Tablet: Difference between revisions
Wikispaces>genewardsmith **Imported revision 257695286 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 257934916 - 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>2011-09- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-25 09:24:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>257934916</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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Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions: | Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions: | ||
1-5/4-3/2-7/4 => [1 | 1-5/4-3/2-7/4 => [1 1 1] | ||
1-8/7-4/3-8/5 => [3 | 1-8/7-4/3-8/5 => [3 3 3]) | ||
1-5/4-3/2-12/7 => [0 | 1-5/4-3/2-12/7 => [0 1 3] | ||
1-6/5-3/2-7/4 | 1-6/5-3/2-7/4 => [0 3 1] | ||
35/32-5/4-3/2-7/4 => [1 | 35/32-5/4-3/2-7/4 => [1 0, 0] | ||
1-5/4-35/24-7/4 => [3, 0 | 1-5/4-35/24-7/4 => [3, 0 0] | ||
35/32-5/4-3/2-12/7 => [0 | 35/32-5/4-3/2-12/7 => [0 0 2] | ||
Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 => [-1, 3, 3]. | Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 => [-1 3 3]. | ||
If we have a 3-tuple c representing a keenanismic tetrad, define k(c) by reversing the above correspondence, except that we assign k([3 3 3]) = [1,6/5,3/2,12/7]. If w is a keenanismic tetrad, define c(w) by taking the product w[1]*w[2]*w[3]*w[4], finding the monzo m, and returning [2-m[2] 2-m[3] 2-m[4]]. Now we may define note(n, c) by setting u = k(c), v = <4 6 9 11|u[1], i = v mod 4, and then setting note(n, c) = 2^((n-v-i)/4) u[i+1]. We then have our basic identity <4 6 9 11|note(n, c) = n. | |||
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Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y&gt;. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z&gt; + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:<br /> | Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y&gt;. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z&gt; + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:<br /> | ||
<br /> | <br /> | ||
1-5/4-3/2-7/4 =&gt; [1 | 1-5/4-3/2-7/4 =&gt; [1 1 1]<br /> | ||
1-8/7-4/3-8/5 =&gt; [3 | 1-8/7-4/3-8/5 =&gt; [3 3 3])<br /> | ||
1-5/4-3/2-12/7 =&gt; [0 | 1-5/4-3/2-12/7 =&gt; [0 1 3]<br /> | ||
1-6/5-3/2-7/4 | 1-6/5-3/2-7/4 =&gt; [0 3 1]<br /> | ||
35/32-5/4-3/2-7/4 =&gt; [1 | 35/32-5/4-3/2-7/4 =&gt; [1 0, 0]<br /> | ||
1-5/4-35/24-7/4 =&gt; [3, 0 | 1-5/4-35/24-7/4 =&gt; [3, 0 0]<br /> | ||
35/32-5/4-3/2-12/7 =&gt; [0 | 35/32-5/4-3/2-12/7 =&gt; [0 0 2]<br /> | ||
<br /> | |||
Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 =&gt; [-1 3 3].<br /> | |||
<br /> | <br /> | ||
If we have a 3-tuple c representing a keenanismic tetrad, define k(c) by reversing the above correspondence, except that we assign k([3 3 3]) = [1,6/5,3/2,12/7]. If w is a keenanismic tetrad, define c(w) by taking the product w[1]*w[2]*w[3]*w[4], finding the monzo m, and returning [2-m[2] 2-m[3] 2-m[4]]. Now we may define note(n, c) by setting u = k(c), v = &lt;4 6 9 11|u[1], i = v mod 4, and then setting note(n, c) = 2^((n-v-i)/4) u[i+1]. We then have our basic identity &lt;4 6 9 11|note(n, c) = n.</body></html></pre></div> | |||