Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 497655448 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 497800246 - 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>2014-03-23 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-23 20:54:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>497800246</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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In terms of the rational intonation of the modes of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = <2 0 11 12 7|. Note that pajara∨245/242 = V, so it is up in pajara. | In terms of the rational intonation of the modes of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = <2 0 11 12 7|. Note that pajara∨245/242 = V, so it is up in pajara. | ||
If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]) | If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]).</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>Fokker blocks</title></head><body><!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Expanding the definition">Expanding the definition</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#Scale properties of Fokker blocks">Scale properties of Fokker blocks</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#The Fokblock function and modal UDP notation">The Fokblock function and modal UDP notation</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <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>Fokker blocks</title></head><body><!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Expanding the definition">Expanding the definition</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#Scale properties of Fokker blocks">Scale properties of Fokker blocks</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#The Fokblock function and modal UDP notation">The Fokblock function and modal UDP notation</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | ||
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In terms of the rational intonation of the modes of a Fokker arena, this definition of &quot;chroma positive&quot; is the correct one if we want increasing &quot;up&quot; values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix <a class="wiki_link" href="/collapar">collapar</a>, a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = &lt;2 0 11 12 7|. Note that pajara∨245/242 = V, so it is up in pajara.<br /> | In terms of the rational intonation of the modes of a Fokker arena, this definition of &quot;chroma positive&quot; is the correct one if we want increasing &quot;up&quot; values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix <a class="wiki_link" href="/collapar">collapar</a>, a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = &lt;2 0 11 12 7|. Note that pajara∨245/242 = V, so it is up in pajara.<br /> | ||
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If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance <a class="wiki_link" href="/nofives">nofives</a> is Fokblock([64/63, 729/686, 5], [3, 4, 0]) | If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance <a class="wiki_link" href="/nofives">nofives</a> is Fokblock([64/63, 729/686, 5], [3, 4, 0]).</body></html></pre></div> | ||