Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 497169860 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 497220454 - 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>2014-03-20 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-20 17:51:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>497220454</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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By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1). | By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1). | ||
The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1))º/i^(r-1).</pre></div> | The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1))º/i^(r-1). | ||
=The Fokblock function and modal UDP notation= | |||
Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of [[modal UDP notation]], the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. | |||
Returning to our pajmagorpor22 example, we have that pajmagorpor22 = Fokblock([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13]). It is also equal to Fokblock([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13]), reflecting the fact that pajara has a period of half on octave, ie that P1 = 2. Hence the pajara MOS mode is 7|3(2) in UDP notation. Finding the others by the fact thatfor them Pk=1 and ak=U, we have that the mode, in product word form, is (pajara 7|3(2))*(magic 9|12)*(orwell 4|17)*(porcupine 13|8). We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ai from the corresponding U and Pi as Pi*U, and so display S in terms of Fokblock. | |||
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: | <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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The <strong>Fokker block</strong> is one of the most notable inventions of the physicist and music theorist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a>. While the idea generalizes easily to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a> and tempered groups, for ease of exposition we will suppose that we are in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> situation with n=pi(p) primes up to an including p.<br /> | The <strong>Fokker block</strong> is one of the most notable inventions of the physicist and music theorist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a>. While the idea generalizes easily to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a> and tempered groups, for ease of exposition we will suppose that we are in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> situation with n=pi(p) primes up to an including p.<br /> | ||
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A Fokker block is not just a scale, but a little scale universe of tempered versions of that scale which identify various steps of the scale, as depicted below.<br /> | A Fokker block is not just a scale, but a little scale universe of tempered versions of that scale which identify various steps of the scale, as depicted below.<br /> | ||
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<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:48:&lt;a href=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 170px; width: 714px;&quot; /&gt;&lt;/a&gt; --><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" rel="nofollow"><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="external image 2000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="external image 2000px-Pajmagorpor22_temperament_support_lattice.svg.png" style="height: 170px; width: 714px;" /></a><!-- ws:end:WikiTextRemoteImageRule:48 --><br /> | ||
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One has first the <a class="wiki_link" href="/pajmagorpor22">original JI scale</a>. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are <a class="wiki_link" href="/pajmagorpor22_225">225/224</a>, <a class="wiki_link" href="/pajmagorpor22_100">100/99</a>, <a class="wiki_link" href="/pajmagorpor22_176">176/175</a>, and <a class="wiki_link" href="/pajmagorpor22_385">385/384</a>. The next level gives <a class="wiki_link" href="/pajmagorpor22apollo">apollo</a>, <a class="wiki_link" href="/pajmagorpor22minerva">minerva</a>, <a class="wiki_link" href="/pajmagorpor22marvel">marvel</a>, <a class="wiki_link" href="/pajmagorpor22ares">ares</a>, <a class="wiki_link" href="/pajmagorpor22supermagic">supermagic</a>, and <a class="wiki_link" href="/pajmagorpor22zeus">zeus</a>. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with.<br /> | One has first the <a class="wiki_link" href="/pajmagorpor22">original JI scale</a>. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are <a class="wiki_link" href="/pajmagorpor22_225">225/224</a>, <a class="wiki_link" href="/pajmagorpor22_100">100/99</a>, <a class="wiki_link" href="/pajmagorpor22_176">176/175</a>, and <a class="wiki_link" href="/pajmagorpor22_385">385/384</a>. The next level gives <a class="wiki_link" href="/pajmagorpor22apollo">apollo</a>, <a class="wiki_link" href="/pajmagorpor22minerva">minerva</a>, <a class="wiki_link" href="/pajmagorpor22marvel">marvel</a>, <a class="wiki_link" href="/pajmagorpor22ares">ares</a>, <a class="wiki_link" href="/pajmagorpor22supermagic">supermagic</a>, and <a class="wiki_link" href="/pajmagorpor22zeus">zeus</a>. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with.<br /> | ||
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By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1).<br /> | By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1).<br /> | ||
<br /> | <br /> | ||
The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1))º/i^(r-1).</body></html></pre></div> | The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1))º/i^(r-1).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="The Fokblock function and modal UDP notation"></a><!-- ws:end:WikiTextHeadingRule:28 -->The Fokblock function and modal UDP notation</h1> | |||
Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of <a class="wiki_link" href="/modal%20UDP%20notation">modal UDP notation</a>, the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. <br /> | |||
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Returning to our pajmagorpor22 example, we have that pajmagorpor22 = Fokblock([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13]). It is also equal to Fokblock([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13]), reflecting the fact that pajara has a period of half on octave, ie that P1 = 2. Hence the pajara MOS mode is 7|3(2) in UDP notation. Finding the others by the fact thatfor them Pk=1 and ak=U, we have that the mode, in product word form, is (pajara 7|3(2))*(magic 9|12)*(orwell 4|17)*(porcupine 13|8). We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ai from the corresponding U and Pi as Pi*U, and so display S in terms of Fokblock.<br /> | |||
<br /> | |||
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> | |||