Tablet: Difference between revisions
Wikispaces>genewardsmith **Imported revision 256944158 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 257695286 - 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>2011-09-23 23:19:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>257695286</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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This tablet is based on the [[tutonic sextad]], 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 9/9-9/8-9/8-9/8-9/8-8/7, 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 [[Chromatic pairs#Tutone|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale. | This tablet is based on the [[tutonic sextad]], 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 9/9-9/8-9/8-9/8-9/8-8/7, 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 [[Chromatic pairs#Tutone|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale. | ||
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>.</pre></div> | 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 keenanismic tablet= | |||
This is based on the five [[keenanismic tetrads]], 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 [[Dyadic chord|dyadic]]. 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. | |||
Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions: | |||
1-5/4-3/2-7/4 => [1, 1, 1] | |||
1-8/7-4/3-8/5 => [3, 3, 3]) | |||
1-5/4-3/2-12/7 => [0, 1, 3] | |||
1-6/5-3/2-7/4] => [0, 3, 1] | |||
35/32-5/4-3/2-7/4 => [1, 0, 0] | |||
1-5/4-35/24-7/4 => [3, 0, 0] | |||
35/32-5/4-3/2-12/7 => [0, 0, 2] | |||
Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 => [-1, 3, 3]. | |||
</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:16:&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:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#What is a tablet?">What is a tablet?</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#The 5-limit 3et tablet">The 5-limit 3et tablet</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#The 7-limit 4et tablet">The 7-limit 4et tablet</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#The 7-limit 5et tablet">The 7-limit 5et tablet</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#The 13-limit 7et tablet">The 13-limit 7et tablet</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#The meantone add6/9 tablet">The meantone add6/9 tablet</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#The tutone tutonic tablet">The tutone tutonic tablet</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#The keenanismic tablet">The keenanismic tablet</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:25 --><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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This tablet is based on the <a class="wiki_link" href="/tutonic%20sextad">tutonic sextad</a>, 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 9/9-9/8-9/8-9/8-9/8-8/7, 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 <a class="wiki_link" href="/Chromatic%20pairs#Tutone">tutone temperament</a>, a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.<br /> | This tablet is based on the <a class="wiki_link" href="/tutonic%20sextad">tutonic sextad</a>, 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 9/9-9/8-9/8-9/8-9/8-8/7, 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 <a class="wiki_link" href="/Chromatic%20pairs#Tutone">tutone temperament</a>, a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.<br /> | ||
<br /> | <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;.</body></html></pre></div> | 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 /> | |||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="The keenanismic tablet"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 /> | |||
<br /> | |||
Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y&gt;. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z&gt; + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:<br /> | |||
<br /> | |||
1-5/4-3/2-7/4 =&gt; [1, 1, 1]<br /> | |||
1-8/7-4/3-8/5 =&gt; [3, 3, 3])<br /> | |||
1-5/4-3/2-12/7 =&gt; [0, 1, 3]<br /> | |||
1-6/5-3/2-7/4] =&gt; [0, 3, 1]<br /> | |||
35/32-5/4-3/2-7/4 =&gt; [1, 0, 0]<br /> | |||
1-5/4-35/24-7/4 =&gt; [3, 0, 0]<br /> | |||
35/32-5/4-3/2-12/7 =&gt; [0, 0, 2]<br /> | |||
<br /> | |||
Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 =&gt; [-1, 3, 3].</body></html></pre></div> | |||