Tour of regular temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 541030568 - 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>2015- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-27 12:50:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>548848370</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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===[[Valinorismic temperaments]]=== | ===[[Valinorismic temperaments]]=== | ||
These temper out the valinorsma, 176/175. | These temper out the valinorsma, 176/175. | ||
===[[Rastmic temperaments]]=== | |||
These temper out the rastma, 243/242. | |||
===[[Werckismic temperaments]]=== | ===[[Werckismic temperaments]]=== | ||
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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><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Regul%C3%A4re%20Temperaturen">Deutsch</a><br /> | <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><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Regul%C3%A4re%20Temperaturen">Deutsch</a><br /> | ||
</span><br /> | </span><br /> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:240:&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:240 --><!-- ws:start:WikiTextTocRule:241: --><a href="#Regular temperaments">Regular temperaments</a><!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: --><!-- ws:end:WikiTextTocRule:243 --><!-- ws:start:WikiTextTocRule:244: --> | <a href="#Equal temperaments">Equal temperaments</a><!-- ws:end:WikiTextTocRule:244 --><!-- ws:start:WikiTextTocRule:245: --> | <a href="#Rank 2 (including &quot;linear&quot;) temperaments">Rank 2 (including &quot;linear&quot;) temperaments</a><!-- ws:end:WikiTextTocRule:245 --><!-- ws:start:WikiTextTocRule:246: --><!-- ws:end:WikiTextTocRule:246 --><!-- ws:start:WikiTextTocRule:247: 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ws:start:WikiTextTocRule:288: --><!-- ws:end:WikiTextTocRule:288 --><!-- ws:start:WikiTextTocRule:289: --><!-- ws:end:WikiTextTocRule:289 --><!-- ws:start:WikiTextTocRule:290: --><!-- ws:end:WikiTextTocRule:290 --><!-- ws:start:WikiTextTocRule:291: --><!-- ws:end:WikiTextTocRule:291 --><!-- ws:start:WikiTextTocRule:292: --> | <a href="#Temperaments for a given comma">Temperaments for a given comma</a><!-- ws:end:WikiTextTocRule:292 --><!-- ws:start:WikiTextTocRule:293: --><!-- ws:end:WikiTextTocRule:293 --><!-- ws:start:WikiTextTocRule:294: --><!-- ws:end:WikiTextTocRule:294 --><!-- ws:start:WikiTextTocRule:295: --><!-- ws:end:WikiTextTocRule:295 --><!-- ws:start:WikiTextTocRule:296: --><!-- ws:end:WikiTextTocRule:296 --><!-- ws:start:WikiTextTocRule:297: --><!-- ws:end:WikiTextTocRule:297 --><!-- ws:start:WikiTextTocRule:298: --><!-- ws:end:WikiTextTocRule:298 --><!-- ws:start:WikiTextTocRule:299: --><!-- ws:end:WikiTextTocRule:299 --><!-- ws:start:WikiTextTocRule:300: --><!-- 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ws:start:WikiTextTocRule:314: --><!-- ws:end:WikiTextTocRule:314 --><!-- ws:start:WikiTextTocRule:315: --><!-- ws:end:WikiTextTocRule:315 --><!-- ws:start:WikiTextTocRule:316: --><!-- ws:end:WikiTextTocRule:316 --><!-- ws:start:WikiTextTocRule:317: --><!-- ws:end:WikiTextTocRule:317 --><!-- ws:start:WikiTextTocRule:318: --> | <a href="#Rank 3 temperaments">Rank 3 temperaments</a><!-- ws:end:WikiTextTocRule:318 --><!-- ws:start:WikiTextTocRule:319: --><!-- ws:end:WikiTextTocRule:319 --><!-- ws:start:WikiTextTocRule:320: --><!-- ws:end:WikiTextTocRule:320 --><!-- ws:start:WikiTextTocRule:321: --><!-- ws:end:WikiTextTocRule:321 --><!-- ws:start:WikiTextTocRule:322: --><!-- ws:end:WikiTextTocRule:322 --><!-- ws:start:WikiTextTocRule:323: --><!-- ws:end:WikiTextTocRule:323 --><!-- ws:start:WikiTextTocRule:324: --><!-- ws:end:WikiTextTocRule:324 --><!-- ws:start:WikiTextTocRule:325: --><!-- ws:end:WikiTextTocRule:325 --><!-- ws:start:WikiTextTocRule:326: --><!-- ws:end:WikiTextTocRule:326 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ws:start:WikiTextTocRule:354: --> | <a href="#Rank 4 temperaments">Rank 4 temperaments</a><!-- ws:end:WikiTextTocRule:354 --><!-- ws:start:WikiTextTocRule:355: --> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:355 --><!-- ws:start:WikiTextTocRule:356: --> | <a href="#Commatic realms">Commatic realms</a><!-- ws:end:WikiTextTocRule:356 --><!-- ws:start:WikiTextTocRule:357: --><!-- ws:end:WikiTextTocRule:357 --><!-- ws:start:WikiTextTocRule:358: --><!-- ws:end:WikiTextTocRule:358 --><!-- ws:start:WikiTextTocRule:359: --><!-- ws:end:WikiTextTocRule:359 --><!-- ws:start:WikiTextTocRule:360: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:360 --><!-- ws:start:WikiTextTocRule:361: --> | ||
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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 or ED2) 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 or ED2) 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:362:&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:362 --></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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These temper out the valinorsma, 176/175.<br /> | These temper out the valinorsma, 176/175.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:216:&lt;h3&gt; --><h3 id="toc108"><a name="Rank 3 temperaments--Werckismic temperaments"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:216:&lt;h3&gt; --><h3 id="toc108"><a name="Rank 3 temperaments--Rastmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:216 --><a class="wiki_link" href="/Rastmic%20temperaments">Rastmic temperaments</a></h3> | ||
These temper out the rastma, 243/242.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:218:&lt;h3&gt; --><h3 id="toc109"><a name="Rank 3 temperaments--Werckismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:218 --><a class="wiki_link" href="/Werckismic%20temperaments">Werckismic temperaments</a></h3> | |||
These temper out the werckisma, 441/440.<br /> | These temper out the werckisma, 441/440.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:220:&lt;h3&gt; --><h3 id="toc110"><a name="Rank 3 temperaments--Swetismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:220 --><a class="wiki_link" href="/Swetismic%20temperaments">Swetismic temperaments</a></h3> | ||
These temper out the swetisma, 540/539.<br /> | These temper out the swetisma, 540/539.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:222:&lt;h3&gt; --><h3 id="toc111"><a name="Rank 3 temperaments--Lehmerismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:222 --><a class="wiki_link" href="/Lehmerismic%20temperaments">Lehmerismic temperaments</a></h3> | ||
These temper out the lehmerisma, 3025/3024.<br /> | These temper out the lehmerisma, 3025/3024.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:224:&lt;h3&gt; --><h3 id="toc112"><a name="Rank 3 temperaments--Kalismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:224 --><a class="wiki_link" href="/Kalismic%20temperaments">Kalismic temperaments</a></h3> | ||
These temper out the kalisma, 9801/9800.<br /> | These temper out the kalisma, 9801/9800.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:226:&lt;h1&gt; --><h1 id="toc113"><a name="Rank 4 temperaments"></a><!-- ws:end:WikiTextHeadingRule:226 --><a class="wiki_link" href="/Rank%20four%20temperaments">Rank 4 temperaments</a></h1> | ||
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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:228:&lt;h1&gt; --><h1 id="toc114"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:228 --><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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:230:&lt;h1&gt; --><h1 id="toc115"><a name="Commatic realms"></a><!-- ws:end:WikiTextHeadingRule:230 -->Commatic realms</h1> | ||
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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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:232:&lt;h2&gt; --><h2 id="toc116"><a name="Commatic realms-Orgonia"></a><!-- ws:end:WikiTextHeadingRule:232 --><a class="wiki_link" href="/Orgonia">Orgonia</a></h2> | ||
Orgonia is 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 /> | Orgonia is 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:234:&lt;h2&gt; --><h2 id="toc117"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:234 --><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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:236:&lt;h2&gt; --><h2 id="toc118"><a name="Commatic realms-The Archipelago"></a><!-- ws:end:WikiTextHeadingRule:236 --><a class="wiki_link" href="/The%20Archipelago">The Archipelago</a></h2> | ||
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:238:&lt;h1&gt; --><h1 id="toc119"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:238 -->Links</h1> | ||
<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">Regular temperaments - Wikipedia</a></li><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/lintemps.html" rel="nofollow" target="_blank">List of temperaments</a> in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala" rel="nofollow" target="_blank">Scala</a> with ready to use values</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><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/lintemps.html" rel="nofollow" target="_blank">List of temperaments</a> in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala" rel="nofollow" target="_blank">Scala</a> with ready to use values</li></ul></body></html></pre></div> | ||