Generator preimage: Difference between revisions

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Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [<12 19 28 34|, <19 30 44 53|]. Then a corresponding list of transversal generators is [49/48, 36/35]. 49/48 corresponds to one step of 12et, and zero steps of 19et, whereas 36/35 is zero steps of 12et, and one step of 19et. This gives us a septimal meantone transversal of the 7-limit where 3/2 is represented by (49/48)^7 * (36/35)^11, and 2 is represented by (49/48)^12 * (36/35)^19. A more familiar septimal meantone transversal starts from the normal val list, [<1 0 -4 -13|, <0 1 4 10|], which corresponds to the transversal generators [2, 3].

Given a list of transversal generators, we may append a [[comma-basis]] for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain

Given a list of transversal generators, we may append a [[comma basis]] for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain

[<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|]

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We can find transveral generators for V by the following procedure:

<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma-basis for V</li><li>[[Saturation|Saturate]] (or defactor) the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>

<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li>[[Saturation|Saturate]] (or defactor) the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>

== Example ==

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</math>

And here's the comma-basis for <math>V</math>:

And here's the comma basis for <math>V</math>:

<math>

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</math>

Beginning with <math>i = 1</math>, we'll create a matrix out of <math>U[i]</math> with denominators cleared and the comma-basis appended (transposed so each comma is a row):

Beginning with <math>i = 1</math>, we'll create a matrix out of <math>U[i]</math> with denominators cleared and the comma basis appended (transposed so each comma is a row):

<math>

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So indeed that gives the period for meantone, 2/1.

Let's repeat the latter steps but now with <math>i = 2</math>. Here's <math>U[2]</math> with denominators cleared and the comma-basis appended:

Let's repeat the latter steps but now with <math>i = 2</math>. Here's <math>U[2]</math> with denominators cleared and the comma basis appended:

<math>

@@ Line 13: / Line 13: @@
 Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [&lt;12 19 28 34|, &lt;19 30 44 53|]. Then a corresponding list of transversal generators is [49/48, 36/35]. 49/48 corresponds to one step of 12et, and zero steps of 19et, whereas 36/35 is zero steps of 12et, and one step of 19et. This gives us a septimal meantone transversal of the 7-limit where 3/2 is represented by (49/48)^7 * (36/35)^11, and 2 is represented by (49/48)^12 * (36/35)^19. A more familiar septimal meantone transversal starts from the normal val list, [&lt;1 0 -4 -13|, &lt;0 1 4 10|], which corresponds to the transversal generators [2, 3].
-Given a list of transversal generators, we may append a [[comma-basis]] for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain
+Given a list of transversal generators, we may append a [[comma basis]] for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain
 [&lt;12 19 28 34|, &lt;19 30 44 53|, &lt;-4 -6 -9 -11|, &lt;-5 -8 -12 -14|]
@@ Line 24: / Line 24: @@
 We can find transveral generators for V by the following procedure:
-<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma-basis for V</li><li>[[Saturation|Saturate]] (or defactor) the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>
+<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li>[[Saturation|Saturate]] (or defactor) the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>
 == Example ==
@@ Line 60: / Line 60: @@
 </math>
-And here's the comma-basis for <span><math>V</math></span>:
+And here's the comma basis for <span><math>V</math></span>:
 <math>
@@ Line 70: / Line 70: @@
 </math>
-Beginning with <span><math>i = 1</math></span>, we'll create a matrix out of <span><math>U[i]</math></span> with denominators cleared and the comma-basis appended (transposed so each comma is a row):
+Beginning with <span><math>i = 1</math></span>, we'll create a matrix out of <span><math>U[i]</math></span> with denominators cleared and the comma basis appended (transposed so each comma is a row):
 <math>
@@ Line 140: / Line 140: @@
 So indeed that gives the period for meantone, 2/1.
-Let's repeat the latter steps but now with <span><math>i = 2</math></span>. Here's <span><math>U[2]</math></span> with denominators cleared and the comma-basis appended:
+Let's repeat the latter steps but now with <span><math>i = 2</math></span>. Here's <span><math>U[2]</math></span> with denominators cleared and the comma basis appended:
 <math>