Tour of regular temperaments: Difference between revisions
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-25 09:37:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>204987784</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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===[[Quartonic temperaments]]=== | ===[[Quartonic temperaments]]=== | ||
These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | ||
===[[Avicennmic temperaments]]=== | |||
These temper out the avicennma, Avicenna's enharmonic diesis. | |||
===[[Starling temperaments]]=== | ===[[Starling temperaments]]=== | ||
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* [[http://en.wikipedia.org/wiki/Regular_temperament|Regular temperaments - Wikipedia]]</pre></div> | * [[http://en.wikipedia.org/wiki/Regular_temperament|Regular temperaments - Wikipedia]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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"><html><head><title>Tour of Regular Temperaments</title></head><body><!-- ws:start:WikiTextTocRule: | <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"><html><head><title>Tour of Regular Temperaments</title></head><body><!-- ws:start:WikiTextTocRule:140:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><a href="#Equal temperaments">Equal temperaments</a><!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --> | <a href="#Rank 2 (including &quot;linear&quot;) temperaments">Rank 2 (including &quot;linear&quot;) temperaments</a><!-- ws:end:WikiTextTocRule:142 --><!-- ws:start:WikiTextTocRule:143: --><!-- ws:end:WikiTextTocRule:143 --><!-- ws:start:WikiTextTocRule:144: --><!-- ws:end:WikiTextTocRule:144 --><!-- ws:start:WikiTextTocRule:145: --><!-- ws:end:WikiTextTocRule:145 --><!-- ws:start:WikiTextTocRule:146: --><!-- ws:end:WikiTextTocRule:146 --><!-- ws:start:WikiTextTocRule:147: --><!-- ws:end:WikiTextTocRule:147 --><!-- ws:start:WikiTextTocRule:148: --><!-- ws:end:WikiTextTocRule:148 --><!-- ws:start:WikiTextTocRule:149: --><!-- ws:end:WikiTextTocRule:149 --><!-- ws:start:WikiTextTocRule:150: --><!-- ws:end:WikiTextTocRule:150 --><!-- ws:start:WikiTextTocRule:151: --><!-- ws:end:WikiTextTocRule:151 --><!-- ws:start:WikiTextTocRule:152: --><!-- 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ws:start:WikiTextTocRule:166: --><!-- ws:end:WikiTextTocRule:166 --><!-- ws:start:WikiTextTocRule:167: --><!-- ws:end:WikiTextTocRule:167 --><!-- ws:start:WikiTextTocRule:168: --><!-- ws:end:WikiTextTocRule:168 --><!-- ws:start:WikiTextTocRule:169: --><!-- ws:end:WikiTextTocRule:169 --><!-- ws:start:WikiTextTocRule:170: --><!-- ws:end:WikiTextTocRule:170 --><!-- ws:start:WikiTextTocRule:171: --><!-- ws:end:WikiTextTocRule:171 --><!-- ws:start:WikiTextTocRule:172: --><!-- ws:end:WikiTextTocRule:172 --><!-- ws:start:WikiTextTocRule:173: --><!-- ws:end:WikiTextTocRule:173 --><!-- ws:start:WikiTextTocRule:174: --><!-- ws:end:WikiTextTocRule:174 --><!-- ws:start:WikiTextTocRule:175: --><!-- ws:end:WikiTextTocRule:175 --><!-- ws:start:WikiTextTocRule:176: --><!-- ws:end:WikiTextTocRule:176 --><!-- ws:start:WikiTextTocRule:177: --><!-- ws:end:WikiTextTocRule:177 --><!-- ws:start:WikiTextTocRule:178: --> | <a href="#Rank 3 temperaments">Rank 3 temperaments</a><!-- ws:end:WikiTextTocRule:178 --><!-- ws:start:WikiTextTocRule:179: --><!-- ws:end:WikiTextTocRule:179 --><!-- ws:start:WikiTextTocRule:180: --><!-- ws:end:WikiTextTocRule:180 --><!-- ws:start:WikiTextTocRule:181: --><!-- ws:end:WikiTextTocRule:181 --><!-- ws:start:WikiTextTocRule:182: --><!-- ws:end:WikiTextTocRule:182 --><!-- ws:start:WikiTextTocRule:183: --><!-- ws:end:WikiTextTocRule:183 --><!-- ws:start:WikiTextTocRule:184: --><!-- ws:end:WikiTextTocRule:184 --><!-- ws:start:WikiTextTocRule:185: --><!-- ws:end:WikiTextTocRule:185 --><!-- ws:start:WikiTextTocRule:186: --><!-- ws:end:WikiTextTocRule:186 --><!-- ws:start:WikiTextTocRule:187: --><!-- ws:end:WikiTextTocRule:187 --><!-- ws:start:WikiTextTocRule:188: --><!-- ws:end:WikiTextTocRule:188 --><!-- ws:start:WikiTextTocRule:189: --><!-- ws:end:WikiTextTocRule:189 --><!-- ws:start:WikiTextTocRule:190: --><!-- ws:end:WikiTextTocRule:190 --><!-- ws:start:WikiTextTocRule:191: --><!-- ws:end:WikiTextTocRule:191 --><!-- ws:start:WikiTextTocRule:192: --><!-- 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ws:start:WikiTextTocRule:206: --> | <a href="#Rank 4 temperaments">Rank 4 temperaments</a><!-- ws:end:WikiTextTocRule:206 --><!-- ws:start:WikiTextTocRule:207: --> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:207 --><!-- ws:start:WikiTextTocRule:208: --> | <a href="#Miscellaneous">Miscellaneous</a><!-- ws:end:WikiTextTocRule:208 --><!-- ws:start:WikiTextTocRule:209: --><!-- ws:end:WikiTextTocRule:209 --><!-- ws:start:WikiTextTocRule:210: --><!-- ws:end:WikiTextTocRule:210 --><!-- ws:start:WikiTextTocRule:211: --> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 --><a class="wiki_link" href="/edo">Equal temperaments</a></h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 --><a class="wiki_link" href="/edo">Equal temperaments</a></h1> | ||
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<a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents.<br /> | <a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:212:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@lineartemperaments&quot; title=&quot;Anchor: lineartemperaments&quot;/&gt; --><a name="lineartemperaments"></a><!-- ws:end:WikiTextAnchorRule:212 --></h1> | ||
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<a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> has given us a <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and we also have a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit temperaments.<br /> | <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> has given us a <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and we also have a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit temperaments.<br /> | ||
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These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br /> | These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Starling temperaments"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:58 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | ||
These temper out the avicennma, Avicenna's enharmonic diesis.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Starling temperaments"></a><!-- ws:end:WikiTextHeadingRule:60 --><a class="wiki_link" href="/Starling%20temperaments">Starling temperaments</a></h3> | |||
Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br /> | Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Marvel temperaments"></a><!-- ws:end:WikiTextHeadingRule:62 --><a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a></h3> | ||
These temper out 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.<br /> | These temper out 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h3&gt; --><h3 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Hemifamity temperaments"></a><!-- ws:end:WikiTextHeadingRule:64 --><a class="wiki_link" href="/Hemifamity%20temperaments">Hemifamity temperaments</a></h3> | ||
The hemifamity temperaments temper out the hemifamity comma, 5120/5103.<br /> | The hemifamity temperaments temper out the hemifamity comma, 5120/5103.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Porwell temperaments"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Porwell%20temperaments">Porwell temperaments</a></h3> | ||
The porwell temperaments temper out the porwell comma, 6144/6125.<br /> | The porwell temperaments temper out the porwell comma, 6144/6125.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h3&gt; --><h3 id="toc34"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Breedsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:68 --><a class="wiki_link" href="/Breedsmic%20temperaments">Breedsmic temperaments</a></h3> | ||
A breedsmic temperament is one which tempers out the breedsma, 2401/2400. Some which do not get discussed elsewhere are collected on a page <a class="wiki_link" href="/Breedsmic%20temperaments">here</a>.<br /> | A breedsmic temperament is one which tempers out the breedsma, 2401/2400. Some which do not get discussed elsewhere are collected on a page <a class="wiki_link" href="/Breedsmic%20temperaments">here</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc35"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Ragismic microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:70 --><a class="wiki_link" href="/Ragismic%20microtemperaments">Ragismic microtemperaments</a></h3> | ||
A ragismic temperament is one which tempers out 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric.<br /> | A ragismic temperament is one which tempers out 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:72:&lt;h3&gt; --><h3 id="toc36"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:72 --><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3> | ||
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Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">maqam music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br /> | Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">maqam music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:74 -->Rank 3 temperaments</h1> | ||
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Even less familiar than rank 2 temperaments are the <a class="wiki_link" href="/Planar%20Temperament">rank 3 temperaments</a>, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.<br /> | Even less familiar than rank 2 temperaments are the <a class="wiki_link" href="/Planar%20Temperament">rank 3 temperaments</a>, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h3&gt; --><h3 id="toc38"><a name="Rank 3 temperaments--Marvel family"></a><!-- ws:end:WikiTextHeadingRule:76 --><a class="wiki_link" href="/Marvel%20family">Marvel family</a></h3> | ||
The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.<br /> | The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h3&gt; --><h3 id="toc39"><a name="Rank 3 temperaments--Starling family"></a><!-- ws:end:WikiTextHeadingRule:78 --><a class="wiki_link" href="/Starling%20family">Starling family</a></h3> | ||
Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is <a class="wiki_link" href="/77edo">77edo</a>, but 31, 46 or 58 also work nicely.<br /> | Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is <a class="wiki_link" href="/77edo">77edo</a>, but 31, 46 or 58 also work nicely.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h3&gt; --><h3 id="toc40"><a name="Rank 3 temperaments--Gamelismic family"></a><!-- ws:end:WikiTextHeadingRule:80 --><a class="wiki_link" href="/Gamelismic%20family">Gamelismic family</a></h3> | ||
Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, 1029/1024. <br /> | Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, 1029/1024. <br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:82:&lt;h3&gt; --><h3 id="toc41"><a name="Rank 3 temperaments--Breed family"></a><!-- ws:end:WikiTextHeadingRule:82 --><a class="wiki_link" href="/Breed%20family">Breed family</a></h3> | ||
Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.<br /> | Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:84:&lt;h3&gt; --><h3 id="toc42"><a name="Rank 3 temperaments--Ragisma family"></a><!-- ws:end:WikiTextHeadingRule:84 --><a class="wiki_link" href="/Ragisma%20family">Ragisma family</a></h3> | ||
The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.<br /> | The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:86:&lt;h3&gt; --><h3 id="toc43"><a name="Rank 3 temperaments--Hemifamity family"></a><!-- ws:end:WikiTextHeadingRule:86 --><a class="wiki_link" href="/Hemifamity%20family">Hemifamity family</a></h3> | ||
The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br /> | The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:88:&lt;h3&gt; --><h3 id="toc44"><a name="Rank 3 temperaments--Porwell family"></a><!-- ws:end:WikiTextHeadingRule:88 --><a class="wiki_link" href="/Porwell%20family">Porwell family</a></h3> | ||
The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br /> | The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br /> | ||
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The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br /> | The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br /> | ||
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The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br /> | The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br /> | ||
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These temper out 245/243.<br /> | These temper out 245/243.<br /> | ||
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These temper out 875/864.<br /> | These temper out 875/864.<br /> | ||
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These temper out 1728/1715.<br /> | These temper out 1728/1715.<br /> | ||
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These temper out the nuwell comma, 2430/2401.<br /> | These temper out the nuwell comma, 2430/2401.<br /> | ||
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The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br /> | The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br /> | ||
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The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br /> | The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br /> | ||
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The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br /> | The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br /> | ||
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These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | ||
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These are the rank three temperaments tempering out the dischisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | These are the rank three temperaments tempering out the dischisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | ||
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These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> | ||
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These are the rank three temperaments tempering out the porcupine comma or aximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. <br /> | These are the rank three temperaments tempering out the porcupine comma or aximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. <br /> | ||
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Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br /> | Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br /> | ||
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Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br /> | Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br /> | ||
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Semaphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like &quot;semi-fourth&quot;.<br /> | Semaphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like &quot;semi-fourth&quot;.<br /> | ||
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These temper out the werckisma, 441/440.<br /> | These temper out the werckisma, 441/440.<br /> | ||
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These temper out the swetisma, 540/539.<br /> | These temper out the swetisma, 540/539.<br /> | ||
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These temper out the lehmerisma, 3025/3024.<br /> | These temper out the lehmerisma, 3025/3024.<br /> | ||
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These temper out the kalisma, 9801/9800.<br /> | These temper out the kalisma, 9801/9800.<br /> | ||
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Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example <a class="wiki_link" href="/Hobbits">hobbit scales</a> can be constructed for them.<br /> | Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example <a class="wiki_link" href="/Hobbits">hobbit scales</a> can be constructed for them.<br /> | ||
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A wide-open field. These are regular temperaments of various ranks which temper <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>.<br /> | A wide-open field. These are regular temperaments of various ranks which temper <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:136:&lt;h2&gt; --><h2 id="toc68"><a name="Miscellaneous-The Archipelago"></a><!-- ws:end:WikiTextHeadingRule:136 --><a class="wiki_link" href="/The%20Archipelago">The Archipelago</a></h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:138:&lt;h2&gt; --><h2 id="toc69"><a name="Miscellaneous-Links"></a><!-- ws:end:WikiTextHeadingRule:138 -->Links</h2> | ||
<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">Regular temperaments - Wikipedia</a></li></ul></body></html></pre></div> | <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">Regular temperaments - Wikipedia</a></li></ul></body></html></pre></div> | ||