Harmonic template
A harmonic template is a geometrical construction equivalent to a rank-2 mapping in regular temperament theory. Erv Wilson used harmonic templates to map scales onto a two-dimensional keyboard.[1][2] For example, the harmonic template:
. . . 2/1 . . . . 7/4 . . . 3/2 . . 5/4 . . . . . . . . . . 1/1 . . .
shows where 5/4, 3/2, 7/4, and 2/1 map onto the keyboard. Other ratios are mapped by writing them as a product of these octave-reduced harmonics, and moving by the step in the template for each factor (with 1/1 taken as the origin). So 15/8 = (5/4)⋅(3/2) is mapped to the sum of the positions of 5/4 and 3/2:
. 15/8 . 2/1 . . . . 7/4 . . . 3/2 . . 5/4 . . . . . . . . . . 1/1 . . .
The 22-tone scale in [3] is thus mapped as:
. 15/8 35/18 2/1 . . 5/3 27/16 7/4 9/5 . 40/27 3/2 14/9 8/5 5/4 35/27 4/3 112/81 64/45 10/9 9/8 7/6 6/5 . . 1/1 28/27 16/15 .
This construction is precisely equivalent to using the RTT mapping:
[math]\displaystyle{ \displaystyle \begin{bmatrix} 2 & 3 & 3 & 6 \\ 5 & 8 & 12 & 14 \\ \end{bmatrix} }[/math]
where the columns are the x and y coordinates that 2/1, 3/1, 5/1, and 7/1 would be mapped to on the keyboard. For example, 15/8 is mapped to:
[math]\displaystyle{ \displaystyle \begin{bmatrix} 2 & 3 & 3 & 6 \\ 5 & 8 & 12 & 14 \\ \end{bmatrix} \cdot \begin{bmatrix} -3 \\ 1 \\ 1 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 5 \\ \end{bmatrix} }[/math]
that is to x = 0, y = 5, agreeing with the geometrical construction.
The translation from harmonic template to RTT mapping is completely mechanical; every harmonic template can be directly expressed as an RTT mapping.
Temperaments
The following table shows the temperaments corresponding to some Wilson keyboard mappings:
| Harmonic template | Mapping | Commas | Temperament | Reference |
|---|---|---|---|---|
. . 7/4 . 2/1
11/8 . . . .
. . . 3/2 .
. . 5/4 . .
. . . . .
. . 1/1 . .
|
[math]\displaystyle{ \begin{bmatrix} 2 & 3 & 4 & 4 & 4 \\ 5 & 8 & 12 & 15 & 19 \\ \end{bmatrix} }[/math] | 81/80, 99/98, 126/125 |
Meantone | D'alessandro[4] |
7/4 . 2/1 .
. . . .
. 3/2 . .
5/4 . . .
. . . 11/8
1/1 . . .
|
[math]\displaystyle{ \begin{bmatrix} 2 & 3 & 4 & 4 & 9 \\ 5 & 8 & 12 & 15 & 16 \\ \end{bmatrix} }[/math] | 81/80, 126/125, 385/384 |
Meanpop | Inverted D'alessandro[4] |
. . 2/1 . .
. . . . .
. 3/2 . . 7/4
. . . . .
. . 5/4 . 11/8
1/1 . . . .
|
[math]\displaystyle{ \begin{bmatrix} 2 & 3 & 6 & 8 & 10 \\ 5 & 8 & 11 & 13 & 16 \\ \end{bmatrix} }[/math] | 100/99, 225/224, 245/242 |
Andromeda | Partch[3] |
. . . 2/1
. . . 7/4
. . 3/2 .
5/4 . . 11/8
. . . .
. 1/1 . .
|
[math]\displaystyle{ \begin{bmatrix} 2 & 3 & 3 & 6 & 8 \\ 5 & 8 & 12 & 14 & 17 \\ \end{bmatrix} }[/math] | 55/54, 64/63, 99/98 |
Suprapyth | Pascal[4] |
. . 7/4 . .
. . . . .
. . 13/8 . 2/1
. . . . .
. . 3/2 . .
. . . . .
. 5/4 . . .
. . . . .
. . . . .
1/1 . . 11/8 .
|
[math]\displaystyle{ \begin{bmatrix} 4 & 6 & 9 & 10 & 15 & 14 \\ 7 & 12 & 17 & 23 & 21 & 28 \\ \end{bmatrix} }[/math] | 169/168, 225/224, 325/324, 385/384 |
Catakleismic | Hebdomekontany[5] |
References
- ↑ Terumi Narushima, Microtonality and the Tuning Systems of Erv Wilson, Routledge (2017)
- ↑ Naren Ratan, Another look at Wilson's keyboard mapping system, Xenharmonikon Online (2026)
- ↑ 3.0 3.1 Erv Wilson, On the development of intonational systems by extended linear mapping, Xenharmonikon 3 (1975)
- ↑ 4.0 4.1 4.2 Erv Wilson, D'alessandro, like a Hurricane, Xenharmonikon 12 (1989)
- ↑ Erv Wilson, Hebdomekontany Notes