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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 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"/>
 == References ==
@@ Line 209: / Line 231: @@
 [[Category:Erv Wilson]]

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The '''Gral method''' is a method used by [[Erv Wilson]] for keyboard mapping and finding [[MOS|moments of symmetry]].

== Lattice bases ==

For concreteness, consider a two-dimensional keyboard.
The keys on the keyboard are labeled by pairs of integers <math>(x, y)</math>, their <math>x</math> and <math>y</math> coordinates.
The set of all these integer pairs is called a lattice.<ref name="cassels1971introduction"/>
For two lattice points <math>(a, b)</math> and <math>(c, d)</math>, if we can write any lattice point <math>(x, y)</math> as

<math>\displaystyle
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
=
p
\begin{bmatrix}
a \\
b \\
\end{bmatrix}
+
q
\begin{bmatrix}
c \\
d \\
\end{bmatrix}
</math>

with integer <math>p</math> and <math>q</math>,
we say <math>(a, b)</math> and <math>(c, d)</math> form a basis for the lattice.
Two points <math>(a, b)</math> and <math>(c, d)</math> form a basis if and only if
<math>ad - bc = \pm 1 </math>.

== Problem ==

[[File:Gral-diagram.svg|thumb|200px|Figure 1: Lattice geometry for the Gral method.]]

Say we have a lattice with points <math>(x, y)</math>.
Consider a quantity <math>q</math> which depends linearly on <math>x</math> and <math>y</math>, so

<math>
q = s x + t y
</math>

for some step sizes <math>s</math> and <math>t</math>.

Given two numbers <math>g</math> and <math>h</math>,
we are looking for a lattice basis <math>(a, b)</math>, <math>(c, d)</math>
such that the value of <math>q</math> is <math>g</math> at <math>(a, b)</math> and <math>h</math> at <math>(c, d)</math>, that is

<math>\displaystyle
\begin{align}
g &= a s + b t \\
h &= c s + d t
\end{align}
</math>

and for which the step sizes <math>s</math> and <math>t</math> are positive.

The step sizes are determined by the numbers <math>g</math>, <math>h</math> and the basis <math>(a, b)</math>, <math>(c, d)</math> as

<math>\displaystyle
\begin{align}
s &= \frac{dg - bh}{\Delta} \\
t &= \frac{-cg + ah}{\Delta} \\
\end{align}
</math>

where <math>\Delta = ad - bc</math>,
and <math>\Delta = \pm 1</math> since <math>(a, b)</math>, <math>(c, d)</math> is a lattice basis.

If <math>\Delta = -1</math>, <math>c > 0</math>, and <math>d > 0</math>,
the positive step size conditions <math>s > 0</math>, <math>t > 0</math> are equivalent to

<math>\displaystyle
\frac{a}{c} < \frac{g}{h} < \frac{b}{d}
</math>

Figure 1 shows the problem setup geometrically.

== Algorithm ==

Given numbers <math>g</math> and <math>h</math>, the Gral method finds all lattice bases <math>(a, b)</math>, <math>(c, d)</math> such that
* <math>ad - bc = -1</math>
* <math>a, b, c, d \geq 0</math>
* <math>\frac{a}{c} < \frac{g}{h} < \frac{b}{d}</math>

The algorithm is a binary search for the number <math>g/h</math>,
starting with the interval <math>[0/1, 1/0]</math> and 'bisecting' each interval <math>[a/c, b/d]</math> with the mediant <math>(a+b)/(c+d)</math>.<ref name="knuth1994concrete"/>
Each interval <math>[a/c, b/d]</math> we encounter in the search has <math>ad - bc = -1</math>,
so <math>(a, b)</math>, <math>(c, d)</math> form a basis,
and <math>a/c < g/h < b/d</math> since <math>g/h</math> is in each search interval.

== Application to finding MOS ==

The Gral method finds all [[MOS]] formed when stacking a generator of pitch <math>g</math> cents within a period of pitch <math>h</math> cents.
Applying the Gral method to <math>g</math> and <math>h</math>,
each interval <math>[a/c, b/d]</math> found gives a [[MOS]] with <math>n = c + d</math> notes,
where the generator is at scale degree <math>m = a + b</math>; call this an <math>m/n</math> MOS.

* The step sizes <math>s = -d g + b h</math> and <math>t = c g - a h</math> are the MOS step sizes.
* The MOS has <math>c</math> steps of <math>s</math> and <math>d</math> steps of <math>t</math>.
* The interval <math>[a/c, b/d]</math> gives the range of <math>g/h</math> which will produce an <math>m/n</math> MOS when stacking.

In this way the Gral method directly gives the generator range and step size formulae from the [[Generator ranges of MOS]] page.

For example, take <math>g = 707.22</math> cents and <math>h = 1200</math> cents.
The Gral method can usefully be shown in a table as follows:

<div style="text-align:center">
{| class="wikitable"
! Left !! Right !! Mediant !! style="width:2.5em" | <math>a</math> !! style="width:2.5em" | <math>b</math> !! style="width:2.5em" | <math>c</math> !! style="width:2.5em" | <math>d</math> !! <math>s</math> !! <math>t</math>
|-
| 0/1 || 1/0 || 1/1 || 0 || 1 || 1 || 0 || 1200.00 || 707.22
|-
| || 1/1 || 1/2 || 0 || 1 || 1 || 1 || 492.78 || 707.22
|-
| 1/2 || || 2/3 || 1 || 1 || 2 || 1 || 492.78 || 214.44
|-
| || 2/3 || 3/5 || 1 || 2 || 2 || 3 || 278.34 || 214.44
|-
| || 3/5 || 4/7 || 1 || 3 || 2 || 5 || 63.90 || 214.44
|-
| 4/7 || || 7/12 || 4 || 3 || 7 || 5 || 63.90 || 150.54
|-
| 7/12 || || 10/17 || 7 || 3 || 12 || 5 || 63.90 || 86.64
|-
| 10/17 || || 13/22 || 10 || 3 || 17 || 5 || 63.90 || 22.74
|-
| || 13/22 || 23/39 || 10 || 13 || 17 || 22 || 41.16 || 22.74
|-
| || 23/39 || 33/56 || 10 || 23 || 17 || 39 || 18.42 || 22.74
|-
| 33/56 || || 56/95 || 33 || 23 || 56 || 39 || 18.42 || 4.32
|}
</div>

Here Left and Right are the endpoints of the search interval;
a blank means the same value as the row above
(this lets you see at a glance which endpoint moved to form each row).
The Mediant column is the mediant of the search interval.
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 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"/>
We can understand why the Gral method calculates MOS by thinking about keyboard mapping.<ref name="ratan2026another"/>

== Application to keyboard mapping ==

One way to map a scale onto a two-dimensional keyboard is to always go up <math>s</math> scale degrees when moving one key along a row,
and <math>t</math> scale degrees when moving one key up a column, for some step sizes <math>s</math> and <math>t</math>.
If we choose scale degrees <math>m</math> and <math>n</math> which we want to be mapped to a basis on the keyboard,
the Gral method applied with <math>g = m</math> and <math>h = n</math> tells us which bases with non-negative coordinates will give positive step sizes <math>s</math> and <math>t</math>.<ref name="ratan2026another"/>

For example, if we want scale degrees <math>m = 11</math> and <math>n = 19</math> to be mapped to a basis on the keyboard,
we can apply the Gral method to 11 and 19:

<div style="text-align:center">
{| class="wikitable"
! Left !! Right !! Mediant !! style="width:2.5em" | <math>a</math> !! style="width:2.5em" | <math>b</math> !! style="width:2.5em" | <math>c</math> !! style="width:2.5em" | <math>d</math> !! style="width:2.5em" | <math>s</math> !! style="width:2.5em" | <math>t</math>
|-
| 0/1 || 1/0 || 1/1 || 0 || 1 || 1 || 0 || 19 || 11
|-
| || 1/1 || 1/2 || 0 || 1 || 1 || 1 || 8 || 11
|-
| 1/2 || || 2/3 || 1 || 1 || 2 || 1 || 8 || 3
|-
| || 2/3 || 3/5 || 1 || 2 || 2 || 3 || 5 || 3
|-
| || 3/5 || 4/7 || 1 || 3 || 2 || 5 || 2 || 3
|-
| 4/7 || || 7/12 || 4 || 3 || 7 || 5 || 2 || 1
|-
| || 7/12 || 11/19 || 4 || 7 || 7 || 12 || 1 || 1
|}
</div>

The row with mediant 4/7, for example,
tells us that mapping the scale by going up 2 scale degrees when moving along a row and 3 scale degrees when moving up a column
will place scale degree 11 at <math>(1, 3)</math> and scale degree 19 at <math>(2, 5)</math>.
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"/>

== References ==
<references>
<ref name="cassels1971introduction">
J.W.S. Cassels, [https://archive.org/details/introductiontoge0099jwsc An Introduction To The Geometry Of Numbers]. Springer-Verlag, 1971.
</ref>
<ref name="knuth1994concrete">
D.E. Knuth, O. Patashnik, and R.L. Graham, [https://seriouscomputerist.atariverse.com/media/pdf/book/Concrete%20Mathematics.pdf Concrete Mathematics]. Addison-Wesley, 1994.
</ref>
<ref name="wilson-temperament">
Erv Wilson, [https://www.anaphoria.com/DiophantineTripletsTEMPER.pdf Diophantine triplets of temperament derived intervals]. The Wilson Archives.
</ref>
<ref name="wilson-recurrent">
Erv Wilson, [https://www.anaphoria.com/DiophantineTripletsRECURR.pdf Diophantine triplets of recurrent derived intervals]. The Wilson Archives.
</ref>
<ref name="ratan2026another">
Naren Ratan, [https://www.xenharmonikon.org/2026/05/19/another-look-at-wilsons-keyboard-mapping-system/ Another look at Wilson's keyboard mapping system], Xenharmonikon Online, 2026.
</ref>
<ref name="wilson-gralspectrum">
Erv Wilson, [https://www.anaphoria.com/gralspectrum.pdf The Gral Spectrum]. The Wilson Archives.
</ref>
<ref name="wilson-gralkeyboard">
Erv Wilson, [https://www.anaphoria.com/gralkeyboard.pdf Gral Keyboard Guide]. The Wilson Archives.
</ref>
</references>

[[Category:Erv Wilson]]

Gral method - Revision history

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