Interseptimal interval: Difference between revisions

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In the theory of [[Margo Schulter]], //interseptimal// is a category of intervals which occupy regions intermediate between two septimal ratios such as [[8_7|8/7]] and [[7_6|7/6]], or [[12_7|12/7]] and [[7_4|7/4]]. There are four interseptimal regions given below, with approximate cents ranges from Schulter's article [[http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt|Regions of the Interval Spectrum]]:
# Maj2-min3 -- intermediate between 8/7 and 7/6 -- 240¢-260¢
# Maj3-4 -- intermediate between [[9_7|9/7]] and [[21_16|21/16]] -- 440¢-468¢
# 5-min6 -- intermediate between [[32_21|32/21]] and [[14_9|14/9]] -- 732¢-760¢
# Maj6-min7 -- intermediate between 12/7 and 7/4 -- 940¢-960¢

Interseptimal intervals are well-represented in [[24edo]] at 250¢, 450¢, 750¢ and 950¢. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic.

==Examples== 
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.

===Maj2-min3 - 240¢-260¢=== 
||~ Interval ||~ Cents Value ||~ Prime Limit (if applicable) ||
|| 147/128 || 239.607 || 7 ||
|| 1\[[5edo]] || 240.000 || - ||
|| 54/47 || 240.358 || 47 ||
|| 23/20 || 241.961 || 23 ||
|| 1152/1001 || 243.238 || 13 ||
|| 38/33 || 244.240 || 19 ||
|| [[15_13|15/13]] || 247.741 || 13 ||
|| 6\[[29edo]] || 248.276 || - ||
|| 5\[[24edo]] || 250.000 || - ||
|| 52/45 || 250.304 || 13 ||
|| 37/32 || 251.344 || 37 ||
|| 4\[[19edo]] || 252.632 || - ||
|| 22/19 || 253.805 || 19 ||
|| 29/25 || 256.950 || 29 ||
|| 3\[[14edo]] || 257.143 || - ||
|| 297/256 || 257.183 || 11 ||
|| 36/31 || 258.874 || 31 ||
|| 5\[[23edo]] || 260.870 || - ||

===Maj3-4 - 440-468=== 

||~ Interval ||~ Cents Value ||~ Prime Limit (if applicable) ||
|| 5\[[88cET]] or 11\[[30edo]] || 440.000 || - ||
|| 40/31 || 441.278 || 31 ||
|| 7\[[19edo]] || 442.015 || - ||
|| 31/24 || 443.081 || 31 ||
|| 10\[[27edo]] || 444.444 || - ||
|| [[22_17|22/17]] || 446.363 || 17 ||
|| [[35_27|35/27]] || 449.275 || 7 ||
|| 3\[[24edo]] || 450.000 || - ||
|| 48/37 || 450.611 || 37 ||
|| [[13_10|13/10]] || 454.214 || 13 ||
|| 11\[[29edo]] || 455.172 || - ||
|| 8\[[21edo]] || 457.143 || - ||
|| 56/43 || 457.308 || 43 ||
|| 43/33 || 458.245 || 43 ||
|| 30/23 || 459.994 || 23 ||
|| 5\[[13edo]] || 461.538 || - ||
|| 47/36 || 461.597 || 47 ||
|| [[49_64|49/64]] || 462.348 || 7 ||
|| [[17_13|17/13]] || 464.428 || 17 ||
|| 12\[[31edo]] || 464.516 || - ||
|| 7\[[18edo]] || 466.667 || - ||
|| 38/29 || 467.936 || 29 ||


See: [[Interval Category]], [[Gallery of Just Intervals]]

Original HTML content:

<html><head><title>Interseptimal</title></head><body>In the theory of <a class="wiki_link" href="/Margo%20Schulter">Margo Schulter</a>, <em>interseptimal</em> is a category of intervals which occupy regions intermediate between two septimal ratios such as <a class="wiki_link" href="/8_7">8/7</a> and <a class="wiki_link" href="/7_6">7/6</a>, or <a class="wiki_link" href="/12_7">12/7</a> and <a class="wiki_link" href="/7_4">7/4</a>. There are four interseptimal regions given below, with approximate cents ranges from Schulter's article <a class="wiki_link_ext" href="http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt" rel="nofollow">Regions of the Interval Spectrum</a>:<br />
<ol><li>Maj2-min3 -- intermediate between 8/7 and 7/6 -- 240¢-260¢</li><li>Maj3-4 -- intermediate between <a class="wiki_link" href="/9_7">9/7</a> and <a class="wiki_link" href="/21_16">21/16</a> -- 440¢-468¢</li><li>5-min6 -- intermediate between <a class="wiki_link" href="/32_21">32/21</a> and <a class="wiki_link" href="/14_9">14/9</a> -- 732¢-760¢</li><li>Maj6-min7 -- intermediate between 12/7 and 7/4 -- 940¢-960¢</li></ol><br />
Interseptimal intervals are well-represented in <a class="wiki_link" href="/24edo">24edo</a> at 250¢, 450¢, 750¢ and 950¢. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2>
 Some interseptimal intervals in all four ranges, both just and tempered, are listed below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Examples-Maj2-min3 - 240¢-260¢"></a><!-- ws:end:WikiTextHeadingRule:2 -->Maj2-min3 - 240¢-260¢</h3>
 

<table class="wiki_table">
    <tr>
        <th>Interval<br />
</th>
        <th>Cents Value<br />
</th>
        <th>Prime Limit (if applicable)<br />
</th>
    </tr>
    <tr>
        <td>147/128<br />
</td>
        <td>239.607<br />
</td>
        <td>7<br />
</td>
    </tr>
    <tr>
        <td>1\<a class="wiki_link" href="/5edo">5edo</a><br />
</td>
        <td>240.000<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>54/47<br />
</td>
        <td>240.358<br />
</td>
        <td>47<br />
</td>
    </tr>
    <tr>
        <td>23/20<br />
</td>
        <td>241.961<br />
</td>
        <td>23<br />
</td>
    </tr>
    <tr>
        <td>1152/1001<br />
</td>
        <td>243.238<br />
</td>
        <td>13<br />
</td>
    </tr>
    <tr>
        <td>38/33<br />
</td>
        <td>244.240<br />
</td>
        <td>19<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/15_13">15/13</a><br />
</td>
        <td>247.741<br />
</td>
        <td>13<br />
</td>
    </tr>
    <tr>
        <td>6\<a class="wiki_link" href="/29edo">29edo</a><br />
</td>
        <td>248.276<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>5\<a class="wiki_link" href="/24edo">24edo</a><br />
</td>
        <td>250.000<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>52/45<br />
</td>
        <td>250.304<br />
</td>
        <td>13<br />
</td>
    </tr>
    <tr>
        <td>37/32<br />
</td>
        <td>251.344<br />
</td>
        <td>37<br />
</td>
    </tr>
    <tr>
        <td>4\<a class="wiki_link" href="/19edo">19edo</a><br />
</td>
        <td>252.632<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>22/19<br />
</td>
        <td>253.805<br />
</td>
        <td>19<br />
</td>
    </tr>
    <tr>
        <td>29/25<br />
</td>
        <td>256.950<br />
</td>
        <td>29<br />
</td>
    </tr>
    <tr>
        <td>3\<a class="wiki_link" href="/14edo">14edo</a><br />
</td>
        <td>257.143<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>297/256<br />
</td>
        <td>257.183<br />
</td>
        <td>11<br />
</td>
    </tr>
    <tr>
        <td>36/31<br />
</td>
        <td>258.874<br />
</td>
        <td>31<br />
</td>
    </tr>
    <tr>
        <td>5\<a class="wiki_link" href="/23edo">23edo</a><br />
</td>
        <td>260.870<br />
</td>
        <td>-<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Maj3-4 - 440-468"></a><!-- ws:end:WikiTextHeadingRule:4 -->Maj3-4 - 440-468</h3>
 <br />


<table class="wiki_table">
    <tr>
        <th>Interval<br />
</th>
        <th>Cents Value<br />
</th>
        <th>Prime Limit (if applicable)<br />
</th>
    </tr>
    <tr>
        <td>5\<a class="wiki_link" href="/88cET">88cET</a> or 11\<a class="wiki_link" href="/30edo">30edo</a><br />
</td>
        <td>440.000<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>40/31<br />
</td>
        <td>441.278<br />
</td>
        <td>31<br />
</td>
    </tr>
    <tr>
        <td>7\<a class="wiki_link" href="/19edo">19edo</a><br />
</td>
        <td>442.015<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>31/24<br />
</td>
        <td>443.081<br />
</td>
        <td>31<br />
</td>
    </tr>
    <tr>
        <td>10\<a class="wiki_link" href="/27edo">27edo</a><br />
</td>
        <td>444.444<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/22_17">22/17</a><br />
</td>
        <td>446.363<br />
</td>
        <td>17<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/35_27">35/27</a><br />
</td>
        <td>449.275<br />
</td>
        <td>7<br />
</td>
    </tr>
    <tr>
        <td>3\<a class="wiki_link" href="/24edo">24edo</a><br />
</td>
        <td>450.000<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>48/37<br />
</td>
        <td>450.611<br />
</td>
        <td>37<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/13_10">13/10</a><br />
</td>
        <td>454.214<br />
</td>
        <td>13<br />
</td>
    </tr>
    <tr>
        <td>11\<a class="wiki_link" href="/29edo">29edo</a><br />
</td>
        <td>455.172<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>8\<a class="wiki_link" href="/21edo">21edo</a><br />
</td>
        <td>457.143<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>56/43<br />
</td>
        <td>457.308<br />
</td>
        <td>43<br />
</td>
    </tr>
    <tr>
        <td>43/33<br />
</td>
        <td>458.245<br />
</td>
        <td>43<br />
</td>
    </tr>
    <tr>
        <td>30/23<br />
</td>
        <td>459.994<br />
</td>
        <td>23<br />
</td>
    </tr>
    <tr>
        <td>5\<a class="wiki_link" href="/13edo">13edo</a><br />
</td>
        <td>461.538<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>47/36<br />
</td>
        <td>461.597<br />
</td>
        <td>47<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/49_64">49/64</a><br />
</td>
        <td>462.348<br />
</td>
        <td>7<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/17_13">17/13</a><br />
</td>
        <td>464.428<br />
</td>
        <td>17<br />
</td>
    </tr>
    <tr>
        <td>12\<a class="wiki_link" href="/31edo">31edo</a><br />
</td>
        <td>464.516<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>7\<a class="wiki_link" href="/18edo">18edo</a><br />
</td>
        <td>466.667<br />
</td>
        <td>-<br />
</td>
    </tr>
    <tr>
        <td>38/29<br />
</td>
        <td>467.936<br />
</td>
        <td>29<br />
</td>
    </tr>
</table>

<br />
<br />
See: <a class="wiki_link" href="/Interval%20Category">Interval Category</a>, <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a></body></html>

Interseptimal interval: Difference between revisions

Revision as of 19:51, 10 October 2011

IMPORTED REVISION FROM WIKISPACES

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