Regular temperament theory notes

These notes present regular temperament theory as an extension of Erv Wilson's keyboard mapping theory. This gives a concrete, playable interpretation of some otherwise rather abstract parts of the theory. The notes use ordinary matrix notation and standard mathematical terminology.

Problem

Our musical problem is to play a just intonation scale, or some approximation of it, on a two-dimensional keyboard. For example, we might lay out this 19-tone scale as follows: 

64/45   8/5    9/5    2/1     .      .

  .   112/81  14/9    7/4   35/18    .

16/15   6/5    4/3    3/2   27/16  15/8

  .    28/27   7/6   35/27    .      .

  .      .     1/1    9/8    5/4     .
Figure 1: Mapping of a 19-tone scale on a two-dimensional keyboard

Harmonic template mapping

It's helpful for navigation if the same interval always makes the same shape on the keyboard. For example, in the keyboard mapping of Figure 1, the interval 5/4 is found between:

and in all cases these notes are separated by a move $[2, 0]$ on the keyboard, that is two columns across and on the same row.

One way to ensure this property is to map each ratio based on its prime factors as follows. Write say $35/18$ as

$$ \frac{35}{18} = 2^{-1} \cdot 3^{-2} \cdot 5^{1} \cdot 7^{1} $$

and collect the exponents into a vector

$$ V = \begin{bmatrix} -1 \\ -2 \\ 1 \\ 1 \end{bmatrix} $$

Then find the $x$ and $y$ coordinates on the keyboard, $X = [x, y]$, by multiplying by some matrix $A$,

$$ X = A V $$

The keyboard mapping of Figure 1 uses this method with

$$ A = \begin{bmatrix} 1 & 2 & 4 & 3 \\ 4 & 6 & 8 & 11 \end{bmatrix} $$

Calculating the keyboard position for 35/18, we get

$$ \begin{aligned} X &= A V \\ &= \begin{bmatrix} 1 & 2 & 4 & 3 \\ 4 & 6 & 8 & 11 \end{bmatrix} \begin{bmatrix} -1 \\ -2 \\ 1 \\ 1 \end{bmatrix} \\ &= \begin{bmatrix} 2 \\ 3 \end{bmatrix} \end{aligned} $$

and in the keyboard mapping 35/18 really does get mapped to $x = 2$, $y = 3$ (where the origin is the position of 1/1).

The matrix $A$ is an algebraic version of the geometrical harmonic template used by Erv Wilson.

Pitch grids

We might further want the steps in pitch when moving along a row, or up a column, to be approximately equal. This means playing the same shape anywhere on the keyboard gives a similar sounding chord.

Looking at our example keyboard mapping, the cents values for the scale and the $x$ and $y$ step sizes to reach each note are: 

  610   814  1018  1200     .     .
    .   561   765   969  1151     .
  112   316   498   702   906  1088
    .    63   267   449     .     .
    .     .     0   204   386     .

Scale cents

    .   204   204   182     .     .
    .     .   204   204   182     .
    .   204   182   204   204   182
    .     .   204   182     .     .
    .     .     .   204   182     .

$x$ step sizes

    .   253   253   231     .     .
    .   245   267   267   245     .
    .   253   231   253     .     .
    .     .   267   245     .     .
    .     .     .     .     .     .

$y$ step sizes

Figure 2: The $x$ and $y$ step sizes for the keyboard mapping of Figure 1

The range of step sizes is fairly small. The $x$ step sizes (from moving along a row) are all either 182 cents or 204 cents (10/9 or 9/8), giving a range of 22 cents. The $y$ step sizes lie between 231 cents and 267 cents, a range of 36 cents.

We can control the range of step sizes by asking for all notes to lie within some tolerance $\epsilon$ cents of a grid of pitches

$$ p = s x + t y $$

where $x$ and $y$ are the (integer) coordinates on the keyboard and $s$ and $t$ are some step sizes in cents. For example, our keyboard mapping has all notes within 20 cents of the pitch grid with step sizes $s = 195.32$ cents and $t = 251.17$ cents: 

 610   814  1018  1200     .     .
   .   561   765   969  1151     .
 112   316   498   702   906  1088
   .    63   267   449     .     .
   .     .     0   204   386     .

Actual cents

 614   809  1005  1200  1395  1591
 363   558   754   949  1144  1339
 112   307   502   698   893  1088
-139    56   251   446   642   837
-391  -195     0   195   391   586

Pitch grid

Figure 3: A pitch grid (right) within 20 cents of the just scale (left)

The ranges of $x$ and $y$ step sizes are necessarily less than $4 \epsilon$ cents, and are typically smaller; in our case they are 22 cents and 36 cents, both less than $4 \cdot 20 = 80$ cents.

Temperaments

While we can very well play our keyboard mapping of the just scale, it's now a short step to play the pitches of the pitch grid itself. On the pitch grid, the same shape always plays exactly the same chord; it's isomorphic. The pitch grid also assigns pitches to keys that were unmapped for our just scale, so we can move around (transpose) shapes freely.

Physically you can think of a pitch grid as a guitar with frets all separated by the same interval $s$ and strings all separated by the same interval $t$. I play such a guitar with frets spaced by 68 cent 'third tones' and strings tuned in 317 cent minor thirds.

Temperament optimization

It's convenient if our scale fits in a small area on the keyboard, so we can reach its notes. For a given error tolerance $\epsilon$, we might ask for both the $A$ which maps the scale into the smallest area on the keyboard, and the pitch grid steps $s$ and $t$ which approximate the scale to within $\epsilon$ cents. This is an optimization problem we can solve on a computer.

Table 1: Temperament optimization for a 19-tone scale
Error tolerance $\epsilon$ Keyboard mapping $A$ $s$ $t$
20 
64/45   8/5    9/5    2/1     .      .

  .   112/81  14/9    7/4   35/18    .

16/15   6/5    4/3    3/2   27/16  15/8

  .    28/27   7/6   35/27    .      .

  .      .     1/1    9/8    5/4     .
$ \begin{bmatrix} 1 & 2 & 4 & 3 \\ 4 & 6 & 8 & 11 \end{bmatrix} $ 195.32 251.17
15 
  .    27/16   7/4    9/5   15/8   35/18   2/1     .

  .     .      .      3/2   14/9    8/5     .      .

 9/8   7/6    6/5     5/4   35/27   4/3  112/81  64/45

  .     .     1/1    28/27  16/15    .      .      .
$ \begin{bmatrix} 4 & 5 & 9 & 8 \\ 3 & 5 & 7 & 9 \end{bmatrix} $ 58.82 321.57
10 
  .      .    15/8    2/1     .      .      .      .      .

  .      .      .      .      .    35/18    .      .      .

 5/4    4/3   64/45   3/2    8/5   27/16   9/5     .      .

  .      .      .    35/27 112/81    .    14/9     .     7/4

  .     1/1   16/15   9/8    6/5     .      .      .      .

  .      .      .      .    28/27    .     7/6     .      .
$ \begin{bmatrix} 2 & 4 & 3 & 11 \\ 4 & 6 & 10 & 9 \end{bmatrix} $ 103.34 248.22

Table 1 shows three keyboard mappings and their associated matrices $A$ and pitch grid step sizes $s$ and $t$, calculated by minimizing the scale's area on the keyboard with a given error tolerance $\epsilon$. To get a lower error, we need a more spread-out keyboard mapping. We say a temperament has a high complexity if it maps the scale into a large area. So a lower error comes with a higher complexity.

Commas

Sometimes two notes in the just scale are mapped to the same key on the keyboard. In this case we have a few musical choices; we could for example use a split key to play both notes, or choose different notes in different octaves. When using the tempered pitches of the pitch grid, notes mapped to the same key get exactly the same pitch, so we cannot distinguish them.

If two ratios $r_{1}$ and $r_{2}$ are mapped to the same key, their ratio $r_{2}/r_{1}$ is mapped to $[0, 0]$ (since $X = AV$ is linear). In this case $c = r_{2}/r_{1}$ is called a tempered-out comma (or simply a comma), as is any ratio with $X = AV = [0, 0]$.

Some people make compositional use of the tempered-out commas, for example when coming up with chord progressions. On the other hand, as soon as you have a keyboard mapping or pitch grid, you can set about playing it; thinking about commas is not required to apply the theory musically.

Naren Ratan