapping for 17, at -6 generators. The word ''mineral'' is related to ''mine'' and ''miner'', a pun on ''minor''.  
EDOs: 7, 13, 33, 46, 125, 171
Badness: 0.0252


=Domain=
[[Subgroup]]: 2.3.5.7
Domain temperament adds the landscape comma, 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[Chromatic pairs#Terrain|terrain]].


Commas: 250047/250000, 645700815/645657712
[[Comma list]]: 4375/4374, 2100875/2097152


POTE generator: ~10/9 = 182.467
{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}


Map: [&lt;3 14 26 28|, &lt;0 -17 -35 -36|]
[[Optimal tuning]]s:  
EDOS: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038
* [[WE]]: ~2 = 1200.0963{{c}}, ~10/9 = 182.4727{{c}}
Badness: 0.0140</pre></div>
: [[error map]]: {{val| +0.096 -0.015 -0.057 -0.176 }}
<h4>Original HTML content:</h4>
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4592{{c}}
<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;Minortonic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This tempers out the minortone comma, |-16 35 -17&amp;gt;. The head of the family is minortonic temperament, with generator a minor tone.&lt;br /&gt;
: error map: {{val| 0.000 -0.149 -0.243 -0.468 }}
&lt;br /&gt;
 
Comma: |-16 35 -17&amp;gt;&lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 182.466&lt;br /&gt;
[[Badness]] (Sintel): 0.637
&lt;br /&gt;
 
Map: [&amp;lt;1 16 32|, &amp;lt;0 -17 -35|]&lt;br /&gt;
=== Mineral ===
EDOs: 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextTocRule:4:&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:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#Mitonic"&gt;Mitonic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt; | &lt;a href="#Domain"&gt;Domain&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;
Comma list: 441/440, 4375/4374, 16384/16335
&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Mitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Mitonic&lt;/h1&gt;
{{Mapping|legend=0| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}
As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, |-16 35 -17&amp;gt;. Flipping that gives the 5-limit wedgie &amp;lt;&amp;lt;17 35 16||, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning:  
However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt;. The wedgie is now &amp;lt;&amp;lt;17 35 -21 16 -81 -147||, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.&lt;br /&gt;
* WE: ~2 = 1200.8759{{c}}, ~10/9 = 182.4631{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4818{{c}}
Commas: 4375/4374, 2100875/2097152&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
POTE generator: ~10/9 = 182.458&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.95
Map: [&amp;lt;1 16 32 -15|, &amp;lt;0 -17 -35 21|]&lt;br /&gt;
 
EDOs: 7, 13, 33, 46, 125, 171&lt;br /&gt;
==== 13-limit ====
Badness: 0.0252&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Domain"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Domain&lt;/h1&gt;
Comma list: 364/363, 441/440, 3584/3575, 4375/4374
Domain temperament adds the landscape comma, 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament &lt;a class="wiki_link" href="/Chromatic%20pairs#Terrain"&gt;terrain&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}
Commas: 250047/250000, 645700815/645657712&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
POTE generator: ~10/9 = 182.467&lt;br /&gt;
* WE: ~2 = 1199.8808{{c}}, ~10/9 = 182.4633{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
Map: [&amp;lt;3 14 26 28|, &amp;lt;0 -17 -35 -36|]&lt;br /&gt;
 
EDOS: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038&lt;br /&gt;
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
Badness: 0.0140&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Sintel): 1.37
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575
 
{{Mapping|legend=0| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8923{{c}}, ~10/9 = 182.4650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
 
Badness (Sintel): 1.01
 
=== Ore ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 4375/4374
 
{{Mapping|legend=0| 1 -1 -3 6 3 | 0 17 35 -21 3 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3408{{c}}, ~10/9 = 182.5004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4518{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.77
 
==== Goldmine ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 385/384, 1331/1323
 
{{Mapping|legend=0| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
 
Optimal tunings:  
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}
 
{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}
 
Badness (Sintel): 1.62
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
 
{{Mapping|legend=0| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}
 
Badness (Sintel): 1.40
 
==== Orefield ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 1331/1323, 3267/3250
 
{{Mapping|legend=0| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
 
Optimal tunings:
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.91
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
 
{{Mapping|legend=0| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.45
 
=== Seminar ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 2100875/2097152
 
{{Mapping|legend=0| 2 -2 -6 12 13 | 0 17 35 -21 -20 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0535{{c}}, ~10/9 = 182.4735{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4578{{c}}
 
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 3466cddee, 3808bcddeee, …, 5176bccdddeeee }}
 
Badness (Sintel): 0.886
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
 
Comma list: 1089/1088, 1225/1224, 1701/1700, 262395/262144
 
{{Mapping|legend=0| 2 -2 -6 12 13 4 | 0 17 35 -21 -20 6 }}
 
Optimal tunings:  
* WE: ~99/70 = 600.0371{{c}}, ~10/9 = 182.4699{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4589{{c}}
 
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 1414de }}
 
Badness (Sintel): 0.831
 
== Domain ==
Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[terrain]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, 645700815/645657712
 
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
: mapping generators: ~63/50, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0013{{c}}, ~10/9 = 182.4672{{c}}
: [[error map]]: {{val| +0.004 -0.016 +0.028 -0.016 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 182.4668{{c}}
: error map: {{val| 0.000 -0.020 +0.024 -0.021 }}
 
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}
 
[[Badness]] (Sintel): 0.354
 
=== Semidomain ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 151263/151250, 1771561/1771470
 
{{Mapping|legend=0| 6 -6 -18 -16 -13 | 0 17 35 36 37 }}
: mapping generators: ~55/49, ~10/9
 
Optimal tunings:
* WE: ~55/49 = 200.0009{{c}}, ~10/9 = 182.4669{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~10/9 = 182.4675{{c}}
 
{{Optimal ET sequence|legend=0| 342, 1164, 1506, 1848, 4038, 5886, 9924 }}
 
Badness (Sintel): 0.360
 
[[Category:Minortone family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Minortone_family"