Tablet: 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>2011-09-25 09:43:24 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-25 20:34:12 UTC</tt>.<br>

: The original revision id was <tt>~~257936816~~</tt>.<br>

: The original revision id was <tt>258067172</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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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 and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c).

If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c). Now we may define note(n, c) by setting u = kord(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.

Now we may define note(n, c) by setting u = kord(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.

Given any keenanismic or otonal/utonal tetrad, and counting octave equivalent tetrads as the same, any tetrad will share a common triad with either four or five other tetrads. If we make tetrads the verticies of a graph where the edges join two tetrads if and only if they share a common triad, we obtain an infinite, but locally finite, connected graph. It is possible to get from one tetrad to any other solely by means of chord relationships where there is a common triad.

If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a [[http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html|positive definite quaratic form]] on 3-tuples Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root sqrt(Q(a-b)) is a Eulidean measure of distance, and sqrt(8) is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.

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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 />

If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c).~~<br />~~

If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c). Now we may define note(n, c) by setting u = kord(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.<br />

Now we may define note(n, c) by setting u = kord(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>

<br />

Given any keenanismic or otonal/utonal tetrad, and counting octave equivalent tetrads as the same, any tetrad will share a common triad with either four or five other tetrads. If we make tetrads the verticies of a graph where the edges join two tetrads if and only if they share a common triad, we obtain an infinite, but locally finite, connected graph. It is possible to get from one tetrad to any other solely by means of chord relationships where there is a common triad.<br />

<br />

If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a <a class="wiki_link_ext" href="http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html" rel="nofollow">positive definite quaratic form</a> on 3-tuples Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root sqrt(Q(a-b)) is a Eulidean measure of distance, and sqrt(8) is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.</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>2011-09-25 09:43:24 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-25 20:34:12 UTC</tt>.<br>
-: The original revision id was <tt>257936816</tt>.<br>
+: The original revision id was <tt>258067172</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 165: / Line 165: @@
 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].
-If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c).
+If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c). Now we may define note(n, c) by setting u = kord(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.
-Now we may define note(n, c) by setting u = kord(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.
+Given any keenanismic or otonal/utonal tetrad, and counting octave equivalent tetrads as the same, any tetrad will share a common triad with either four or five other tetrads. If we make tetrads the verticies of a graph where the edges join two tetrads if and only if they share a common triad, we obtain an infinite, but locally finite, connected graph. It is possible to get from one tetrad to any other solely by means of chord relationships where there is a common triad.
+If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a [[http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html|positive definite quaratic form]] on 3-tuples Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root sqrt(Q(a-b)) is a Eulidean measure of distance, and sqrt(8) is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.
@@ Line 302: / Line 305: @@
 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 =&amp;gt; [-1 3 3].&lt;br /&gt;
 &lt;br /&gt;
-If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c).&lt;br /&gt;
+If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, 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 q(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]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c). Now we may define note(n, c) by setting u = kord(c), v = &amp;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 &amp;lt;4 6 9 11|note(n, c) = n.&lt;br /&gt;
-Now we may define note(n, c) by setting u = kord(c), v = &amp;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 &amp;lt;4 6 9 11|note(n, c) = n.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;br /&gt;
+Given any keenanismic or otonal/utonal tetrad, and counting octave equivalent tetrads as the same, any tetrad will share a common triad with either four or five other tetrads. If we make tetrads the verticies of a graph where the edges join two tetrads if and only if they share a common triad, we obtain an infinite, but locally finite, connected graph. It is possible to get from one tetrad to any other solely by means of chord relationships where there is a common triad.&lt;br /&gt;
+&lt;br /&gt;
+If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html" rel="nofollow"&gt;positive definite quaratic form&lt;/a&gt; on 3-tuples Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root sqrt(Q(a-b)) is a Eulidean measure of distance, and sqrt(8) is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.&lt;/body&gt;&lt;/html&gt;</pre></div>