Xenharmonic Wiki

Associated temperament

Revision as of 16:18, 7 February 2012 by Wikispaces>genewardsmith (**Imported revision 299452532 - Original comment: **)

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This revision was by author genewardsmith and made on 2012-02-07 16:18:42 UTC.
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Original Wikitext content:

By an //associated temperament// to a p-limit comma is meant a p-limit temperament which shares the same [[optimal patent val]] as the [[Rank and codimension|codimension one]] temperament tempering out that comma. By definition, the optimal patent val defines the unique rank one associated temperament. For rank two temperaments, it is possible for more than one temperament to be associated, and some of these are listed below. The column headings are the comma being associated, the optimal patent val, the rank two temperament, and a comma basis for the rank two temperament.

=7 limit=
|| Comma || OPV || Temperament || Basis ||
|| 28/27 || [[15edo]] || [[blacksmith]] || 28/27, 49/48 ||
|| 1029/1000 || [[55edo]] || [[liese]] || 81/80, 686/675 ||
|| 36/35 || [[12edo]] || [[diminished]] || 36/35, 50/49 ||
|| 36/35 || [[12edo]] || [[august]] || 36/35, 128/125 ||
|| 36/35 || [[12edo]] || [[dominant]] || 36/35, 64/63 ||
|| 525/512 || [[45edo]] || [[flattone]] || 81/80, 525/512 ||
|| 49/48 || [[19edo]] || [[keemun]] || 49/48, 126/125 ||
|| 49/48 || [[19edo]] || [[godzilla]] || 49/48, 81/80 ||
|| 50/49 || [[48edo]] || [[doublewide]] || 50/49, 875/864 ||
|| 64/63 || [[49edo]] || [[superpyth]] || 64/63, 245/243 ||
|| 875/864 || [[41edo]] || [[magic]] || 225/224, 245/243 ||
|| 3125/3087 || [[94edo]] || [[garibaldi]] || 225/224, 3125/3087 ||
|| 2430/2401 || [[137edo]] || [[orwell]] || 225/224, 1728/1715 ||
|| 245/243 || [[283edo]] || [[Escapade family#Escaped|escaped]] || 245/243, 65625/65536 ||
|| 126/125 || [[185edo]] || [[valentine]] || 126/125, 1029/1024 ||
|| 1728/1715 || [[111edo]] || [[Hemifamity temperaments#Buzzard|buzzard]] || 1728/1715, 5120/5103 ||
|| 1728/1715 || [[111edo]] || [[semisept]] || 1728/1715, 3136/3125 ||
|| 1029/1024 || [[190edo]] || [[unidec]] || 1029/1024, 4375/4374 ||
|| 225/224 || [[197edo]] || [[catakleismic]] || 225/224, 4375/4374 ||
|| 3136/3125 || [[446edo]] || [[Hemimean clan#Sengagen|sengagen]] || 3136/3125, 420175/419904 ||
|| 5120/5103 || [[391edo]] || [[Hemifamity temperaments#Alphaquarter|alphaquarter]] || 5120/5103, 29360128/29296875 ||
|| 5120/5103 || [[391edo]] || [[Hemifamity temperaments#Septiquarter|septiquarter]] || 5120/5103, 420175/419904 ||
|| 6144/6125 || [[381edo]] || [[Porwell temperaments#Nessafof|nessafof]] || 6144/6125, 250047/250000 ||
|| 65625/65536 || [[171edo]] || [[Breedsmic temperaments#Tertiaseptal|tertiaseptal]] || 2401/2400, 65625/65536 ||
|| 703125/702464 || [[2185edo]] || [[Ragismic microtemperaments#Enneadecal|enneadecal]] || 4375/4374, 703125/702464 ||
|| 4375/4374 || [[8419edo]] || [[Ragismic microtemperaments#Semidimi|semidimi]] || 4375/4374, 3955078125/3954653486 ||
|| 250047/250000 || [[12555edo]] || || 250047/250000, 281484423828125/281474976710656 ||

=11 limit=
|| Comma || OPV || Temperament || Basis ||
|| 33/32 || [[16edo]] || [[Pelogic family#Armodue|armodue]] || 33/32 36/35 45/44 ||
|| 77/75 || [[39edo]] || [[triforce]] || 49/48, 56/55, 77/75 ||
|| 352/343 || [[22edo]] || [[hedgehog]] || 50/49, 55/54, 99/98 ||
|| 45/44 || [[45edo]] || [[flattone]] || 45/44, 81/80, 385/384 ||
|| 55/54 || [[51edo]] || [[porky]] || 55/54, 100/99, 225/224 ||
|| 56/55 || [[36edo]] || [[Pythagorean family#Catler temperament|catcall]] || 56/55, 81/80, 128/125 ||
|| 245/242 || [[91edo]] || [[septimin]] || 225/224, 245/242, 385/384 ||

Original HTML content:

<html><head><title>Associated rank two temperaments</title></head><body>By an <em>associated temperament</em> to a p-limit comma is meant a p-limit temperament which shares the same <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> as the <a class="wiki_link" href="/Rank%20and%20codimension">codimension one</a> temperament tempering out that comma. By definition, the optimal patent val defines the unique rank one associated temperament. For rank two temperaments, it is possible for more than one temperament to be associated, and some of these are listed below. The column headings are the comma being associated, the optimal patent val, the rank two temperament, and a comma basis for the rank two temperament.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x7 limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->7 limit</h1>


<table class="wiki_table">
    <tr>
        <td>Comma<br />
</td>
        <td>OPV<br />
</td>
        <td>Temperament<br />
</td>
        <td>Basis<br />
</td>
    </tr>
    <tr>
        <td>28/27<br />
</td>
        <td><a class="wiki_link" href="/15edo">15edo</a><br />
</td>
        <td><a class="wiki_link" href="/blacksmith">blacksmith</a><br />
</td>
        <td>28/27, 49/48<br />
</td>
    </tr>
    <tr>
        <td>1029/1000<br />
</td>
        <td><a class="wiki_link" href="/55edo">55edo</a><br />
</td>
        <td><a class="wiki_link" href="/liese">liese</a><br />
</td>
        <td>81/80, 686/675<br />
</td>
    </tr>
    <tr>
        <td>36/35<br />
</td>
        <td><a class="wiki_link" href="/12edo">12edo</a><br />
</td>
        <td><a class="wiki_link" href="/diminished">diminished</a><br />
</td>
        <td>36/35, 50/49<br />
</td>
    </tr>
    <tr>
        <td>36/35<br />
</td>
        <td><a class="wiki_link" href="/12edo">12edo</a><br />
</td>
        <td><a class="wiki_link" href="/august">august</a><br />
</td>
        <td>36/35, 128/125<br />
</td>
    </tr>
    <tr>
        <td>36/35<br />
</td>
        <td><a class="wiki_link" href="/12edo">12edo</a><br />
</td>
        <td><a class="wiki_link" href="/dominant">dominant</a><br />
</td>
        <td>36/35, 64/63<br />
</td>
    </tr>
    <tr>
        <td>525/512<br />
</td>
        <td><a class="wiki_link" href="/45edo">45edo</a><br />
</td>
        <td><a class="wiki_link" href="/flattone">flattone</a><br />
</td>
        <td>81/80, 525/512<br />
</td>
    </tr>
    <tr>
        <td>49/48<br />
</td>
        <td><a class="wiki_link" href="/19edo">19edo</a><br />
</td>
        <td><a class="wiki_link" href="/keemun">keemun</a><br />
</td>
        <td>49/48, 126/125<br />
</td>
    </tr>
    <tr>
        <td>49/48<br />
</td>
        <td><a class="wiki_link" href="/19edo">19edo</a><br />
</td>
        <td><a class="wiki_link" href="/godzilla">godzilla</a><br />
</td>
        <td>49/48, 81/80<br />
</td>
    </tr>
    <tr>
        <td>50/49<br />
</td>
        <td><a class="wiki_link" href="/48edo">48edo</a><br />
</td>
        <td><a class="wiki_link" href="/doublewide">doublewide</a><br />
</td>
        <td>50/49, 875/864<br />
</td>
    </tr>
    <tr>
        <td>64/63<br />
</td>
        <td><a class="wiki_link" href="/49edo">49edo</a><br />
</td>
        <td><a class="wiki_link" href="/superpyth">superpyth</a><br />
</td>
        <td>64/63, 245/243<br />
</td>
    </tr>
    <tr>
        <td>875/864<br />
</td>
        <td><a class="wiki_link" href="/41edo">41edo</a><br />
</td>
        <td><a class="wiki_link" href="/magic">magic</a><br />
</td>
        <td>225/224, 245/243<br />
</td>
    </tr>
    <tr>
        <td>3125/3087<br />
</td>
        <td><a class="wiki_link" href="/94edo">94edo</a><br />
</td>
        <td><a class="wiki_link" href="/garibaldi">garibaldi</a><br />
</td>
        <td>225/224, 3125/3087<br />
</td>
    </tr>
    <tr>
        <td>2430/2401<br />
</td>
        <td><a class="wiki_link" href="/137edo">137edo</a><br />
</td>
        <td><a class="wiki_link" href="/orwell">orwell</a><br />
</td>
        <td>225/224, 1728/1715<br />
</td>
    </tr>
    <tr>
        <td>245/243<br />
</td>
        <td><a class="wiki_link" href="/283edo">283edo</a><br />
</td>
        <td><a class="wiki_link" href="/Escapade%20family#Escaped">escaped</a><br />
</td>
        <td>245/243, 65625/65536<br />
</td>
    </tr>
    <tr>
        <td>126/125<br />
</td>
        <td><a class="wiki_link" href="/185edo">185edo</a><br />
</td>
        <td><a class="wiki_link" href="/valentine">valentine</a><br />
</td>
        <td>126/125, 1029/1024<br />
</td>
    </tr>
    <tr>
        <td>1728/1715<br />
</td>
        <td><a class="wiki_link" href="/111edo">111edo</a><br />
</td>
        <td><a class="wiki_link" href="/Hemifamity%20temperaments#Buzzard">buzzard</a><br />
</td>
        <td>1728/1715, 5120/5103<br />
</td>
    </tr>
    <tr>
        <td>1728/1715<br />
</td>
        <td><a class="wiki_link" href="/111edo">111edo</a><br />
</td>
        <td><a class="wiki_link" href="/semisept">semisept</a><br />
</td>
        <td>1728/1715, 3136/3125<br />
</td>
    </tr>
    <tr>
        <td>1029/1024<br />
</td>
        <td><a class="wiki_link" href="/190edo">190edo</a><br />
</td>
        <td><a class="wiki_link" href="/unidec">unidec</a><br />
</td>
        <td>1029/1024, 4375/4374<br />
</td>
    </tr>
    <tr>
        <td>225/224<br />
</td>
        <td><a class="wiki_link" href="/197edo">197edo</a><br />
</td>
        <td><a class="wiki_link" href="/catakleismic">catakleismic</a><br />
</td>
        <td>225/224, 4375/4374<br />
</td>
    </tr>
    <tr>
        <td>3136/3125<br />
</td>
        <td><a class="wiki_link" href="/446edo">446edo</a><br />
</td>
        <td><a class="wiki_link" href="/Hemimean%20clan#Sengagen">sengagen</a><br />
</td>
        <td>3136/3125, 420175/419904<br />
</td>
    </tr>
    <tr>
        <td>5120/5103<br />
</td>
        <td><a class="wiki_link" href="/391edo">391edo</a><br />
</td>
        <td><a class="wiki_link" href="/Hemifamity%20temperaments#Alphaquarter">alphaquarter</a><br />
</td>
        <td>5120/5103, 29360128/29296875<br />
</td>
    </tr>
    <tr>
        <td>5120/5103<br />
</td>
        <td><a class="wiki_link" href="/391edo">391edo</a><br />
</td>
        <td><a class="wiki_link" href="/Hemifamity%20temperaments#Septiquarter">septiquarter</a><br />
</td>
        <td>5120/5103, 420175/419904<br />
</td>
    </tr>
    <tr>
        <td>6144/6125<br />
</td>
        <td><a class="wiki_link" href="/381edo">381edo</a><br />
</td>
        <td><a class="wiki_link" href="/Porwell%20temperaments#Nessafof">nessafof</a><br />
</td>
        <td>6144/6125, 250047/250000<br />
</td>
    </tr>
    <tr>
        <td>65625/65536<br />
</td>
        <td><a class="wiki_link" href="/171edo">171edo</a><br />
</td>
        <td><a class="wiki_link" href="/Breedsmic%20temperaments#Tertiaseptal">tertiaseptal</a><br />
</td>
        <td>2401/2400, 65625/65536<br />
</td>
    </tr>
    <tr>
        <td>703125/702464<br />
</td>
        <td><a class="wiki_link" href="/2185edo">2185edo</a><br />
</td>
        <td><a class="wiki_link" href="/Ragismic%20microtemperaments#Enneadecal">enneadecal</a><br />
</td>
        <td>4375/4374, 703125/702464<br />
</td>
    </tr>
    <tr>
        <td>4375/4374<br />
</td>
        <td><a class="wiki_link" href="/8419edo">8419edo</a><br />
</td>
        <td><a class="wiki_link" href="/Ragismic%20microtemperaments#Semidimi">semidimi</a><br />
</td>
        <td>4375/4374, 3955078125/3954653486<br />
</td>
    </tr>
    <tr>
        <td>250047/250000<br />
</td>
        <td><a class="wiki_link" href="/12555edo">12555edo</a><br />
</td>
        <td><br />
</td>
        <td>250047/250000, 281484423828125/281474976710656<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x11 limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11 limit</h1>


<table class="wiki_table">
    <tr>
        <td>Comma<br />
</td>
        <td>OPV<br />
</td>
        <td>Temperament<br />
</td>
        <td>Basis<br />
</td>
    </tr>
    <tr>
        <td>33/32<br />
</td>
        <td><a class="wiki_link" href="/16edo">16edo</a><br />
</td>
        <td><a class="wiki_link" href="/Pelogic%20family#Armodue">armodue</a><br />
</td>
        <td>33/32 36/35 45/44<br />
</td>
    </tr>
    <tr>
        <td>77/75<br />
</td>
        <td><a class="wiki_link" href="/39edo">39edo</a><br />
</td>
        <td><a class="wiki_link" href="/triforce">triforce</a><br />
</td>
        <td>49/48, 56/55, 77/75<br />
</td>
    </tr>
    <tr>
        <td>352/343<br />
</td>
        <td><a class="wiki_link" href="/22edo">22edo</a><br />
</td>
        <td><a class="wiki_link" href="/hedgehog">hedgehog</a><br />
</td>
        <td>50/49, 55/54, 99/98<br />
</td>
    </tr>
    <tr>
        <td>45/44<br />
</td>
        <td><a class="wiki_link" href="/45edo">45edo</a><br />
</td>
        <td><a class="wiki_link" href="/flattone">flattone</a><br />
</td>
        <td>45/44, 81/80, 385/384<br />
</td>
    </tr>
    <tr>
        <td>55/54<br />
</td>
        <td><a class="wiki_link" href="/51edo">51edo</a><br />
</td>
        <td><a class="wiki_link" href="/porky">porky</a><br />
</td>
        <td>55/54, 100/99, 225/224<br />
</td>
    </tr>
    <tr>
        <td>56/55<br />
</td>
        <td><a class="wiki_link" href="/36edo">36edo</a><br />
</td>
        <td><a class="wiki_link" href="/Pythagorean%20family#Catler temperament">catcall</a><br />
</td>
        <td>56/55, 81/80, 128/125<br />
</td>
    </tr>
    <tr>
        <td>245/242<br />
</td>
        <td><a class="wiki_link" href="/91edo">91edo</a><br />
</td>
        <td><a class="wiki_link" href="/septimin">septimin</a><br />
</td>
        <td>225/224, 245/242, 385/384<br />
</td>
    </tr>
</table>

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