Tablet: Difference between revisions
Wikispaces>genewardsmith **Imported revision 258067534 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 258483110 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-26 21:15:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>258483110</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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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>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i>. | 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>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i>. | ||
=The orwell nonad tablet= | |||
The [[orwell tetrad|orwell nonad]] 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. | |||
If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define aa transversal for note(n, c) by determining if i is even or odd, and setting | |||
note(n, c) = |(u-i)/9-i/2 -i/2-c 0 i/2+c> | |||
if i is even, and | |||
note(n, c) = |(u-i)/9-(i+9)/2 (1-i)/2-c 1 (i+1)/2+c> | |||
if i is odd. We then have <9 14 21 25|note(n, c) = n. | |||
=The keenanismic tablet= | =The keenanismic tablet= | ||
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</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tablets</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tablets</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#What is a tablet?">What is a tablet?</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#The 5-limit 3et tablet">The 5-limit 3et tablet</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#The 7-limit 4et tablet">The 7-limit 4et tablet</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#The 7-limit 5et tablet">The 7-limit 5et tablet</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#The 13-limit 7et tablet">The 13-limit 7et tablet</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#The meantone add6/9 tablet">The meantone add6/9 tablet</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#The tutone tutonic tablet">The tutone tutonic tablet</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#The orwell nonad tablet">The orwell nonad tablet</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#The keenanismic tablet">The keenanismic tablet</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:28 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="What is a tablet?"></a><!-- ws:end:WikiTextHeadingRule:0 -->What is a tablet?</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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 /> | 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 /> | ||
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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&gt;. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity &lt;12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as &lt;6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i&gt;.<br /> | 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&gt;. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity &lt;12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as &lt;6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i&gt;.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="The keenanismic tablet"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="The orwell nonad tablet"></a><!-- ws:end:WikiTextHeadingRule:14 -->The orwell nonad tablet</h1> | ||
The <a class="wiki_link" href="/orwell%20tetrad">orwell nonad</a> 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. <br /> | |||
<br /> | |||
If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define aa transversal for note(n, c) by determining if i is even or odd, and setting <br /> | |||
note(n, c) = |(u-i)/9-i/2 -i/2-c 0 i/2+c&gt; <br /> | |||
if i is even, and <br /> | |||
note(n, c) = |(u-i)/9-(i+9)/2 (1-i)/2-c 1 (i+1)/2+c&gt;<br /> | |||
if i is odd. We then have &lt;9 14 21 25|note(n, c) = n.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="The keenanismic tablet"></a><!-- ws:end:WikiTextHeadingRule:16 -->The keenanismic tablet</h1> | |||
This is based on the five <a class="wiki_link" href="/keenanismic%20tetrads">keenanismic tetrads</a>, which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords <a class="wiki_link" href="/Dyadic%20chord">dyadic</a>. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.<br /> | This is based on the five <a class="wiki_link" href="/keenanismic%20tetrads">keenanismic tetrads</a>, which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords <a class="wiki_link" href="/Dyadic%20chord">dyadic</a>. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.<br /> | ||
<br /> | <br /> | ||