Miracle/Chords: Difference between revisions

<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 [[Gamelismic clan#Miracle|miracle temperament]]. They are listed in order of increasing [[Graham complexity]], with hash complexity used to break ties. For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2^s for all s∊S provides a unique odd number encoding chord, and the logarithm base two of this number is the hash complexity, listed under the heading "Hash" below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity. The heading "transversal" shows a JI chord whose intervals temper to the marvel chord. Under "type" it says otonal if it is an essentially just chord best understood in terms of the overtone series, utonal if it is the inversion of an otonal chord, and ambitonal if it can equally well be seen as otonal or utonal. The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord, so that if for example it says "werckismic" only 441/440 needs to be tempered out. Also, "marvel" denotes 9 odd limit (7-limit) marvel, whereas "unimarvel" means 11-limit marvel. The generator used below is the standard choice for miracle temperament, the secor. 16/15~15/14. However, it should be noted that the choice of generator does not effect the list of chords, only how those chords are interpreted. Does 0-7-13 denote the major triad or the minor triad? It's on the list of chords either way, but with a secor generator it's the major triad.

|| 1 || 0-2-5 || 1-8/7-7/5 || werckismic || 5.209453 ||

|| 2 || 0-3-5 || 1-11/9-7/5 || werckismic || 5.357552 ||

|| 4 || 0-2-7 || 1-8/7-8/5 || utonal || 7.055282 ||

|| 63 || 0-19-22 || 1-9/5-11/10 || otonal || 22.169925 ||

|| 9 || 0-5-7-12 || 1-7/5-8/5-9/8 || marvel || 12.055621 ||

|| 11 || 0-6-9-12 || 1-3/2-11/6-9/8 || rastmic || 12.190133 ||

|| 13 || 0-5-7-13 || 1-7/5-8/5-6/5 || otonal || 13.028079 ||

|| 15 || 0-6-8-13 || 1-3/2-12/7-6/5 || utonal || 13.055452 ||

|| 19 || 0-2-8-14 || 1-8/7-12/7-9/7 || otonal || 14.022801 ||

|| 21 || 0-6-8-14 || 1-3/2-12/7-9/7 || ambitonal || 14.027992 ||

|| 47 || 0-2-8-17 || 1-8/7-12/7-11/7 || otonal || 17.002870 ||

|| 51 || 0-3-9-17 || 1-11/9-11/6-11/7 || utonal || 17.005723 ||

|| 111 || 0-17-19-22 || 1-11/7-9/5-11/10 || werckismic || 22.209454 ||

|| 31 || 0-9-12-15-17 || 1-11/6-9/8-11/8-11/7 || jove || 17.361952 ||

|| 32 || 0-2-5-7-19 || 1-8/7-7/5-8/5-9/5 || werckismic || 19.000454 ||

|| 33 || 0-5-7-12-19 || 1-7/5-8/5-9/8-9/5 || marvel || 19.011667 ||

|| 36 || 0-2-7-14-19 || 1-8/7-8/5-9/7-9/5 || prodigy || 19.044749 ||

|| 37 || 0-5-7-14-19 || 1-7/5-8/5-9/7-9/5 || unimarvel || 19.044824 ||

|| 38 || 0-5-12-14-19 || 1-7/5-9/8-9/7-9/5 || unimarvel || 19.055370 ||

|| 39 || 0-6-12-14-19 || 1-3/2-9/8-9/7-9/5 || utonal || 19.055455 ||

|| 40 || 0-7-12-14-19 || 1-8/5-9/8-9/7-9/5 || marvel || 19.055624 ||

|| 69 || 0-7-13-19-22 || 1-8/5-6/5-9/5-11/10 || otonal || 22.172467 ||

|| 70 || 0-5-14-19-22 || 1-7/5-9/7-9/5-11/10 || swetismic || 22.174936 ||

|| 71 || 0-7-14-19-22 || 1-8/5-9/7-9/5-11/10 || unimarvel || 22.174965 ||

|| 73 || 0-14-17-19-22 || 1-9/7-11/7-9/5-11/10 || jove || 22.214319 ||

|| 1 || 0-3-6-9-12-15 || 1-11/9-3/2-11/6-9/8-11/8 || rastmic || 15.192640 ||

|| 3 || 0-3-9-12-15-17 || 1-11/9-11/6-9/8-11/8-11/7 || jove || 17.362021 ||

|| 4 || 0-2-5-7-14-19 || 1-8/7-7/5-8/5-9/7-9/5 || miracle || 19.044834 ||

|| 5 || 0-5-7-12-14-19 || 1-7/5-8/5-9/8-9/7-9/5 || unimarvel || 19.055709 ||

|| 14 || 0-5-14-17-19-22 || 1-7/5-9/7-11/7-9/5-11/10 || jove || 22.214329 ||

<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 miracle&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="/Gamelismic%20clan#Miracle"&gt;miracle temperament&lt;/a&gt;.  They are listed in order of increasing &lt;a class="wiki_link" href="/Graham%20complexity"&gt;Graham complexity&lt;/a&gt;, with hash complexity used to break ties. For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2^s for all s∊S provides a unique odd number encoding chord, and the logarithm base two of this number is the hash complexity, listed under the heading &amp;quot;Hash&amp;quot; below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity. The heading &amp;quot;transversal&amp;quot; shows a JI chord whose intervals temper to the marvel chord. Under &amp;quot;type&amp;quot; it says otonal if it is an essentially just chord best understood in terms of the overtone series, utonal if it is the inversion of an otonal chord, and ambitonal if it can equally well be seen as otonal or utonal. The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord, so that if for example it says &amp;quot;werckismic&amp;quot; only 441/440 needs to be tempered out. Also, &amp;quot;marvel&amp;quot; denotes 9 odd limit (7-limit) marvel, whereas &amp;quot;unimarvel&amp;quot; means 11-limit marvel. The generator used below is the standard choice for miracle temperament, the secor. 16/15~15/14. However, it should be noted that the choice of generator does not effect the list of chords, only how those chords are interpreted. Does 0-7-13 denote the major triad or the minor triad? It's on the list of chords either way, but with a secor generator it's the major triad.&lt;br /&gt;

        &lt;td&gt;8.049849&lt;br /&gt;

&lt;/td&gt;

        &lt;td&gt;0-5-8&lt;br /&gt;

&lt;/td&gt;

        &lt;td&gt;1-3/2-9/8&lt;br /&gt;

    &lt;/tr&gt;

        &lt;td&gt;1-3/2-6/5&lt;br /&gt;

        &lt;td&gt;21&lt;br /&gt;

        &lt;td&gt;22.000176&lt;br /&gt;

        &lt;td&gt;22.002815&lt;br /&gt;

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