Fokker block: Difference between revisions
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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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 13:03:30 UTC</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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Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. | Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. | ||
=Expanding the definition= | |||
A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways. | |||
Explicitly, S is a [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + N] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[N]) = V(2) = N, so that V = <N ... |, with N a positive integer; in other words, V is an N-edo val. | |||
=Example= | =Example= | ||
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One has first the [[pajmagorpor22|original JI scale]]. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are [[pajmagorpor22_225|225/224]], [[pajmagorpor22_100|100/99]], [[pajmagorpor22_176|176/175]], and [[pajmagorpor22_385|385/384]]. The next level gives [[pajmagorpor22apollo|apollo]], [[pajmagorpor22minerva|minerva]], [[pajmagorpor22marvel|marvel]], [[pajmagorpor22ares|ares]], [[pajmagorpor22supermagic|supermagic]], and [[pajmagorpor22zeus|zeus]]. 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.</pre></div> | One has first the [[pajmagorpor22|original JI scale]]. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are [[pajmagorpor22_225|225/224]], [[pajmagorpor22_100|100/99]], [[pajmagorpor22_176|176/175]], and [[pajmagorpor22_385|385/384]]. The next level gives [[pajmagorpor22apollo|apollo]], [[pajmagorpor22minerva|minerva]], [[pajmagorpor22marvel|marvel]], [[pajmagorpor22ares|ares]], [[pajmagorpor22supermagic|supermagic]], and [[pajmagorpor22zeus|zeus]]. 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.</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:24:&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:24 --><!-- ws:start:WikiTextTocRule:25: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Expanding the definition">Expanding the definition</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | ||
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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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Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. <br /> | Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Expanding the definition"></a><!-- ws:end:WikiTextHeadingRule:12 -->Expanding the definition</h1> | ||
<!-- ws:start:WikiTextHeadingRule: | A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways.<br /> | ||
<br /> | |||
Explicitly, S is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + N] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[N]) = V(2) = N, so that V = &lt;N ... |, with N a positive integer; in other words, V is an N-edo val.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:14 -->Example</h1> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Example-Using a Fokker group basis"></a><!-- ws:end:WikiTextHeadingRule:16 -->Using a Fokker group basis</h2> | |||
Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = &lt;22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {&lt;&lt;1 9 -2 -6 12 -6 -13 -30 -45 -10||, &lt;&lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||, &lt;&lt;6 10 10 8 2 -1 -8 -5 -16 -12||, &lt;&lt;2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, magic = pajara+hedgehog-suprapyth-pajarous, orwell = pajara+hedgehog-suprapyth, porcupine = suprapyth+pajarous; hence, S is a Fokker block.<br /> | Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = &lt;22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {&lt;&lt;1 9 -2 -6 12 -6 -13 -30 -45 -10||, &lt;&lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||, &lt;&lt;6 10 10 8 2 -1 -8 -5 -16 -12||, &lt;&lt;2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, magic = pajara+hedgehog-suprapyth-pajarous, orwell = pajara+hedgehog-suprapyth, porcupine = suprapyth+pajarous; hence, S is a Fokker block.<br /> | ||
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If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.<br /> | If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Example-Generator range and the first definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:18 -->Generator range and the first definition of a Fokker block</h2> | ||
From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose first row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 36/35, 385/384, 175/176, 100/99 and 224/225, inverting and transposing, we obtain [&lt;22 35 51 62 76|, &lt;12 19 28 34 42|, &lt;3 5 7 9 10|, &lt;9 14 21 25 31|, &lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that<br /> | From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose first row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 36/35, 385/384, 175/176, 100/99 and 224/225, inverting and transposing, we obtain [&lt;22 35 51 62 76|, &lt;12 19 28 34 42|, &lt;3 5 7 9 10|, &lt;9 14 21 25 31|, &lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that<br /> | ||
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is the periodic scale with which we began this analysis.<br /> | is the periodic scale with which we began this analysis.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Example-Product words and the fourth definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:20 -->Product words and the fourth definition of a Fokker block</h2> | ||
Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &quot;product&quot; means product in a quite literal sense, since these can be construed as wedge product words. <br /> | Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &quot;product&quot; means product in a quite literal sense, since these can be construed as wedge product words. <br /> | ||
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As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking <a class="wiki_link" href="/The%20dual">dual</a> we obtain the wedgie for zeus, which is &lt;&lt;&lt;2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = &lt;&lt;1 -3 5 -1 -7 5 -5 20 8 -20||, x = &lt;&lt;-3 5 -9 1 15 -6 12 -35 -15 34||, y = &lt;&lt;4 2 -1 3 -6 -13 -9 -8 0 12||, and z = &lt;&lt;-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.<br /> | As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking <a class="wiki_link" href="/The%20dual">dual</a> we obtain the wedgie for zeus, which is &lt;&lt;&lt;2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = &lt;&lt;1 -3 5 -1 -7 5 -5 20 8 -20||, x = &lt;&lt;-3 5 -9 1 15 -6 12 -35 -15 34||, y = &lt;&lt;4 2 -1 3 -6 -13 -9 -8 0 12||, and z = &lt;&lt;-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Example-The tempered scales of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:22 -->The tempered scales of a Fokker block</h2> | ||
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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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.</body></html></pre></div> | 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.</body></html></pre></div> | ||