Tour of regular temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 505791746 - 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>2014-05- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-04 16:49:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>506390598</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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===[[Landscape microtemperaments]]=== | ===[[Landscape microtemperaments]]=== | ||
A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000. | A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000. | ||
==[[Wizmic microtemperaments]]== | |||
A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 > = 420175/419904. | |||
===[[31 comma temperaments]]=== | ===[[31 comma 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:236:&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:236 --><!-- ws:start:WikiTextTocRule:237: --><a href="#Regular temperaments">Regular temperaments</a><!-- 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="#Equal temperaments">Equal temperaments</a><!-- ws:end:WikiTextTocRule:240 --><!-- ws:start:WikiTextTocRule:241: --> | <a href="#Rank 2 (including &quot;linear&quot;) temperaments">Rank 2 (including &quot;linear&quot;) temperaments</a><!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: 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ws:start:WikiTextTocRule:284: --><!-- ws:end:WikiTextTocRule:284 --><!-- ws:start:WikiTextTocRule:285: --><!-- ws:end:WikiTextTocRule:285 --><!-- ws:start:WikiTextTocRule:286: --><!-- ws:end:WikiTextTocRule:286 --><!-- ws:start:WikiTextTocRule:287: --><!-- ws:end:WikiTextTocRule:287 --><!-- ws:start:WikiTextTocRule:288: --> | <a href="#Temperaments for a given comma">Temperaments for a given comma</a><!-- 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: --><!-- 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: --><!-- 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ws:start:WikiTextTocRule:310: --><!-- ws:end:WikiTextTocRule:310 --><!-- ws:start:WikiTextTocRule:311: --><!-- ws:end:WikiTextTocRule:311 --><!-- ws:start:WikiTextTocRule:312: --><!-- ws:end:WikiTextTocRule:312 --><!-- ws:start:WikiTextTocRule:313: --> | <a href="#Rank 3 temperaments">Rank 3 temperaments</a><!-- ws:end:WikiTextTocRule:313 --><!-- 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: --><!-- 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 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ws:end:WikiTextTocRule:336 --><!-- ws:start:WikiTextTocRule:337: --><!-- ws:end:WikiTextTocRule:337 --><!-- ws:start:WikiTextTocRule:338: --><!-- ws:end:WikiTextTocRule:338 --><!-- ws:start:WikiTextTocRule:339: --><!-- ws:end:WikiTextTocRule:339 --><!-- ws:start:WikiTextTocRule:340: --><!-- ws:end:WikiTextTocRule:340 --><!-- ws:start:WikiTextTocRule:341: --><!-- ws:end:WikiTextTocRule:341 --><!-- ws:start:WikiTextTocRule:342: --><!-- ws:end:WikiTextTocRule:342 --><!-- ws:start:WikiTextTocRule:343: --><!-- ws:end:WikiTextTocRule:343 --><!-- ws:start:WikiTextTocRule:344: --><!-- ws:end:WikiTextTocRule:344 --><!-- ws:start:WikiTextTocRule:345: --><!-- ws:end:WikiTextTocRule:345 --><!-- ws:start:WikiTextTocRule:346: --><!-- ws:end:WikiTextTocRule:346 --><!-- ws:start:WikiTextTocRule:347: --><!-- ws:end:WikiTextTocRule:347 --><!-- ws:start:WikiTextTocRule:348: --> | <a href="#Rank 4 temperaments">Rank 4 temperaments</a><!-- ws:end:WikiTextTocRule:348 --><!-- ws:start:WikiTextTocRule:349: --> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:349 --><!-- ws:start:WikiTextTocRule:350: --> | <a href="#Commatic realms">Commatic realms</a><!-- ws:end:WikiTextTocRule:350 --><!-- ws:start:WikiTextTocRule:351: --><!-- ws:end:WikiTextTocRule:351 --><!-- ws:start:WikiTextTocRule:352: --><!-- ws:end:WikiTextTocRule:352 --><!-- ws:start:WikiTextTocRule:353: --><!-- ws:end:WikiTextTocRule:353 --><!-- ws:start:WikiTextTocRule:354: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:354 --><!-- ws:start:WikiTextTocRule:355: --> | ||
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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:356:&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:356 --></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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A landscape temperament is one which tempers out |-4 6 -6 3&gt; = 250047/250000.<br /> | A landscape temperament is one which tempers out |-4 6 -6 3&gt; = 250047/250000.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:138:&lt;h3&gt; --><h3 id=" | <!-- ws:start:WikiTextHeadingRule:138:&lt;h2&gt; --><h2 id="toc69"><a name="Temperaments for a given comma-Wizmic microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:138 --><a class="wiki_link" href="/Wizmic%20microtemperaments">Wizmic microtemperaments</a></h2> | ||
A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 &gt; = 420175/419904.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:140:&lt;h3&gt; --><h3 id="toc70"><a name="Temperaments for a given comma-Wizmic microtemperaments-31 comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:140 --><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:142:&lt;h3&gt; --><h3 id="toc71"><a name="Temperaments for a given comma-Wizmic microtemperaments-Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:142 --><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3> | ||
Various 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">makam (maqam) music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br /> | Various 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">makam (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:144:&lt;h3&gt; --><h3 id="toc72"><a name="Temperaments for a given comma-Wizmic microtemperaments-Very low accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:144 --><a class="wiki_link" href="/Very%20low%20accuracy%20temperaments">Very low accuracy temperaments</a></h3> | ||
All hope abandon ye who enter here.<br /> | All hope abandon ye who enter here.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:146:&lt;h3&gt; --><h3 id="toc73"><a name="Temperaments for a given comma-Wizmic microtemperaments-Very high accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:146 --><a class="wiki_link" href="/Very%20high%20accuracy%20temperaments">Very high accuracy temperaments</a></h3> | ||
Microtemperaments which don't fit in elsewhere.<br /> | Microtemperaments which don't fit in elsewhere.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:148:&lt;h3&gt; --><h3 id="toc74"><a name="Temperaments for a given comma-Wizmic microtemperaments-High badness temperaments"></a><!-- ws:end:WikiTextHeadingRule:148 --><a class="wiki_link" href="/High%20badness%20temperaments">High badness temperaments</a></h3> | ||
High in badness, but worth cataloging for one reason or another.<br /> | High in badness, but worth cataloging for one reason or another.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:150:&lt;h3&gt; --><h3 id="toc75"><a name="Temperaments for a given comma-Wizmic microtemperaments-11-limit comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:150 --><a class="wiki_link" href="/11-limit%20comma%20temperaments">11-limit comma temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:152:&lt;h1&gt; --><h1 id="toc76"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:152 -->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:154:&lt;h3&gt; --><h3 id="toc77"><a name="Rank 3 temperaments--Marvel family"></a><!-- ws:end:WikiTextHeadingRule:154 --><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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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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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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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, 2749edo will certainly do the trick. Breed has generators of 2/1, 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, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.<br /> | ||
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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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The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.<br /> | The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.<br /> | ||
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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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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 the senga, 686/675.<br /> | These temper out the senga, 686/675.<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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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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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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These temper out the tolerma, 179200/177147.<br /> | These temper out the tolerma, 179200/177147.<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 maximal 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 maximal 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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Semiphore 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 /> | Semiphore 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 mint temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br /> | The mint 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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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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:228:&lt;h2&gt; --><h2 id="toc114"><a name="Commatic realms-Orgonia"></a><!-- ws:end:WikiTextHeadingRule:228 --><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:230:&lt;h2&gt; --><h2 id="toc115"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:230 --><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:234:&lt;h1&gt; --><h1 id="toc117"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:234 -->Links</h1> | ||
<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> | ||