Gral method: Difference between revisions

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The columns a, b, c, d give the basis <math>(a,b)</math>, <math>(c,d)</math> corresponding to the search interval <math>[a/c,b/d]</math>.

The columns s and t give the resulting step sizes (for which <math>as+ bt = g</math> and <math>cs + dt = h</math>).

The row with mediant 10/17, for example, tells us that we get a 17 note MOS with 12 steps of 63.90 cents and 5 steps of 86.64 cents.

The corresponding search interval <math>[7/12, 3/5]</math> tells us that any generator <math>g</math>

between <math>1200 \cdot 7/12 = 700</math> cents and <math>1200 \cdot 3/5 = 720</math> cents will give a 10/17 MOS.

The Gral method is a very convenient way to calculate MOS, and it was extensively used by Wilson.<ref name="wilson-temperament"/><ref name="wilson-recurrent"/>

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Wilson describes each basis as a different keyboard named with the mediant,

so the basis <math>(1, 3)</math>, <math>(2, 5)</math> is called the 4/7 keyboard.<ref name="wilson-gralspectrum"/><ref name="wilson-gralkeyboard"/>

== Labeling the MOS ==

For a given interval <math>[a/c, b/d]</math> found by the Gral method,

we might label the corresponding MOS by either of

* The mediant <math>m/n = (a + b)/(c + d)</math>

* The step counts <math>c</math> and <math>d</math>

So for the MOS corresponding to the row with mediant 10/17 discussed above, we might call it a 10/17 MOS, or a 12<math>s</math> 5<math>t</math> MOS.

We can convert between these two labels:

* Given the mediant <math>m/n</math>, we have <math>c = m^{-1}\ (\mathrm{mod}\ n)</math> and <math>d = n - c</math>.

* Given the step counts <math>c</math> and <math>d</math>, we have <math>n = c + d</math> and <math>m = c^{-1}\ (\mathrm{mod}\ n)</math>.

Here <math>m = c^{-1}\ (\mathrm{mod}\ n)</math> means <math>m</math> is the modular inverse of <math>c</math> with respect to <math>n</math>;

in Python this is written <code>m = pow(c, -1, n)</code>.

The modular inverse comes up because <math>ad - bc = -1</math> is equivalent to <math>cm - an = 1</math>,

so <math>cm \equiv 1\ (\mathrm{mod}\ n)</math>.

== References ==

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[[Category:Erv Wilson]]

[[Category:MOS scale]]

@@ Line 142: / Line 142: @@
 The columns a, b, c, d give the basis <math>(a,b)</math>, <math>(c,d)</math> corresponding to the search interval <math>[a/c,b/d]</math>.
 The columns s and t give the resulting step sizes (for which <math>as+ bt = g</math> and <math>cs + dt = h</math>).
 The row with mediant 10/17, for example, tells us that we get a 17 note MOS with 12 steps of 63.90 cents and 5 steps of 86.64 cents.
+The corresponding search interval <math>[7/12, 3/5]</math> tells us that any generator <math>g</math>
+between <math>1200 \cdot 7/12 = 700</math> cents and <math>1200 \cdot 3/5 = 720</math> cents will give a 10/17 MOS.
 The Gral method is a very convenient way to calculate MOS, and it was extensively used by Wilson.<ref name="wilson-temperament"/><ref name="wilson-recurrent"/>
@@ Line 182: / Line 185: @@
 Wilson describes each basis as a different keyboard named with the mediant,
 so the basis <math>(1, 3)</math>, <math>(2, 5)</math> is called the 4/7 keyboard.<ref name="wilson-gralspectrum"/><ref name="wilson-gralkeyboard"/>
+== Labeling the MOS ==
+For a given interval <math>[a/c, b/d]</math> found by the Gral method,
+we might label the corresponding MOS by either of
+* The mediant <math>m/n = (a + b)/(c + d)</math>
+* The step counts <math>c</math> and <math>d</math>
+So for the MOS corresponding to the row with mediant 10/17 discussed above, we might call it a 10/17 MOS, or a 12<math>s</math> 5<math>t</math> MOS.
+We can convert between these two labels:
+* Given the mediant <math>m/n</math>, we have <math>c = m^{-1}\ (\mathrm{mod}\ n)</math> and <math>d = n - c</math>.
+* Given the step counts <math>c</math> and <math>d</math>, we have <math>n = c + d</math> and <math>m = c^{-1}\ (\mathrm{mod}\ n)</math>.
+Here <math>m = c^{-1}\ (\mathrm{mod}\ n)</math> means <math>m</math> is the modular inverse of <math>c</math> with respect to <math>n</math>;
+in Python this is written <code>m = pow(c, -1, n)</code>.
+The modular inverse comes up because <math>ad - bc = -1</math> is equivalent to <math>cm - an = 1</math>,
+so <math>cm \equiv 1\ (\mathrm{mod}\ n)</math>.
 == References ==
@@ Line 209: / Line 231: @@
 [[Category:Erv Wilson]]
+[[Category:MOS scale]]