Tablet: Difference between revisions

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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 [[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.

This is a tablet for the rank 3 13-limit temperament [[Hemifamity family#Pele|pele]]. It is based on the following 71 chords, in the 5-limit transversal:  

&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x4et tablets-The 7-limit 4et tablet"&gt;The 7-limit 4et tablet&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x4et tablets-The keenanismic tablet"&gt;The keenanismic tablet&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x5et tablets"&gt;5et tablets&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x5et tablets-The 7-limit 5et tablet"&gt;The 7-limit 5et tablet&lt;/a&gt;&lt;/div&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 [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.&lt;br /&gt;

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This is a tablet for the rank 3 13-limit temperament &lt;a class="wiki_link" href="/Hemifamity%20family#Pele"&gt;pele&lt;/a&gt;. It is based on the following 71 chords, in the 5-limit transversal: &lt;br /&gt;

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