Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 314428418 - 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>2012-03-25 18: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-25 23:18:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>314501072</tt>.<br> | ||
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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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Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the [[Vals and Tuning Space|val]] v = <v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2<0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = <22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit. | Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the [[Vals and Tuning Space|val]] v = <v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2<0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = <22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit. | ||
Now choose an example step for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then <v|m>=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be ∓1. It is therefore a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1. | Now choose an example step for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then <v|m>=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be ∓1. It is therefore a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1. By inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume. | ||
These unimodular matricies define a [[http://en.wikipedia.org/wiki/Change_of_basis|change of basis]] for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as | These unimodular matricies define a [[http://en.wikipedia.org/wiki/Change_of_basis|change of basis]] for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as | ||
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The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski reduced bases for Fokker groups of certain vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski reduced bases for Fokker groups of certain vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | ||
</pre></div> | |||
=Scale properties of Fokker blocks= | |||
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 parallepiped, 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> | |||
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<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:28:&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:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Expanding the definition">Expanding the definition</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- 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: --> | <a href="#Scale properties of Fokker blocks">Scale properties of Fokker blocks</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | ||
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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>, 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>, 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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Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> v = &lt;v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2&lt;0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = &lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit.<br /> | Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> v = &lt;v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2&lt;0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = &lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit.<br /> | ||
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Now choose an example step for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then &lt;v|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be ∓1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.<br /> | Now choose an example step for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then &lt;v|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be ∓1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1. By inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume.<br /> | ||
<br /> | <br /> | ||
These unimodular matricies define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow">change of basis</a> for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as<br /> | These unimodular matricies define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow">change of basis</a> for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as<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:45:&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:45 --><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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This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the <a class="wiki_link" href="/transversal">transversal</a> we may obtain by <a class="wiki_link" href="/The%20wedgie#Truncation of wedgies">truncation</a>. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is <a class="wiki_link" href="/The%20dual">dual</a> to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way.<br /> | This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the <a class="wiki_link" href="/transversal">transversal</a> we may obtain by <a class="wiki_link" href="/The%20wedgie#Truncation of wedgies">truncation</a>. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is <a class="wiki_link" href="/The%20dual">dual</a> to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way.<br /> | ||
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The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val &lt;22 35 51 62|. Using a basis for the Fokker group, for instance the one listed <a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">here</a>, pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS.</body></html></pre></div> | The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val &lt;22 35 51 62|. Using a basis for the Fokker group, for instance the one listed <a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">here</a>, pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Scale properties of Fokker blocks"></a><!-- ws:end:WikiTextHeadingRule:26 -->Scale properties of Fokker blocks</h1> | |||
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 parallepiped, 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 /> | |||
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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> | |||