User:Frostburn/Geometric algebra for regular temperaments

This is a work-in-progress for my (Frostburn) thoughts about using geometric algebra to work with regular temperaments. I'm trying to write this in such a way that both geometers and musicians can follow the line of thought.

Vals

The simplest kinds of tunings only have a finite set of notes like our western 12-tone equal temperament. To understand such scales harmonically we need to know how to translate musically meaningful fractions such as 3/2 or 5/3 into numbers of steps in the scale of interest. Let's start with 5 equal divisions of the octave (the octave is the fraction 2/1). How should we represent the next prime 3/1? If 5 steps is the octave then the closest we can get to 3/1 is with 8 steps. The next prime 5/1 takes 12 steps of our 5-tone scale. These step counts are collected into a vector [math]\displaystyle{ \lt 5, 8, 12 ] }[/math] known as a val where the prime factorization of intervals of interest are collected into vectors known as monzos: [math]\displaystyle{ 3/2 \mapsto [ -1, 1, 0 \gt }[/math] and [math]\displaystyle{ 5/3 \mapsto [ 0, -1, 1 \gt }[/math]. We can now use the dot product to work out how many steps we need to represent 3/2 (known as the fifth) [math]\displaystyle{ \lt 5, 8, 12 | -1, 1, 0 \gt = 8 - 5 = 3 }[/math] steps of 5-tone equal temperament. Similarly 5/3 (known as the major sixth) equals [math]\displaystyle{ \lt 5, 8, 12 | 0, -1, 1 \gt = 12 - 8 = 4 }[/math] steps.

Combining vals

Notice that it doesn't matter if we scale our val. [math]\displaystyle{ \lt 10, 16, 24 ] }[/math] represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament [math]\displaystyle{ \lt 7, 11, 16 ] }[/math] and add it to [math]\displaystyle{ \lt 5, 8, 12 ] }[/math]. The result is [math]\displaystyle{ \lt 12, 19, 28 ] }[/math] which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get [math]\displaystyle{ \lt 17, 27, 40 ] }[/math] which is different from the optimal (patent) val for 17-tone equal temperament [math]\displaystyle{ \lt 17, 27, \mathbf{39} ] }[/math].

Geometric interpretation

Because vals can be multiplied by an arbitrary scalar they represent lines that pass through the origin. Because one endpoint of the line is fixed the resulting space of interest is 2-dimensional in the case of three component vals. This is known as the val space and is isomorphic to the projective tuning space.

3D lines that pass through the origin find representation in the Geometric Algebra [math]\displaystyle{ \mathcal G(3,0) }[/math]. Our vals are now vectors, or points if you think projectively, in this space.

[math]\displaystyle{ \lt 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} }[/math]

I will use the term pseudoscalar to refer to the combination of all basis vectors and denote it with i.

[math]\displaystyle{ i := e1e2e3 }[/math]

I will use the term pseudovector to refer to vectors multiplied by i.

Combining vals geometrically

We can now use geometric algebraic operations to combine vals.

[math]\displaystyle{ \begin{align} \overleftarrow{5} \wedge \overleftarrow{7} &= (5e_1 + 8e_2 + 12e_3) \wedge (7e_1 + 11e_2 + 16e_3) \\ &= (5 \times 11 - 8 \times 7)e_{12} + (5 \times 16 - 12 \times 7)e_{13} + (8 \times 16 - 12 \times 11)e_{23} \\ &= -4e_1i + 4e_2i - e_3i \\ \end{align} }[/math]

Readers familiar with monzos will recognize this as the syntonic comma 81/80 defining the Meantone temperament. Another example would be [math]\displaystyle{ \overleftarrow{7} \wedge \overleftarrow{8} = e_1i - 5e_2i + 3e_3i }[/math] looking like the monzo for the maximal diesis 250/243 defining the Porcupine temperament.

Geometric interpretation of temperaments

Let's use the vector notation with right-facing arrows for monzos.

[math]\displaystyle{ -3e_1 + e_2 + e_3 =: \overrightarrow{15/8} }[/math]

Strictly speaking these do not belong in the algebra [math]\displaystyle{ \mathcal G(3,0) }[/math] because their size does matter, but the numerical math works out.

[math]\displaystyle{ \overleftarrow{12} \cdot \overrightarrow{15/8} = 11 }[/math]

Indeed 15/8 (the major seventh) is worth 11 steps of 12-TET. However the commas defining temperaments do belong to the algebra. It makes no difference if you temper out 81/80 or its square 6561/6400 [math]\displaystyle{ = 2 \cdot \overrightarrow{81/80} }[/math]. You still get the Meantone temperament. As suggested by the observations above we will define rank-2 temperaments in 3 dimensions to be comma monzos interpreted as pseudovectors. Thus Meantone [math]\displaystyle{ = i \cdot \overrightarrow{81/80} }[/math]. The geometric interpretation of a temperament is therefore a plane spanned by two vals, or a line that passes through two points thinking projectively. We can project the just intonation point to such a line to find the mapping that is as close to just intonation as possible. In the case of Meantone:

[math]\displaystyle{ (\lt 1200.0, 1901.9, 2786.3] \cdot \overrightarrow{81/80}i) (\overrightarrow{81/80}i)^{-1} \approx \lt 1202.6, 1899.3, 2787.0] }[/math]

We can then normalize the first component for pure octaves to get [math]\displaystyle{ \lt 1200.0, 1895, 2780.9] }[/math]. We're again abusing [math]\displaystyle{ \mathcal G(3,0) }[/math]. Tuning maps such as the JIP do not belong in the algebra. Their size matters. Another factor we're missing is weights. When we projected the JIP to Meantone we measured distances in cents which is not always perceptually optimal. To get a POTE tuning we would need to Tenney-weigh the JIP and inverse-weigh the coordinates of [math]\displaystyle{ \overrightarrow{81/80} }[/math] before calculating the projection and inverse-weight the result. For the same result instead of inverse-weighing the pseudovector we could calculate Meantone [math]\displaystyle{ \overleftarrow{5} \wedge \overleftarrow{7} }[/math] in weighted coordinates.