Breedsmic temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2010-07-~~04 05~~:39:02 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-08 23:57:30 UTC</tt>.<br>

: The original revision id was <tt>~~151486757~~</tt>.<br>

: The original revision id was <tt>152022519</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>

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===Tertiaseptal===

Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

~~===Neptune===~~

Neptune adds 48828125/48771072 = |-12 -5 11 -2> to the list of commas, and may be described as the 68&171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as <<40 22 21 -3 ...|| or 68&103, and 171 can still be used as a tuning, with val <171 271 397 480 591|.

~~An article on Neptune as an analog of miracle can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/6001|here]].~~

===Harry===

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Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br />

<br />

<h3 id="toc1"><a name="x--~~Neptune~~"></a>Neptune</h3>

<h3 id="toc1"><a name="x--Harry"></a>Harry</h3>

Neptune adds 48828125/48771072 = |-12 -5 11 -2&gt; to the list of commas, and may be described as the 68&amp;171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. <a class="wiki_link" href="/171edo">171edo</a> makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.<br />

~~<br />~~

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &lt;&lt;40 22 21 -3 ...|| or 68&amp;103, and 171 can still be used as a tuning, with val &lt;171 271 397 480 591|.<br />

~~<br />~~

~~An article on Neptune as an analog of miracle can be found <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/6001" rel="nofollow">here</a>.<br />~~

~~<br />~~

<h3 id="toc2"><a name="x--Harry"></a>Harry</h3>

Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;72 temperament, with wedgie &lt;&lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.<br />

<br />

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Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. <a class="wiki_link" href="/130edo">130edo</a> is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.<br />

<br />

<h3 id="~~toc3~~"><a name="x--Quasiorwell"></a>Quasiorwell</h3>

<h3 id="toc2"><a name="x--Quasiorwell"></a>Quasiorwell</h3>

In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br />

<br />

Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-07-04 05:39:02 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-08 23:57:30 UTC</tt>.<br>
-: The original revision id was <tt>151486757</tt>.<br>
+: The original revision id was <tt>152022519</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>
@@ Line 12: / Line 12: @@
 ===Tertiaseptal===
 Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
-===Neptune===
-Neptune adds 48828125/48771072 = |-12 -5 11 -2&gt; to the list of commas, and may be described as the 68&amp;171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.
-Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &lt;&lt;40 22 21 -3 ...|| or 68&amp;103, and 171 can still be used as a tuning, with val &lt;171 271 397 480  591|.
-An article on Neptune as an analog of miracle can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/6001|here]].
 ===Harry===
@@ Line 39: / Line 32: @@
 Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Neptune"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Neptune&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Harry"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harry&lt;/h3&gt;
-Neptune adds 48828125/48771072 = |-12 -5 11 -2&amp;gt; to the list of commas, and may be described as the 68&amp;amp;171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.&lt;br /&gt;
-&lt;br /&gt;
-Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &amp;lt;&amp;lt;40 22 21 -3 ...|| or 68&amp;amp;103, and 171 can still be used as a tuning, with val &amp;lt;171 271 397 480  591|.&lt;br /&gt;
-&lt;br /&gt;
-An article on Neptune as an analog of miracle can be found &lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/6001" rel="nofollow"&gt;here&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--Harry"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Harry&lt;/h3&gt;
 Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;amp;72 temperament, with wedgie &amp;lt;&amp;lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 53: / Line 39: @@
 Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &amp;lt;&amp;lt;12 34 20 30 52 ...|| as the octave wedgie. &lt;a class="wiki_link" href="/130edo"&gt;130edo&lt;/a&gt; is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--Quasiorwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Quasiorwell&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--Quasiorwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Quasiorwell&lt;/h3&gt;
 In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&amp;gt;. It has a wedgie &amp;lt;&amp;lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.&lt;br /&gt;
 &lt;br /&gt;
 Adding 3025/3024 extends to the 11-limit and gives &amp;lt;&amp;lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>