Gral method
The Gral method is a method used by Erv Wilson for keyboard mapping and finding moments of symmetry.
Lattice bases
For concreteness, consider a two-dimensional keyboard. The keys on the keyboard are labeled by pairs of integers [math]\displaystyle{ (x, y) }[/math], their [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] coordinates. The set of all these integer pairs is called a lattice.[1] For two lattice points [math]\displaystyle{ (a, b) }[/math] and [math]\displaystyle{ (c, d) }[/math], if we can write any lattice point [math]\displaystyle{ (x, y) }[/math] as
[math]\displaystyle{ \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]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math], we say [math]\displaystyle{ (a, b) }[/math] and [math]\displaystyle{ (c, d) }[/math] form a basis for the lattice. Two points [math]\displaystyle{ (a, b) }[/math] and [math]\displaystyle{ (c, d) }[/math] form a basis if and only if [math]\displaystyle{ ad - bc = \pm 1 }[/math].
Problem
Say we have a lattice with points [math]\displaystyle{ (x, y) }[/math]. Consider a quantity [math]\displaystyle{ q }[/math] which depends linearly on [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], so
[math]\displaystyle{ q = s x + t y }[/math]
for some step sizes [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math].
Given two numbers [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math], we are looking for a lattice basis [math]\displaystyle{ (a, b) }[/math], [math]\displaystyle{ (c, d) }[/math] such that the value of [math]\displaystyle{ q }[/math] is [math]\displaystyle{ g }[/math] at [math]\displaystyle{ (a, b) }[/math] and [math]\displaystyle{ h }[/math] at [math]\displaystyle{ (c, d) }[/math], that is
[math]\displaystyle{ \displaystyle \begin{align} g &= a s + b t \\ h &= c s + d t \end{align} }[/math]
and for which the step sizes [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math] are positive.
The step sizes are determined by the numbers [math]\displaystyle{ g }[/math], [math]\displaystyle{ h }[/math] and the basis [math]\displaystyle{ (a, b) }[/math], [math]\displaystyle{ (c, d) }[/math] as
[math]\displaystyle{ \displaystyle \begin{align} s &= \frac{dg - bh}{\Delta} \\ t &= \frac{-cg + ah}{\Delta} \\ \end{align} }[/math]
where [math]\displaystyle{ \Delta = ad - bc }[/math], and [math]\displaystyle{ \Delta = \pm 1 }[/math] since [math]\displaystyle{ (a, b) }[/math], [math]\displaystyle{ (c, d) }[/math] is a lattice basis.
If [math]\displaystyle{ \Delta = -1 }[/math], [math]\displaystyle{ c > 0 }[/math], and [math]\displaystyle{ d > 0 }[/math], the positive step size conditions [math]\displaystyle{ s > 0 }[/math], [math]\displaystyle{ t > 0 }[/math] are equivalent to
[math]\displaystyle{ \displaystyle \frac{a}{c} < \frac{g}{h} < \frac{b}{d} }[/math]
Figure 1 shows the problem setup geometrically.
Algorithm
Given numbers [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math], the Gral method finds all lattice bases [math]\displaystyle{ (a, b) }[/math], [math]\displaystyle{ (c, d) }[/math] such that
- [math]\displaystyle{ ad - bc = -1 }[/math]
- [math]\displaystyle{ a, b, c, d \geq 0 }[/math]
- [math]\displaystyle{ \frac{a}{c} < \frac{g}{h} < \frac{b}{d} }[/math]
The algorithm is a binary search for the number [math]\displaystyle{ g/h }[/math], starting with the interval [math]\displaystyle{ [0/1, 1/0] }[/math] and 'bisecting' each interval [math]\displaystyle{ [a/c, b/d] }[/math] with the mediant [math]\displaystyle{ (a+b)/(c+d) }[/math].[2] Each interval [math]\displaystyle{ [a/c, b/d] }[/math] we encounter in the search has [math]\displaystyle{ ad - bc = -1 }[/math], so [math]\displaystyle{ (a, b) }[/math], [math]\displaystyle{ (c, d) }[/math] form a basis, and [math]\displaystyle{ a/c < g/h < b/d }[/math] since [math]\displaystyle{ 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]\displaystyle{ g }[/math] cents within a period of pitch [math]\displaystyle{ h }[/math] cents. Applying the Gral method to [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math], each interval [math]\displaystyle{ [a/c, b/d] }[/math] found gives a MOS with [math]\displaystyle{ n = c + d }[/math] notes, where the generator is at scale degree [math]\displaystyle{ m = a + b }[/math]; call this an [math]\displaystyle{ m/n }[/math] MOS.
- The step sizes [math]\displaystyle{ s = -d g + b h }[/math] and [math]\displaystyle{ t = c g - a h }[/math] are the MOS step sizes.
- The MOS has [math]\displaystyle{ c }[/math] steps of [math]\displaystyle{ s }[/math] and [math]\displaystyle{ d }[/math] steps of [math]\displaystyle{ t }[/math].
- The interval [math]\displaystyle{ [a/c, b/d] }[/math] gives the range of [math]\displaystyle{ g/h }[/math] which will produce an [math]\displaystyle{ 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]\displaystyle{ g = 707.22 }[/math] cents and [math]\displaystyle{ h = 1200 }[/math] cents. The Gral method can usefully be shown in a table as follows:
| Left | Right | Mediant | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ c }[/math] | [math]\displaystyle{ d }[/math] | [math]\displaystyle{ s }[/math] | [math]\displaystyle{ 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 |
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]\displaystyle{ (a,b) }[/math], [math]\displaystyle{ (c,d) }[/math] corresponding to the search interval [math]\displaystyle{ [a/c,b/d] }[/math]. The columns s and t give the resulting step sizes (for which [math]\displaystyle{ as+ bt = g }[/math] and [math]\displaystyle{ 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.[3][4] We can understand why the Gral method calculates MOS by thinking about keyboard mapping.[5]
Application to keyboard mapping
One way to map a scale onto a two-dimensional keyboard is to always go up [math]\displaystyle{ s }[/math] scale degrees when moving one key along a row, and [math]\displaystyle{ t }[/math] scale degrees when moving one key up a column, for some step sizes [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math]. If we choose scale degrees [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] which we want to be mapped to a basis on the keyboard, the Gral method applied with [math]\displaystyle{ g = m }[/math] and [math]\displaystyle{ h = n }[/math] tells us which bases with non-negative coordinates will give positive step sizes [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math].[5]
For example, if we want scale degrees [math]\displaystyle{ m = 11 }[/math] and [math]\displaystyle{ n = 19 }[/math] to be mapped to a basis on the keyboard, we can apply the Gral method to 11 and 19:
| Left | Right | Mediant | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ c }[/math] | [math]\displaystyle{ d }[/math] | [math]\displaystyle{ s }[/math] | [math]\displaystyle{ 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 |
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]\displaystyle{ (1, 3) }[/math] and scale degree 19 at [math]\displaystyle{ (2, 5) }[/math]. Wilson describes each basis as a different keyboard named with the mediant, so the basis [math]\displaystyle{ (1, 3) }[/math], [math]\displaystyle{ (2, 5) }[/math] is called the 4/7 keyboard.[6][7]
References
- ↑ J.W.S. Cassels, An Introduction To The Geometry Of Numbers. Springer-Verlag, 1971.
- ↑ D.E. Knuth, O. Patashnik, and R.L. Graham, Concrete Mathematics. Addison-Wesley, 1994.
- ↑ Erv Wilson, Diophantine triplets of temperament derived intervals. The Wilson Archives.
- ↑ Erv Wilson, Diophantine triplets of recurrent derived intervals. The Wilson Archives.
- ↑ 5.0 5.1 Naren Ratan, Another look at Wilson's keyboard mapping system, Xenharmonikon Online, 2026.
- ↑ Erv Wilson, The Gral Spectrum. The Wilson Archives.
- ↑ Erv Wilson, Gral Keyboard Guide. The Wilson Archives.