Wizard/Chords: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-24 21:25:08 UTC</tt>.<br>

<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;white-space: pre-wrap ! important" class="old-revision-html">Below are listed the [[Dyadic chord|dyadic chords]] of 11-limit [[Marvel temperaments#Wizard|wizard temperament]]. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 385/384 keenanismic, and by 225/224 marvel. Those requiring tempering by any two of 540/539, 385/384 or 225/224 are labeled unimarvel.

The normal mapping for wizard is wiz = [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]. From this we may derive a val v = wiz[1] + 100 wiz[2] = &lt;2 601 -95 1002 -292|. Since v is approximately 100 times the generator chain val, &lt;0 6 -1 10 -3|, it will sort elements of the 11-limit tonality diamond in the same order, but it uniquely encodes each interval. Since it is a val of wizard, we may take a chord, find the smallest element under the mapping, and use it to translate the whole chord starting at zero. The val v is what might be called a quasi higher rank val; it has a comma comma basis 225/224, 385/384, 4000/3993, 1001255211106304, which are the commas of wizard plus a complex interval which can play no role in ordinary circumstances, so that v can be used in place of the mapping for wizard.

Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v. If we look at the highest, rightmost, element of the chord, divide that by 100, round, and multiply by 2, we get the Graham complexity of the chord. Redundantly for the sake of convenience, the Graham complexity is listed in the last column.

Wizard has MOS of size 6, 10, 16, 22, 28, 50 and 72. It may be seen that even ten notes supplies enough chords to be interesting, and that by the time we reach 22 we might feel we are pretty well supplied.

=Triads=

|| Number || Chord || Transversal || Type || Complexity ||

|| 1 || 0-101-200 || 1-8/5-9/7 || marvel || 4 ||

|| 2 || 0-101-300 || 1-8/5-16/11 || utonal || 6 ||

|| 3 || 0-201-300 || 1-20/11-16/11 || otonal || 6 ||

|| 4 || 0-101-399 || 1-8/5-7/6 || keenanismic || 8 ||

|| 5 || 0-200-399 || 1-9/7-7/6 || swetismic || 8 ||

|| 6 || 0-201-399 || 1-20/11-7/6 || swetismic || 8 ||

|| 7 || 0-300-399 || 1-16/11-7/6 || keenanismic || 8 ||

|| 8 || 0-200-599 || 1-9/7-3/2 || utonal || 12 ||

|| 9 || 0-399-599 || 1-7/6-3/2 || otonal || 12 ||

|| 10 || 0-101-698 || 1-8/5-6/5 || otonal || 14 ||

|| 11 || 0-599-698 || 1-3/2-6/5 || utonal || 14 ||

|| 12 || 0-201-897 || 1-20/11-12/11 || otonal || 18 ||

|| 13 || 0-300-897 || 1-16/11-12/11 || otonal || 18 ||

|| 14 || 0-599-897 || 1-3/2-12/11 || utonal || 18 ||

|| 15 || 0-698-897 || 1-6/5-12/11 || utonal || 18 ||

|| 16 || 0-101-998 || 1-8/5-7/4 || keenanismic || 20 ||

|| 17 || 0-300-998 || 1-16/11-7/4 || keenanismic || 20 ||

|| 18 || 0-399-998 || 1-7/6-7/4 || utonal || 20 ||

|| 19 || 0-599-998 || 1-3/2-7/4 || otonal || 20 ||

|| 20 || 0-698-998 || 1-6/5-7/4 || keenanismic || 20 ||

|| 21 || 0-897-998 || 1-12/11-7/4 || keenanismic || 20 ||

|| 22 || 0-101-1097 || 1-8/5-7/5 || otonal || 22 ||

|| 23 || 0-200-1097 || 1-9/7-7/5 || swetismic || 22 ||

|| 24 || 0-399-1097 || 1-7/6-7/5 || utonal || 22 ||

|| 25 || 0-698-1097 || 1-6/5-7/5 || otonal || 22 ||

|| 26 || 0-897-1097 || 1-12/11-7/5 || swetismic || 22 ||

|| 27 || 0-998-1097 || 1-7/4-7/5 || utonal || 22 ||

|| 28 || 0-101-1196 || 1-8/5-9/8 || marvel || 24 ||

|| 29 || 0-200-1196 || 1-9/7-9/8 || utonal || 24 ||

|| 30 || 0-599-1196 || 1-3/2-9/8 || ambitonal || 24 ||

|| 31 || 0-998-1196 || 1-7/4-9/8 || otonal || 24 ||

|| 32 || 0-1097-1196 || 1-7/5-9/8 || marvel || 24 ||

|| 33 || 0-201-1296 || 1-20/11-14/11 || otonal || 26 ||

|| 34 || 0-300-1296 || 1-16/11-14/11 || otonal || 26 ||

|| 35 || 0-399-1296 || 1-7/6-14/11 || utonal || 26 ||

|| 36 || 0-897-1296 || 1-12/11-14/11 || otonal || 26 ||

|| 37 || 0-998-1296 || 1-7/4-14/11 || utonal || 26 ||

|| 38 || 0-1097-1296 || 1-7/5-14/11 || utonal || 26 ||

|| 39 || 0-101-1297 || 1-8/5-9/5 || otonal || 26 ||

|| 40 || 0-200-1297 || 1-9/7-9/5 || utonal || 26 ||

|| 41 || 0-599-1297 || 1-3/2-9/5 || utonal || 26 ||

|| 42 || 0-698-1297 || 1-6/5-9/5 || otonal || 26 ||

|| 43 || 0-1097-1297 || 1-7/5-9/5 || otonal || 26 ||

|| 44 || 0-1196-1297 || 1-9/8-9/5 || utonal || 26 ||

|| 45 || 0-200-1496 || 1-9/7-18/11 || utonal || 30 ||

|| 46 || 0-201-1496 || 1-20/11-18/11 || otonal || 30 ||

|| 47 || 0-300-1496 || 1-16/11-18/11 || otonal || 30 ||

|| 48 || 0-399-1496 || 1-7/6-18/11 || swetismic || 30 ||

|| 49 || 0-599-1496 || 1-3/2-18/11 || utonal || 30 ||

|| 50 || 0-897-1496 || 1-12/11-18/11 || otonal || 30 ||

|| 51 || 0-1097-1496 || 1-7/5-18/11 || swetismic || 30 ||

|| 52 || 0-1196-1496 || 1-9/8-18/11 || utonal || 30 ||

|| 53 || 0-1296-1496 || 1-14/11-18/11 || otonal || 30 ||

|| 54 || 0-1297-1496 || 1-9/5-18/11 || utonal || 30 ||

=Pentads=

|| Number || Chord || Transversal || Type || Complexity ||

|| 1 || 0-101-300-399-998 || 1-8/5-16/11-7/6-7/4 || keenanismic || 20 ||

|| 2 || 0-599-698-897-998 || 1-3/2-6/5-12/11-7/4 || keenanismic || 20 ||

|| 3 || 0-101-200-399-1097 || 1-8/5-9/7-7/6-7/5 || unimarvel || 22 ||

|| 4 || 0-101-399-998-1097 || 1-8/5-7/6-7/4-7/5 || keenanismic || 22 ||

|| 5 || 0-101-698-998-1097 || 1-8/5-6/5-7/4-7/5 || keenanismic || 22 ||

|| 6 || 0-698-897-998-1097 || 1-6/5-12/11-7/4-7/5 || unimarvel || 22 ||

|| 7 || 0-101-200-1097-1196 || 1-8/5-9/7-7/5-9/8 || unimarvel || 24 ||

|| 8 || 0-101-998-1097-1196 || 1-8/5-7/4-7/5-9/8 || unimarvel || 24 ||

|| 9 || 0-201-300-399-1296 || 1-20/11-16/11-7/6-14/11 || unimarvel || 26 ||

|| 10 || 0-201-300-897-1296 || 1-20/11-16/11-12/11-14/11 || otonal || 26 ||

|| 11 || 0-300-399-998-1296 || 1-16/11-7/6-7/4-14/11 || keenanismic || 26 ||

|| 12 || 0-300-897-998-1296 || 1-16/11-12/11-7/4-14/11 || keenanismic || 26 ||

|| 13 || 0-399-998-1097-1296 || 1-7/6-7/4-7/5-14/11 || utonal || 26 ||

|| 14 || 0-897-998-1097-1296 || 1-12/11-7/4-7/5-14/11 || unimarvel || 26 ||

|| 15 || 0-101-200-1097-1297 || 1-8/5-9/7-7/5-9/5 || unimarvel || 26 ||

|| 16 || 0-101-698-1097-1297 || 1-8/5-6/5-7/5-9/5 || otonal || 26 ||

|| 17 || 0-101-200-1196-1297 || 1-8/5-9/7-9/8-9/5 || marvel || 26 ||

|| 18 || 0-200-599-1196-1297 || 1-9/7-3/2-9/8-9/5 || utonal || 26 ||

|| 19 || 0-101-1097-1196-1297 || 1-8/5-7/5-9/8-9/5 || marvel || 26 ||

|| 20 || 0-200-1097-1196-1297 || 1-9/7-7/5-9/8-9/5 || unimarvel || 26 ||

|| 21 || 0-201-300-399-1496 || 1-20/11-16/11-7/6-18/11 || unimarvel || 30 ||

|| 22 || 0-200-399-599-1496 || 1-9/7-7/6-3/2-18/11 || swetismic || 30 ||

|| 23 || 0-201-300-897-1496 || 1-20/11-16/11-12/11-18/11 || otonal || 30 ||

|| 24 || 0-200-399-1097-1496 || 1-9/7-7/6-7/5-18/11 || swetismic || 30 ||

|| 25 || 0-200-599-1196-1496 || 1-9/7-3/2-9/8-18/11 || utonal || 30 ||

|| 26 || 0-200-1097-1196-1496 || 1-9/7-7/5-9/8-18/11 || unimarvel || 30 ||

|| 27 || 0-201-300-1296-1496 || 1-20/11-16/11-14/11-18/11 || otonal || 30 ||

|| 28 || 0-201-399-1296-1496 || 1-20/11-7/6-14/11-18/11 || swetismic || 30 ||

|| 29 || 0-300-399-1296-1496 || 1-16/11-7/6-14/11-18/11 || unimarvel || 30 ||

|| 30 || 0-201-897-1296-1496 || 1-20/11-12/11-14/11-18/11 || otonal || 30 ||

|| 31 || 0-300-897-1296-1496 || 1-16/11-12/11-14/11-18/11 || otonal || 30 ||

|| 32 || 0-399-1097-1296-1496 || 1-7/6-7/5-14/11-18/11 || swetismic || 30 ||

|| 33 || 0-897-1097-1296-1496 || 1-12/11-7/5-14/11-18/11 || swetismic || 30 ||

|| 34 || 0-200-599-1297-1496 || 1-9/7-3/2-9/5-18/11 || utonal || 30 ||

|| 35 || 0-200-1097-1297-1496 || 1-9/7-7/5-9/5-18/11 || swetismic || 30 ||

|| 36 || 0-200-1196-1297-1496 || 1-9/7-9/8-9/5-18/11 || utonal || 30 ||

|| 37 || 0-599-1196-1297-1496 || 1-3/2-9/8-9/5-18/11 || utonal || 30 ||

|| 38 || 0-1097-1196-1297-1496 || 1-7/5-9/8-9/5-18/11 || unimarvel || 30 ||

=Hexads=

|| Number || Chord || Transversal || Type || Complexity ||

|| 1 || 0-101-200-1097-1196-1297 || 1-8/5-9/7-7/5-9/8-9/5 || unimarvel || 26 ||

|| 2 || 0-201-300-399-1296-1496 || 1-20/11-16/11-7/6-14/11 18/11 || unimarvel || 30 ||

|| 3 || 0-201-300-897-1296-1496 || 1-20/11-16/11-12/11-14/11-18/11 || otonal || 30 ||

|| 4 || 0-200-599-1196-1297-1496 || 1-9/7-3/2-9/8-9/5-18/11 || utonal || 30 ||

|| 5 || 0-200-1097-1196-1297-1496 || 1-9/7-7/5-9/8-9/5-18/11 || unimarvel || 30 ||

</pre></div>

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Chords of wizard&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Below are listed the &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;dyadic chords&lt;/a&gt; of 11-limit &lt;a class="wiki_link" href="/Marvel%20temperaments#Wizard"&gt;wizard temperament&lt;/a&gt;. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 385/384 keenanismic, and by 225/224 marvel. Those requiring tempering by any two of 540/539, 385/384 or 225/224 are labeled unimarvel.&lt;br /&gt;

&lt;br /&gt;

The normal mapping for wizard is wiz = [&amp;lt;2 1 5 2 8|, &amp;lt;0 6 -1 10 -3|]. From this we may derive a val v = wiz[1] + 100 wiz[2] = &amp;lt;2 601 -95 1002 -292|. Since v is approximately 100 times the generator chain val, &amp;lt;0 6 -1 10 -3|, it will sort elements of the 11-limit tonality diamond in the same order, but it uniquely encodes each interval. Since it is a val of wizard, we may take a chord, find the smallest element under the mapping, and use it to translate the whole chord starting at zero. The val v is what might be called a quasi higher rank val; it has a comma comma basis 225/224, 385/384, 4000/3993, 1001255211106304, which are the commas of wizard plus a complex interval which can play no role in ordinary circumstances, so that v can be used in place of the mapping for wizard.&lt;br /&gt;

&lt;br /&gt;

Under &amp;quot;Chord&amp;quot; is listed the chord, normalized to start from zero, in the mapping by v. If we look at the highest, rightmost, element of the chord, divide that by 100, round, and multiply by 2, we get the Graham complexity of the chord. Redundantly for the sake of convenience, the Graham complexity is listed in the last column.&lt;br /&gt;

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