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.

=5et tablets=

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&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="What is a tablet?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;What is a tablet?&lt;/h1&gt;

By a &lt;em&gt;tablet&lt;/em&gt; (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tuple" rel="nofollow"&gt;tuple&lt;/a&gt; of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see &lt;a class="wiki_link" href="/Composing%20with%20tablets"&gt;Composing with tablets&lt;/a&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 [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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If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,&lt;br /&gt;

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad, where &amp;lt;5 8 12 14|note(n, t) = n.&lt;br /&gt;

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The meantone add6/9 tablet is based on the &lt;a class="wiki_link" href="/meantone%20add6-9%20quintad"&gt;meantone add6/9 quintad&lt;/a&gt;, which can also be called the add2/9 quintad, the meantone pentatonic scale or Meantone[5]. The tablet is extremly simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where &lt;br /&gt;

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In all cases &amp;lt;5 8|note(n, c) = n. Tempering the the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.&lt;br /&gt;

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This is based on the following twelve chords, which are expressed in terms of the 2.5.7 transversal of the 11-limit rank three temperament portent, which tempers out 385/384, 441/440 and hence also 1029/1024 and 3025/3024. &lt;br /&gt;

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Each of the other eleven chords shares a triad with the first, otonal, pentad, and all of the chords can be related by an infinite but locally finite graph by drawing an edge between chords with a common triad.&lt;br /&gt;

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This tablet is based on the &lt;a class="wiki_link" href="/tutonic%20sextad"&gt;tutonic sextad&lt;/a&gt;, which in terms of the  99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in &lt;a class="wiki_link" href="/Chromatic%20pairs#Tutone"&gt;tutone temperament&lt;/a&gt;, a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.&lt;br /&gt;

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If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = |(u-i)/6-3i 2c+2i&amp;gt;. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity &amp;lt;12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as &amp;lt;6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i&amp;gt;.&lt;br /&gt;

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Let &amp;lt;r e3 e5 e7 e11 e13| denote an otonal 13-limit septad with root given by |* e3 e5 e7 e11 e13&amp;gt; when r is even, which in close position is 9/8-5/4-11/8-3/2-13/8-7/4-2. If r is odd, let it denote a utonal pentad which in close position is 12/11-6/5-4/3-3/2-12/7-24/13-2&lt;br /&gt;

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&amp;lt;7 11 16 20 24 26|note(n, [r e3 e5 e7 e11 e13]) = n.&lt;br /&gt;

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The &lt;a class="wiki_link" href="/orwell%20tetrad"&gt;orwell nonad&lt;/a&gt; is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used. &lt;br /&gt;

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