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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.<ref name=narushima2017microtonality /><ref name=ratan2026another />
For example, the harmonic template:

<pre>
. . . 2/1 .
. . . 7/4 .
. . 3/2 . .
5/4 . . . .
. . . . .
. 1/1 . . .
</pre>

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:

<pre>
. 15/8 . 2/1 .
. . . 7/4 .
. . 3/2 . .
5/4 . . . .
. . . . .
. 1/1 . . .
</pre>

The 22-tone scale in <ref name=wilson1975development /> is thus mapped as:

<pre>
. 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 .
</pre>

This construction is precisely equivalent to using the RTT mapping:

<pre>
2 3 3 6
5 8 12 14
</pre>

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:

<pre>
2 3 3 6 . -3 = 0
5 8 12 14 1 5
1
0
</pre>

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:

{| class="wikitable"
! Harmonic template
! Mapping
! Commas
! Temperament
! Reference
|-
| <pre>
. . 7/4 . 2/1
11/8 . . . .
. . . 3/2 .
. . 5/4 . .
. . . . .
. . 1/1 . .
</pre>
| <pre>
2 3 4 4 4
5 8 12 15 19
</pre>
| <pre>
81/80
99/98
126/125
</pre>
| [[Meantone family#Undecimal meantone (huygens)|Meantone]]
| D'alessandro<ref name=wilson1989dalessandro />
|-
| <pre>
7/4 . 2/1 .
. . . .
. 3/2 . .
5/4 . . .
. . . 11/8
1/1 . . .
</pre>
| <pre>
2 3 4 4 9
5 8 12 15 16
</pre>
| <pre>
81/80
126/125
385/384
</pre>
| [[Meanpop]]
| Inverted D'alessandro<ref name=wilson1989dalessandro />
|-
| <pre>
. . 2/1 . .
. . . . .
. 3/2 . . 7/4
. . . . .
. . 5/4 . 11/8
1/1 . . . .
</pre>
| <pre>
2 3 6 8 10
5 8 11 13 16
</pre>
| <pre>
100/99
225/224
245/242
</pre>
| [[Andromeda]]
| Partch<ref name=wilson1975development />
|-
| <pre>
. . . 2/1
. . . 7/4
. . 3/2 .
5/4 . . 11/8
. . . .
. 1/1 . .
</pre>
| <pre>
2 3 3 6 8
5 8 12 14 17
</pre>
| <pre>
55/54
64/63
99/98
</pre>
| [[Suprapyth]]
| Pascal<ref name=wilson1989dalessandro />
|-
| <pre>
. . 7/4 . .
. . . . .
. . 13/8 . 2/1
. . . . .
. . 3/2 . .
. . . . .
. 5/4 . . .
. . . . .
. . . . .
1/1 . . 11/8 .
</pre>
| <pre>
4 6 9 10 15 14
7 12 17 23 21 28
</pre>
| <pre>
169/168
225/224
325/324
385/384
</pre>
| [[Catakleismic]]
| Hebdomekontany<ref name=wilsonhebdomekontany />
|}

== References ==
<references>
<ref name=narushima2017microtonality>
Terumi Narushima, ''Microtonality and the Tuning Systems of Erv Wilson'', Routledge (2017)
</ref>
<ref name=ratan2026another>
Naren Ratan, [https://www.xenharmonikon.org/2026/05/19/another-look-at-wilsons-keyboard-mapping-system/ Another look at Wilson's keyboard mapping system], Xenharmonikon Online (2026)
</ref>
<ref name=wilson1975development>
Erv Wilson, [https://www.anaphoria.com/xen3b.pdf On the development of intonational systems by extended linear mapping], Xenharmonikon 3 (1975)
</ref>
<ref name=wilson1989dalessandro>
Erv Wilson, [https://www.anaphoria.com/dal.pdf D'alessandro, like a Hurricane], Xenharmonikon 12 (1989)
</ref>
<ref name=wilsonhebdomekontany>
Erv Wilson, [https://www.anaphoria.com/Hebdomekontany.pdf Hebdomekontany Notes]
</ref>
</references>

[[Category:Erv Wilson]]

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