Harmonic template
A harmonic template is a geometrical construction equivalent to a regular temperament theory mapping.
Keyboard mapping
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:
2 3 3 6 5 8 12 14
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:
2 3 3 6 . -3 = 0
5 8 12 14 1 5
1
0
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 . .
|
2 3 4 4 4 5 8 12 15 19 |
81/80 99/98 126/125 |
Meantone | D'alessandro[4] |
7/4 . 2/1 .
. . . .
. 3/2 . .
5/4 . . .
. . . 11/8
1/1 . . .
|
2 3 4 4 9 5 8 12 15 16 |
81/80 126/125 385/384 |
Meanpop | Inverted D'alessandro[4] |
. . 2/1 . .
. . . . .
. 3/2 . . 7/4
. . . . .
. . 5/4 . 11/8
1/1 . . . .
|
2 3 6 8 10 5 8 11 13 16 |
100/99 225/224 245/242 |
Andromeda | Partch[3] |
. . . 2/1
. . . 7/4
. . 3/2 .
5/4 . . 11/8
. . . .
. 1/1 . .
|
2 3 3 6 8 5 8 12 14 17 |
55/54 64/63 99/98 |
Suprapyth | Pascal[4] |
. . 7/4 . .
. . . . .
. . 13/8 . 2/1
. . . . .
. . 3/2 . .
. . . . .
. 5/4 . . .
. . . . .
. . . . .
1/1 . . 11/8 .
|
4 6 9 10 15 14 7 12 17 23 21 28 |
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