Tablet
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[[toc|flat]] =What is a tablet?= By a //tablet// (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 [[http://en.wikipedia.org/wiki/Tuple|tuple]] 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. There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below. =The meantone add6/9 tablet= =The 5-limit 3et tablet= =The 7-limit 4et tablet= Suppose m0, m1, m2 and m3 are four [[monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[The Seven Limit Symmetrical Lattices|cubic lattice of 7-limit tetrads]]. If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then * If u mod 4 = 0, then note(t) = |u/4 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2> * If u mod 4 = 1, then note(t) = |(u-9)/4 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2> * If u mod 4 = 2, then note(t) = |(u-6)/4 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2> * If u mod 4 = 3, then note(t) = |(u-11)/4 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2> If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. 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)] is a 7-limit tetrad. =The 7-limit 5et tablet= If we define u = n - 9a - 5b - 3c then supposing a+b+c is even, * If u mod 5 = 0, then note(t) = |u/5 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2> * If u mod 5 = 1, then note(t) = |(u-16)/5 2+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2> * If u mod 5 = 2, then note(t) = |(u-12)/5 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2> * If u mod 5 = 3, then note(t) = |(u-8)/5 1+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2> * If u mod 5 = 4, then note(t) = |(u-14)/5 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2> 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. =The 13-limit 7et tablet=
Original HTML content:
<html><head><title>Tablets</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#What is a tablet?">What is a tablet?</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#The meantone add6/9 tablet">The meantone add6/9 tablet</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#The 5-limit 3et tablet">The 5-limit 3et tablet</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#The 7-limit 4et tablet">The 7-limit 4et tablet</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#The 7-limit 5et tablet">The 7-limit 5et tablet</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#The 13-limit 7et tablet">The 13-limit 7et tablet</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> <!-- ws:end:WikiTextTocRule:19 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="What is a tablet?"></a><!-- ws:end:WikiTextHeadingRule:0 -->What is a tablet?</h1> By a <em>tablet</em> (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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tuple" rel="nofollow">tuple</a> 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.<br /> <br /> There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="The meantone add6/9 tablet"></a><!-- ws:end:WikiTextHeadingRule:2 -->The meantone add6/9 tablet</h1> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="The 5-limit 3et tablet"></a><!-- ws:end:WikiTextHeadingRule:4 -->The 5-limit 3et tablet</h1> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="The 7-limit 4et tablet"></a><!-- ws:end:WikiTextHeadingRule:6 -->The 7-limit 4et tablet</h1> Suppose m0, m1, m2 and m3 are four <a class="wiki_link" href="/monzo">monzo</a>s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">cubic lattice of 7-limit tetrads</a>. <br /> <br /> If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then<br /> <br /> <ul><li>If u mod 4 = 0, then</li></ul>note(t) = |u/4 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 4 = 1, then</li></ul>note(t) = |(u-9)/4 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2><br /> <ul><li>If u mod 4 = 2, then</li></ul>note(t) = |(u-6)/4 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 4 = 3, then</li></ul>note(t) = |(u-11)/4 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2><br /> <br /> If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. 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)] is a 7-limit tetrad.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="The 7-limit 5et tablet"></a><!-- ws:end:WikiTextHeadingRule:8 -->The 7-limit 5et tablet</h1> <br /> If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,<br /> <br /> <ul><li>If u mod 5 = 0, then</li></ul>note(t) = |u/5 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 5 = 1, then</li></ul>note(t) = |(u-16)/5 2+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 5 = 2, then</li></ul>note(t) = |(u-12)/5 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 5 = 3, then</li></ul>note(t) = |(u-8)/5 1+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 5 = 4, then</li></ul>note(t) = |(u-14)/5 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2><br /> <br /> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="The 13-limit 7et tablet"></a><!-- ws:end:WikiTextHeadingRule:10 -->The 13-limit 7et tablet</h1> </body></html>