Tenney–Euclidean tuning: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-28 16:03:50 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-28 16:07:02 UTC</tt>.

: The original revision id was <tt>~~196931134~~</tt>.

: The original revision id was <tt>196932164</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 72:

<144 228 336|, <228 361 532|, <336 532 784|]/1289

Multiplying <12 19 28| by F gives <12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, <19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0> times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 ~~cennts~~, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0>F, of 1196.778 cents.

Multiplying <12 19 28| by F gives <12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, <19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0> times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0>F, of 1196.778 cents.

We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 19 equal; then U`U, the Frobenius projection map for pajara, is

We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 22 equal; then U`U, the Frobenius projection map for pajara, is

[|36 92 14 32>, |92 269 -32 14>, |14 -32 141 148>, |32 14 148 64>]/305

Line 137:

&lt;144 228 336|, &lt;228 361 532|, &lt;336 532 784|]/1289

Multiplying &lt;12 19 28| by F gives &lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 ~~cennts~~, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&gt;F, of 1196.778 cents.

Multiplying &lt;12 19 28| by F gives &lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&gt;F, of 1196.778 cents.

We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 19 equal; then U`U, the Frobenius projection map for pajara, is

We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 22 equal; then U`U, the Frobenius projection map for pajara, is

[|36 92 14 32&gt;, |92 269 -32 14&gt;, |14 -32 141 148&gt;, |32 14 148 64&gt;]/305

This sends monzos for 50/49, 64/63 etc to the unison monzo, and vals for 10et, 12et and 22et to themselves.</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>2011-01-28 16:03:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-28 16:07:02 UTC</tt>.<br>
-: The original revision id was <tt>196931134</tt>.<br>
+: The original revision id was <tt>196932164</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 72: / Line 72: @@
 &lt;144 228 336|, &lt;228 361 532|, &lt;336 532 784|]/1289
-Multiplying &lt;12 19 28| by F gives &lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cennts, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&gt;F, of 1196.778 cents.
+Multiplying &lt;12 19 28| by F gives &lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&gt;F, of 1196.778 cents.
-We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 19 equal; then U`U, the Frobenius projection map for pajara, is
+We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 22 equal; then U`U, the Frobenius projection map for pajara, is
 [|36 92 14 32&gt;, |92 269 -32 14&gt;, |14 -32 141 148&gt;, |32 14 148 64&gt;]/305
@@ Line 137: / Line 137: @@
 &amp;lt;144 228 336|, &amp;lt;228 361 532|, &amp;lt;336 532 784|]/1289&lt;br /&gt;
 &lt;br /&gt;
-Multiplying &amp;lt;12 19 28| by F gives &amp;lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &amp;lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&amp;gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cennts, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&amp;gt;F, of 1196.778 cents.&lt;br /&gt;
+Multiplying &amp;lt;12 19 28| by F gives &amp;lt;12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, &amp;lt;19 30 44|, by F gives [24360 38570 56840]/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0&amp;gt; times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0&amp;gt;F, of 1196.778 cents.&lt;br /&gt;
 &lt;br /&gt;
-We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 19 equal; then U`U, the Frobenius projection map for pajara, is &lt;br /&gt;
+We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 22 equal; then U`U, the Frobenius projection map for pajara, is &lt;br /&gt;
 &lt;br /&gt;
 [|36 92 14 32&amp;gt;, |92 269 -32 14&amp;gt;, |14 -32 141 148&amp;gt;, |32 14 148 64&amp;gt;]/305&lt;br /&gt;
 &lt;br /&gt;
 This sends monzos for 50/49, 64/63 etc to the unison monzo, and vals for 10et, 12et and 22et to themselves.&lt;/body&gt;&lt;/html&gt;</pre></div>