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= If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted <* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that <r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7> when r is even, and a minor triad when r is odd. If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have * If u mod 3 = 0, then note(n, [r e3 e5]) = |u/3 e3 e5> * If u mod 3 = 1, then note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1> * If u mod 3 = 2, then note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5> On the other hand, if r is odd then * If u mod 3 = 0, then note(n, [r e3 e5]) = |u/3 e3 e3-e5> * If u mod 3 = 1, then note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1> * If u mod 3 = 2, then note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5> Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7|note(n, c)> = n. =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 1+(a-b+c)/2 (a+b-c)/2> * If u mod 4 = 2, then note(t) = |(u-6)/4 1+(-a+b+c)/2 (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 1+(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, and <4 6 9 11|note(n, t)| = n. =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, where <5 8 12 14|note(n, t)> = n. =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> If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted <* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that <r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7> when r is even, and a minor triad when r is odd.<br /> <br /> If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have<br /> <br /> <ul><li>If u mod 3 = 0, then</li></ul>note(n, [r e3 e5]) = |u/3 e3 e5><br /> <br /> <ul><li>If u mod 3 = 1, then</li></ul>note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1><br /> <br /> <ul><li>If u mod 3 = 2, then</li></ul>note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5><br /> <br /> On the other hand, if r is odd then<br /> <br /> <ul><li>If u mod 3 = 0, then</li></ul>note(n, [r e3 e5]) = |u/3 e3 e3-e5><br /> <br /> <ul><li>If u mod 3 = 1, then</li></ul>note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1><br /> <br /> <ul><li>If u mod 3 = 2, then</li></ul>note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5><br /> <br /> Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7|note(n, c)> = n.<br /> <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 1+(a-b+c)/2 (a+b-c)/2><br /> <ul><li>If u mod 4 = 2, then</li></ul>note(t) = |(u-6)/4 1+(-a+b+c)/2 (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 1+(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, and <4 6 9 11|note(n, t)| = n.<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, where <5 8 12 14|note(n, t)> = n.<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>