Tour of regular temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 290296397 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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>2012-01- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-08 20:25:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>290444565</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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===[[Bug family]]=== | ===[[Bug family]]=== | ||
This tempers out 27/25, the large limma or bug comma. | This tempers out 27/25, the large limma or bug comma. | ||
===[[Father family]]=== | |||
This tempers out 16/15, the just diatonic semitone. | |||
===[[Sycamore family]]=== | ===[[Sycamore family]]=== | ||
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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> | ||
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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;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:186:&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:186 --><!-- ws:start:WikiTextTocRule:187: --><a href="#Regular temperaments">Regular temperaments</a><!-- ws:end:WikiTextTocRule:187 --><!-- ws:start:WikiTextTocRule:188: --><!-- ws:end:WikiTextTocRule:188 --><!-- ws:start:WikiTextTocRule:189: --><!-- ws:end:WikiTextTocRule:189 --><!-- ws:start:WikiTextTocRule:190: --> | <a href="#Equal temperaments">Equal temperaments</a><!-- ws:end:WikiTextTocRule:190 --><!-- ws:start:WikiTextTocRule:191: --> | <a href="#Rank 2 (including &quot;linear&quot;) temperaments">Rank 2 (including &quot;linear&quot;) temperaments</a><!-- ws:end:WikiTextTocRule:191 --><!-- ws:start:WikiTextTocRule:192: --><!-- ws:end:WikiTextTocRule:192 --><!-- ws:start:WikiTextTocRule:193: --><!-- ws:end:WikiTextTocRule:193 --><!-- ws:start:WikiTextTocRule:194: --><!-- ws:end:WikiTextTocRule:194 --><!-- ws:start:WikiTextTocRule:195: --><!-- ws:end:WikiTextTocRule:195 --><!-- ws:start:WikiTextTocRule:196: --><!-- ws:end:WikiTextTocRule:196 --><!-- ws:start:WikiTextTocRule:197: --><!-- ws:end:WikiTextTocRule:197 --><!-- ws:start:WikiTextTocRule:198: --><!-- ws:end:WikiTextTocRule:198 --><!-- ws:start:WikiTextTocRule:199: --><!-- ws:end:WikiTextTocRule:199 --><!-- ws:start:WikiTextTocRule:200: --><!-- ws:end:WikiTextTocRule:200 --><!-- ws:start:WikiTextTocRule:201: --><!-- ws:end:WikiTextTocRule:201 --><!-- ws:start:WikiTextTocRule:202: --><!-- ws:end:WikiTextTocRule:202 --><!-- ws:start:WikiTextTocRule:203: --><!-- ws:end:WikiTextTocRule:203 --><!-- ws:start:WikiTextTocRule:204: --><!-- ws:end:WikiTextTocRule:204 --><!-- ws:start:WikiTextTocRule:205: --><!-- ws:end:WikiTextTocRule:205 --><!-- ws:start:WikiTextTocRule:206: --><!-- ws:end:WikiTextTocRule:206 --><!-- ws:start:WikiTextTocRule:207: --><!-- ws:end:WikiTextTocRule:207 --><!-- ws:start:WikiTextTocRule:208: --><!-- ws:end:WikiTextTocRule:208 --><!-- ws:start:WikiTextTocRule:209: --><!-- ws:end:WikiTextTocRule:209 --><!-- ws:start:WikiTextTocRule:210: --><!-- ws:end:WikiTextTocRule:210 --><!-- ws:start:WikiTextTocRule:211: --><!-- ws:end:WikiTextTocRule:211 --><!-- ws:start:WikiTextTocRule:212: --><!-- ws:end:WikiTextTocRule:212 --><!-- ws:start:WikiTextTocRule:213: --><!-- ws:end:WikiTextTocRule:213 --><!-- ws:start:WikiTextTocRule:214: --><!-- ws:end:WikiTextTocRule:214 --><!-- ws:start:WikiTextTocRule:215: --><!-- ws:end:WikiTextTocRule:215 --><!-- ws:start:WikiTextTocRule:216: --><!-- ws:end:WikiTextTocRule:216 --><!-- ws:start:WikiTextTocRule:217: --><!-- ws:end:WikiTextTocRule:217 --><!-- ws:start:WikiTextTocRule:218: --><!-- ws:end:WikiTextTocRule:218 --><!-- ws:start:WikiTextTocRule:219: --><!-- ws:end:WikiTextTocRule:219 --><!-- ws:start:WikiTextTocRule:220: --><!-- ws:end:WikiTextTocRule:220 --><!-- ws:start:WikiTextTocRule:221: --><!-- ws:end:WikiTextTocRule:221 --><!-- ws:start:WikiTextTocRule:222: --><!-- ws:end:WikiTextTocRule:222 --><!-- ws:start:WikiTextTocRule:223: --><!-- ws:end:WikiTextTocRule:223 --><!-- ws:start:WikiTextTocRule:224: --><!-- ws:end:WikiTextTocRule:224 --><!-- ws:start:WikiTextTocRule:225: --><!-- ws:end:WikiTextTocRule:225 --><!-- ws:start:WikiTextTocRule:226: --><!-- ws:end:WikiTextTocRule:226 --><!-- ws:start:WikiTextTocRule:227: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:227 --><!-- ws:start:WikiTextTocRule:228: --> | <a href="#Septisemi temperaments">Septisemi temperaments</a><!-- ws:end:WikiTextTocRule:228 --><!-- ws:start:WikiTextTocRule:229: --><!-- ws:end:WikiTextTocRule:229 --><!-- ws:start:WikiTextTocRule:230: --><!-- ws:end:WikiTextTocRule:230 --><!-- ws:start:WikiTextTocRule:231: --><!-- ws:end:WikiTextTocRule:231 --><!-- ws:start:WikiTextTocRule:232: --><!-- ws:end:WikiTextTocRule:232 --><!-- ws:start:WikiTextTocRule:233: --><!-- ws:end:WikiTextTocRule:233 --><!-- ws:start:WikiTextTocRule:234: --><!-- ws:end:WikiTextTocRule:234 --><!-- ws:start:WikiTextTocRule:235: --><!-- ws:end:WikiTextTocRule:235 --><!-- ws:start:WikiTextTocRule:236: --><!-- ws:end:WikiTextTocRule:236 --><!-- ws:start:WikiTextTocRule:237: --><!-- ws:end:WikiTextTocRule:237 --><!-- ws:start:WikiTextTocRule:238: --><!-- ws:end:WikiTextTocRule:238 --><!-- ws:start:WikiTextTocRule:239: --><!-- ws:end:WikiTextTocRule:239 --><!-- ws:start:WikiTextTocRule:240: --> | <a href="#Rank 3 temperaments">Rank 3 temperaments</a><!-- ws:end:WikiTextTocRule:240 --><!-- ws:start:WikiTextTocRule:241: --><!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: --><!-- ws:end:WikiTextTocRule:243 --><!-- ws:start:WikiTextTocRule:244: --><!-- ws:end:WikiTextTocRule:244 --><!-- ws:start:WikiTextTocRule:245: --><!-- ws:end:WikiTextTocRule:245 --><!-- ws:start:WikiTextTocRule:246: --><!-- ws:end:WikiTextTocRule:246 --><!-- ws:start:WikiTextTocRule:247: --><!-- ws:end:WikiTextTocRule:247 --><!-- ws:start:WikiTextTocRule:248: --><!-- ws:end:WikiTextTocRule:248 --><!-- ws:start:WikiTextTocRule:249: --><!-- ws:end:WikiTextTocRule:249 --><!-- ws:start:WikiTextTocRule:250: --><!-- ws:end:WikiTextTocRule:250 --><!-- ws:start:WikiTextTocRule:251: --><!-- ws:end:WikiTextTocRule:251 --><!-- ws:start:WikiTextTocRule:252: --><!-- ws:end:WikiTextTocRule:252 --><!-- ws:start:WikiTextTocRule:253: --><!-- ws:end:WikiTextTocRule:253 --><!-- ws:start:WikiTextTocRule:254: --><!-- ws:end:WikiTextTocRule:254 --><!-- ws:start:WikiTextTocRule:255: --><!-- ws:end:WikiTextTocRule:255 --><!-- ws:start:WikiTextTocRule:256: --><!-- ws:end:WikiTextTocRule:256 --><!-- ws:start:WikiTextTocRule:257: --><!-- ws:end:WikiTextTocRule:257 --><!-- ws:start:WikiTextTocRule:258: --><!-- ws:end:WikiTextTocRule:258 --><!-- ws:start:WikiTextTocRule:259: --><!-- ws:end:WikiTextTocRule:259 --><!-- ws:start:WikiTextTocRule:260: --><!-- ws:end:WikiTextTocRule:260 --><!-- ws:start:WikiTextTocRule:261: --><!-- ws:end:WikiTextTocRule:261 --><!-- ws:start:WikiTextTocRule:262: --><!-- ws:end:WikiTextTocRule:262 --><!-- ws:start:WikiTextTocRule:263: --><!-- ws:end:WikiTextTocRule:263 --><!-- ws:start:WikiTextTocRule:264: --><!-- ws:end:WikiTextTocRule:264 --><!-- ws:start:WikiTextTocRule:265: --><!-- ws:end:WikiTextTocRule:265 --><!-- ws:start:WikiTextTocRule:266: --><!-- ws:end:WikiTextTocRule:266 --><!-- ws:start:WikiTextTocRule:267: --><!-- ws:end:WikiTextTocRule:267 --><!-- ws:start:WikiTextTocRule:268: --><!-- ws:end:WikiTextTocRule:268 --><!-- ws:start:WikiTextTocRule:269: --><!-- ws:end:WikiTextTocRule:269 --><!-- ws:start:WikiTextTocRule:270: --><!-- ws:end:WikiTextTocRule:270 --><!-- ws:start:WikiTextTocRule:271: --><!-- ws:end:WikiTextTocRule:271 --><!-- ws:start:WikiTextTocRule:272: --><!-- ws:end:WikiTextTocRule:272 --><!-- ws:start:WikiTextTocRule:273: --> | <a href="#Rank 4 temperaments">Rank 4 temperaments</a><!-- ws:end:WikiTextTocRule:273 --><!-- ws:start:WikiTextTocRule:274: --> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:274 --><!-- ws:start:WikiTextTocRule:275: --> | <a href="#Commatic realms">Commatic realms</a><!-- ws:end:WikiTextTocRule:275 --><!-- ws:start:WikiTextTocRule:276: --><!-- ws:end:WikiTextTocRule:276 --><!-- ws:start:WikiTextTocRule:277: --><!-- ws:end:WikiTextTocRule:277 --><!-- ws:start:WikiTextTocRule:278: --><!-- ws:end:WikiTextTocRule:278 --><!-- ws:start:WikiTextTocRule:279: --><!-- ws:end:WikiTextTocRule:279 --><!-- ws:start:WikiTextTocRule:280: --> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Regular temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Regular temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Regular temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Regular temperaments</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. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). 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, although there are other temperaments supported by 12-tET.<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. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). 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, although there are other temperaments supported by 12-tET.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:281:&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:281 --></h1> | ||
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Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Other pages listing them are <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit rank two temperaments. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br /> | Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Other pages listing them are <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit rank two temperaments. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br /> | ||
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This tempers out 27/25, the large limma or bug comma.<br /> | This tempers out 27/25, the large limma or bug comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sycamore family"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Father family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Father%20family">Father family</a></h3> | ||
This tempers out 16/15, the just diatonic semitone.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3> | |||
The sycamore family tempers out the sycamore comma, |-16 -6 11&gt; = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4.<br /> | The sycamore family tempers out the sycamore comma, |-16 -6 11&gt; = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Escapade family"></a><!-- ws:end:WikiTextHeadingRule:52 --><a class="wiki_link" href="/Escapade%20family">Escapade family</a></h3> | ||
This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.<br /> | This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Vulture family"></a><!-- ws:end:WikiTextHeadingRule:54 --><a class="wiki_link" href="/Vulture%20family">Vulture family</a></h3> | ||
This tempers out the vulture comma, |24 -21 4&gt;.<br /> | This tempers out the vulture comma, |24 -21 4&gt;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Vishnuzmic family"></a><!-- ws:end:WikiTextHeadingRule:56 --><a class="wiki_link" href="/Vishnuzmic%20family">Vishnuzmic family</a></h3> | ||
This tempers out the vishnuzma, |23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.<br /> | This tempers out the vishnuzma, |23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Luna Family"></a><!-- ws:end:WikiTextHeadingRule:58 --><a class="wiki_link" href="/Luna%20Family">Luna Family</a></h3> | ||
This tempers out the Luna comma; |38 -2 -15&gt; (274877906944/274658203125)<br /> | This tempers out the Luna comma; |38 -2 -15&gt; (274877906944/274658203125)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Laconic Family"></a><!-- ws:end:WikiTextHeadingRule:60 --><a class="wiki_link" href="/Laconic%20Family">Laconic Family</a></h3> | ||
This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, among others.<br /> | This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, among others.<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-Families-Mutt family"></a><!-- ws:end:WikiTextHeadingRule:62 --><a class="wiki_link" href="/Mutt%20family">Mutt family</a></h3> | ||
This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.<br /> | This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans"></a><!-- ws:end:WikiTextHeadingRule:64 -->Clans</h2> | ||
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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-Clans-Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3> | ||
If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br /> | If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br /> | ||
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Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.<br /> | Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.<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-Clans-Trienstonic clan"></a><!-- ws:end:WikiTextHeadingRule:68 --><a class="wiki_link" href="/Trienstonic%20clan">Trienstonic clan</a></h3> | ||
This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7.<br /> | This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7.<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-Clans-Archytas clan"></a><!-- ws:end:WikiTextHeadingRule:70 --><a class="wiki_link" href="/Archytas%20clan">Archytas clan</a></h3> | ||
This clan tempers out the Archytas comma, 64/63, which is a triprime comma with factors of 2, 3 and 7. The clan consists of rank two temperaments, and should not be confused with the <a class="wiki_link" href="/Archytas%20family">Archytas family</a> of rank three temperaments.<br /> | This clan tempers out the Archytas comma, 64/63, which is a triprime comma with factors of 2, 3 and 7. The clan consists of rank two temperaments, and should not be confused with the <a class="wiki_link" href="/Archytas%20family">Archytas family</a> of rank three temperaments.<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-Clans-Sensamagic clan"></a><!-- ws:end:WikiTextHeadingRule:72 --><a class="wiki_link" href="/Sensamagic%20clan">Sensamagic clan</a></h3> | ||
This clan tempers out 245/243, the sensamagic comma.<br /> | This clan tempers out 245/243, the sensamagic comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h3&gt; --><h3 id="toc37"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Jubilismic clan"></a><!-- ws:end:WikiTextHeadingRule:74 --><a class="wiki_link" href="/Jubilismic%20clan">Jubilismic clan</a></h3> | ||
This tempers out the jubilisma, 50/49, E.A. the difference between 10/7 and 7/5.<br /> | This tempers out the jubilisma, 50/49, E.A. the difference between 10/7 and 7/5.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h3&gt; --><h3 id="toc38"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Hemimean clan"></a><!-- ws:end:WikiTextHeadingRule:76 --><a class="wiki_link" href="/Hemimean%20clan">Hemimean clan</a></h3> | ||
This tempers out the hemimean comma, 3136/3125, a no-threes comma.<br /> | This tempers out the hemimean comma, 3136/3125, a no-threes comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h3&gt; --><h3 id="toc39"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Mirkwai clan"></a><!-- ws:end:WikiTextHeadingRule:78 --><a class="wiki_link" href="/Mirkwai%20clan">Mirkwai clan</a></h3> | ||
This tempers out the mirkwai comma, |0 3 4 -5&gt; = 16875/16807, a no-twos comma (ratio of odd numbers.)<br /> | This tempers out the mirkwai comma, |0 3 4 -5&gt; = 16875/16807, a no-twos comma (ratio of odd numbers.)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:80 -->Temperaments</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:82:&lt;h1&gt; --><h1 id="toc41"><a name="Septisemi temperaments"></a><!-- ws:end:WikiTextHeadingRule:82 --><a class="wiki_link" href="/Septisemi%20temperaments">Septisemi temperaments</a></h1> | ||
These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4.<br /> | These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:84:&lt;h3&gt; --><h3 id="toc42"><a name="Septisemi temperaments--Quartonic temperaments"></a><!-- ws:end:WikiTextHeadingRule:84 --><a class="wiki_link" href="/Quartonic%20temperaments">Quartonic temperaments</a></h3> | ||
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: | <!-- ws:start:WikiTextHeadingRule:86:&lt;h3&gt; --><h3 id="toc43"><a name="Septisemi temperaments--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:86 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | ||
These temper out the avicennma, |-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis.<br /> | These temper out the avicennma, |-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:88:&lt;h3&gt; --><h3 id="toc44"><a name="Septisemi temperaments--Starling temperaments"></a><!-- ws:end:WikiTextHeadingRule:88 --><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 (<span class="commentBody">The difference between three 6/5s, one 7/6, and an octave) </span>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 (<span class="commentBody">The difference between three 6/5s, one 7/6, and an octave) </span>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:90:&lt;h3&gt; --><h3 id="toc45"><a name="Septisemi temperaments--Marvel temperaments"></a><!-- ws:end:WikiTextHeadingRule:90 --><a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a></h3> | ||
These temper out |-5 2 2 -1&gt; = 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 |-5 2 2 -1&gt; = 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:92:&lt;h3&gt; --><h3 id="toc46"><a name="Septisemi temperaments--Hemifamity temperaments"></a><!-- ws:end:WikiTextHeadingRule:92 --><a class="wiki_link" href="/Hemifamity%20temperaments">Hemifamity temperaments</a></h3> | ||
Hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103.<br /> | Hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103.<br /> | ||
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Porwell temperaments temper out the porwell comma, |11 1 -3 -2&gt; = 6144/6125.<br /> | Porwell temperaments temper out the porwell comma, |11 1 -3 -2&gt; = 6144/6125.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:96:&lt;h3&gt; --><h3 id="toc48"><a name="Septisemi temperaments--Hemimage temperaments"></a><!-- ws:end:WikiTextHeadingRule:96 --><a class="wiki_link" href="/Hemimage%20temperaments">Hemimage temperaments</a></h3> | ||
Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.<br /> | Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:98:&lt;h3&gt; --><h3 id="toc49"><a name="Septisemi temperaments--Breedsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:98 --><a class="wiki_link" href="/Breedsmic%20temperaments">Breedsmic temperaments</a></h3> | ||
A breedsmic temperament is one which tempers out the breedsma, |-5 -1 -2 4&gt; = 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, |-5 -1 -2 4&gt; = 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:100:&lt;h3&gt; --><h3 id="toc50"><a name="Septisemi temperaments--Ragismic microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:100 --><a class="wiki_link" href="/Ragismic%20microtemperaments">Ragismic microtemperaments</a></h3> | ||
A ragismic temperament is one which tempers out |-1 -7 4 1&gt; = 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 |-1 -7 4 1&gt; = 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:102:&lt;h3&gt; --><h3 id="toc51"><a name="Septisemi temperaments--31 comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:102 --><a class="wiki_link" href="/31%20comma%20temperaments">31 comma temperaments</a></h3> | ||
These all have period 1/31 of an octave.<br /> | These all have period 1/31 of an octave.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:104:&lt;h3&gt; --><h3 id="toc52"><a name="Septisemi temperaments--Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:104 --><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:106:&lt;h1&gt; --><h1 id="toc53"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:106 -->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:108:&lt;h3&gt; --><h3 id="toc54"><a name="Rank 3 temperaments--Marvel family"></a><!-- ws:end:WikiTextHeadingRule:108 --><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:110:&lt;h3&gt; --><h3 id="toc55"><a name="Rank 3 temperaments--Starling family"></a><!-- ws:end:WikiTextHeadingRule:110 --><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:112:&lt;h3&gt; --><h3 id="toc56"><a name="Rank 3 temperaments--Gamelismic family"></a><!-- ws:end:WikiTextHeadingRule:112 --><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:114:&lt;h3&gt; --><h3 id="toc57"><a name="Rank 3 temperaments--Breed family"></a><!-- ws:end:WikiTextHeadingRule:114 --><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:116:&lt;h3&gt; --><h3 id="toc58"><a name="Rank 3 temperaments--Ragisma family"></a><!-- ws:end:WikiTextHeadingRule:116 --><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:118:&lt;h3&gt; --><h3 id="toc59"><a name="Rank 3 temperaments--Hemifamity family"></a><!-- ws:end:WikiTextHeadingRule:118 --><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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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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:122:&lt;h3&gt; --><h3 id="toc61"><a name="Rank 3 temperaments--Horwell family"></a><!-- ws:end:WikiTextHeadingRule:122 --><a class="wiki_link" href="/Horwell%20family">Horwell family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:124:&lt;h3&gt; --><h3 id="toc62"><a name="Rank 3 temperaments--Hemimage family"></a><!-- ws:end:WikiTextHeadingRule:124 --><a class="wiki_link" href="/Hemimage%20family">Hemimage family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:126:&lt;h3&gt; --><h3 id="toc63"><a name="Rank 3 temperaments--Sensamagic family"></a><!-- ws:end:WikiTextHeadingRule:126 --><a class="wiki_link" href="/Sensamagic%20family">Sensamagic family</a></h3> | ||
These temper out 245/243.<br /> | These temper out 245/243.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:128:&lt;h3&gt; --><h3 id="toc64"><a name="Rank 3 temperaments--Keemic family"></a><!-- ws:end:WikiTextHeadingRule:128 --><a class="wiki_link" href="/Keemic%20family">Keemic family</a></h3> | ||
These temper out 875/864.<br /> | These temper out 875/864.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:130:&lt;h3&gt; --><h3 id="toc65"><a name="Rank 3 temperaments--Sengic family"></a><!-- ws:end:WikiTextHeadingRule:130 --><a class="wiki_link" href="/Sengic%20family">Sengic family</a></h3> | ||
These temper out the senga, 686/675.<br /> | These temper out the senga, 686/675.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:132:&lt;h3&gt; --><h3 id="toc66"><a name="Rank 3 temperaments--Orwellismic family"></a><!-- ws:end:WikiTextHeadingRule:132 --><a class="wiki_link" href="/Orwellismic%20family">Orwellismic family</a></h3> | ||
These temper out 1728/1715.<br /> | These temper out 1728/1715.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:134:&lt;h3&gt; --><h3 id="toc67"><a name="Rank 3 temperaments--Nuwell family"></a><!-- ws:end:WikiTextHeadingRule:134 --><a class="wiki_link" href="/Nuwell%20family">Nuwell family</a></h3> | ||
These temper out the nuwell comma, 2430/2401.<br /> | These temper out the nuwell comma, 2430/2401.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:136:&lt;h3&gt; --><h3 id="toc68"><a name="Rank 3 temperaments--Octagar family"></a><!-- ws:end:WikiTextHeadingRule:136 --><a class="wiki_link" href="/Octagar%20family">Octagar family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:138:&lt;h3&gt; --><h3 id="toc69"><a name="Rank 3 temperaments--Mirkwai family"></a><!-- ws:end:WikiTextHeadingRule:138 --><a class="wiki_link" href="/Mirkwai%20family">Mirkwai family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:140:&lt;h3&gt; --><h3 id="toc70"><a name="Rank 3 temperaments--Hemimean family"></a><!-- ws:end:WikiTextHeadingRule:140 --><a class="wiki_link" href="/Hemimean%20family">Hemimean family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:142:&lt;h3&gt; --><h3 id="toc71"><a name="Rank 3 temperaments--Mirwomo family"></a><!-- ws:end:WikiTextHeadingRule:142 --><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3> | ||
The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br /> | The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:144:&lt;h3&gt; --><h3 id="toc72"><a name="Rank 3 temperaments--Dimcomp family"></a><!-- ws:end:WikiTextHeadingRule:144 --><a class="wiki_link" href="/Dimcomp%20family">Dimcomp family</a></h3> | ||
The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.<br /> | The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:146:&lt;h3&gt; --><h3 id="toc73"><a name="Rank 3 temperaments--Kleismic rank three family"></a><!-- ws:end:WikiTextHeadingRule:146 --><a class="wiki_link" href="/Kleismic%20rank%20three%20family">Kleismic rank three family</a></h3> | ||
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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:148:&lt;h3&gt; --><h3 id="toc74"><a name="Rank 3 temperaments--Diaschismic rank three family"></a><!-- ws:end:WikiTextHeadingRule:148 --><a class="wiki_link" href="/Diaschismic%20rank%20three%20family">Diaschismic rank three family</a></h3> | ||
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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The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br /> | The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br /> | ||
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These temper out the valinorsma, 176/175.<br /> | These temper out the valinorsma, 176/175.<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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:174:&lt;h1&gt; --><h1 id="toc87"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:174 --><a class="wiki_link" href="/Subgroup%20temperaments">Subgroup temperaments</a></h1> | ||
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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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By a <em>commatic realm</em> is meant the whole collection of regular temperaments of various ranks and for both full groups and <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroups</a> tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.<br /> | By a <em>commatic realm</em> is meant the whole collection of regular temperaments of various ranks and for both full groups and <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroups</a> tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.<br /> | ||
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By <em>Orgonia</em> is meant the commatic realm of the <a class="wiki_link" href="/11-limit">11-limit</a> comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.<br /> | By <em>Orgonia</em> is meant the commatic realm of the <a class="wiki_link" href="/11-limit">11-limit</a> comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:180:&lt;h2&gt; --><h2 id="toc90"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:180 --><a class="wiki_link" href="/The%20Biosphere">The Biosphere</a></h2> | ||
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.<br /> | The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.<br /> | ||
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The Archipelago is a name which has been given to the commatic realm of the <a class="wiki_link" href="/13-limit">13-limit</a> comma 676/675.<br /> | The Archipelago is a name which has been given to the commatic realm of the <a class="wiki_link" href="/13-limit">13-limit</a> comma 676/675.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:184:&lt;h2&gt; --><h2 id="toc92"><a name="Commatic realms-Links"></a><!-- ws:end:WikiTextHeadingRule:184 -->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> | ||