Minortonic family: Difference between revisions

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<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;

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

Map: [&lt;1 16 32|, &lt;0 -17 -35|]

[[toc|flat]]

=Mitonic=

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.

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.

Commas: 4375/4374, 2100875/2097152

POTE generator: ~10/9 = 182.458

Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]

EDOs: 7, 13, 33, 46, 125, 171

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]].

POTE generator: ~10/9 = 182.467

Map: [&lt;3 14 26 28|, &lt;0 -17 -35 -36|]

EDOS: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038

Badness: 0.0140</pre></div>

<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;

Comma: |-16 35 -17&amp;gt;&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 182.466&lt;br /&gt;

Map: [&amp;lt;1 16 32|, &amp;lt;0 -17 -35|]&lt;br /&gt;

EDOs: 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091&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;

&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;

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;

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;

Commas: 4375/4374, 2100875/2097152&lt;br /&gt;

POTE generator: ~10/9 = 182.458&lt;br /&gt;

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;

Badness: 0.0252&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;

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;

Commas: 250047/250000, 645700815/645657712&lt;br /&gt;

POTE generator: ~10/9 = 182.467&lt;br /&gt;

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;

Badness: 0.0140&lt;/body&gt;&lt;/html&gt;</pre></div>