Fokker block: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 02:24:36 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 12:22:16 UTC</tt>.<br>

: The original revision id was <tt>~~307349344~~</tt>.<br>

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Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.

Note that because any ak can be chosen satisfying 0 < ak < P, each Fokker block comes associated not only with various modes, but also [[Dome|domes]], which are variations on the same basic theme.

=Second definition of a Fokker block=

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==Generator range and the first definition of a Fokker block==

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 12. 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 [<22 35 51 62 76|, <12 19 28 34 42|, <3 5 7 9 10|, <9 14 21 25 31|, <7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that

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 [<22 35 51 62 76|, <12 19 28 34 42|, <3 5 7 9 10|, <9 14 21 25 31|, <7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)

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==Product words and the fourth definition of a Fokker block==

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 = -<10 16 23 28 34| and b = <12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -<3 5 7 9 10| and d = <19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc.

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 = -<10 16 23 28 34| and b = <12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -<3 5 7 9 10| and d = <19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = <9 14 21 25 31| and f = -<13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = <7 11 16 20 24| and h = -<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 "product" means product in a quite literal sense, since these can be construed as wedge product words.

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<br />

Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.<br />

<br />

Note that because any ak can be chosen satisfying 0 &lt; ak &lt; P, each Fokker block comes associated not only with various modes, but also <a class="wiki_link" href="/Dome">domes</a>, which are variations on the same basic theme. <br />

<br />

<h1 id="toc2"><a name="Second definition of a Fokker block"></a>Second definition of a Fokker block</h1>

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<br />

<h2 id="toc8"><a name="Example-Generator range and the first definition of a Fokker block"></a>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 12. 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 />

<br />

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)<br />

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<br />

<h2 id="toc9"><a name="Example-Product words and the fourth definition of a Fokker block"></a>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: 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. <br />

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 />

<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-03-03 02:24:36 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 12:22:16 UTC</tt>.<br>
-: The original revision id was <tt>307349344</tt>.<br>
+: The original revision id was <tt>307410518</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>
@@ Line 29: / Line 29: @@
 Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.
+Note that because any ak can be chosen satisfying 0 &lt; ak &lt; P, each Fokker block comes associated not only with various modes, but also [[Dome|domes]], which are variations on the same basic theme.
 =Second definition of a Fokker block=
@@ Line 53: / Line 55: @@
 ==Generator range and the first definition of a Fokker block==
-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 12. 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
+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
 S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)
@@ Line 60: / Line 62: @@
 ==Product words and the fourth definition of a Fokker block==
-Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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.
+Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 "product" means product in a quite literal sense, since these can be construed as wedge product words.
@@ Line 96: / Line 98: @@
 &lt;br /&gt;
 Hence S satisfies the conditions for being a &lt;a class="wiki_link" href="/Periodic%20scale"&gt;periodic scale&lt;/a&gt;, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.&lt;br /&gt;
+&lt;br /&gt;
+Note that because any ak can be chosen satisfying 0 &amp;lt; ak &amp;lt; P, each Fokker block comes associated not only with various modes, but also &lt;a class="wiki_link" href="/Dome"&gt;domes&lt;/a&gt;, which are variations on the same basic theme. &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Second definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Second definition of a Fokker block&lt;/h1&gt;
@@ Line 120: / Line 124: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Example-Generator range and the first definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Generator range and the first definition of a Fokker block&lt;/h2&gt;
-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 12. 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 [&amp;lt;22 35 51 62 76|, &amp;lt;12 19 28 34 42|, &amp;lt;3 5 7 9 10|, &amp;lt;9 14 21 25 31|, &amp;lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that&lt;br /&gt;
+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 [&amp;lt;22 35 51 62 76|, &amp;lt;12 19 28 34 42|, &amp;lt;3 5 7 9 10|, &amp;lt;9 14 21 25 31|, &amp;lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that&lt;br /&gt;
 &lt;br /&gt;
 S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)&lt;br /&gt;
@@ Line 127: / Line 131: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Example-Product words and the fourth definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Product words and the fourth definition of a Fokker block&lt;/h2&gt;
-Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 = -&amp;lt;10 16 23 28 34| and b = &amp;lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&amp;lt;3 5 7 9 10| and d = &amp;lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. &lt;br /&gt;
+Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 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 = -&amp;lt;10 16 23 28 34| and b = &amp;lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&amp;lt;3 5 7 9 10| and d = &amp;lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &amp;lt;9 14 21 25 31| and f = -&amp;lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &amp;lt;7 11 16 20 24| and h = -&amp;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 &amp;quot;product&amp;quot; means product in a quite literal sense, since these can be construed as wedge product words. &lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;

Fokker block: Difference between revisions

Wikispaces>genewardsmith