Tablet: Difference between revisions
Wikispaces>genewardsmith **Imported revision 258067172 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 258067534 - 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>2011-09-25 20:34 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-25 20:35:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>258067534</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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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. | 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. | 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 [x y z] by 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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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 /> | 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 /> | ||
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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> | 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 [x y z] by 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> | ||