Fokker block: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-20 13:52:38 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-20 14:10:33 UTC</tt>.

: The original revision id was <tt>~~497165174~~</tt>.

: The original revision id was <tt>497169860</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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.

We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k

We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain <12 19 28 34 42|, -<3 5 7 9 10|, <9 14 21 25 31|, -<7 11 16 20 24|], [<22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that

~~By forming~~ the ~~5x5 matrix whose last row~~ is V, 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 385/384, 175/176, 100/99, 224~~/225~~ and 36/35, inverting and transposing, we obtain <12 19 28 34 42|, -<3 5 7 9 10|, <9 14 21 25 31|, -<7 11 16 20 24|], [<22 35 51 62 76|, . 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+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)

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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.

We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map">Frobenius projection map</a> P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k ~~ ~~

We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map">Frobenius projection map</a> P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that

~~ ~~

~~By forming~~ the ~~5x5 matrix whose last row~~ is V, 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 385/384, 175/176, 100/99, 224~~/225~~ and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . 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+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)

@@ 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>2014-03-20 13:52:38 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-20 14:10:33 UTC</tt>.<br>
-: The original revision id was <tt>497165174</tt>.<br>
+: The original revision id was <tt>497169860</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 66: / Line 66: @@
 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.
-We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k
+We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that
-By forming the 5x5 matrix whose last row is V, 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 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . 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+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)
@@ Line 166: / Line 164: @@
   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. &lt;br /&gt;
 &lt;br /&gt;
-We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the &lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map"&gt;Frobenius projection map&lt;/a&gt; P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k &lt;br /&gt;
+We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the &lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map"&gt;Frobenius projection map&lt;/a&gt; P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain &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|], [&amp;lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that&lt;br /&gt;
-&lt;br /&gt;
-By forming the 5x5 matrix whose last row is V, 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 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain &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|], [&amp;lt;22 35 51 62 76|, . 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+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)&lt;br /&gt;

Fokker block: Difference between revisions

Wikispaces>genewardsmith