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A harmonic template is a geometrical construction equivalent to a regular
A '''harmonic template''' is a geometrical construction equivalent to a [[rank and codimension|rank-2]] [[mapping]] in [[regular temperament theory]]. [[Erv Wilson]] used harmonic templates to map scales onto a two-dimensional keyboard.<ref name=narushima2017microtonality/><ref name=ratan2026another/> For example, the harmonic template:
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>
<pre>
Line 17: Line 10:
</pre>
</pre>


shows where 5/4, 3/2, 7/4, and 2/1 map onto the keyboard. Other ratios are
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:
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>
<pre>
Line 32: Line 21:
</pre>
</pre>


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


<pre>
<pre>
Line 45: Line 34:
This construction is precisely equivalent to using the RTT mapping:
This construction is precisely equivalent to using the RTT mapping:


<pre>
<math>\displaystyle
2 3 3 6
\begin{bmatrix}
5 8 12 14
2 & 3 & 3 & 6 \\
</pre>
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
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:
be mapped to on the keyboard. For example, 15/8 is mapped to:


<pre>
<math>\displaystyle
2 3 3 6 . -3  =  0
\begin{bmatrix}
5 8 12 14   1     5
2 & 3 & 3 & 6 \\
              1
5 & 8 & 12 & 14 \\
              0
\end{bmatrix}
</pre>
\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.
that is to ''x'' = 0, ''y'' = 5, agreeing with the geometrical construction.


The translation from harmonic template to RTT mapping is completely mechanical;
The translation from harmonic template to RTT mapping is completely mechanical; every harmonic template can be directly expressed as an RTT mapping.
every harmonic template can be directly expressed as an RTT mapping.


== Temperaments ==
== Temperaments ==
The following table shows the temperaments corresponding to some Wilson keyboard mappings:
The following table shows the temperaments corresponding to some Wilson keyboard mappings:


{| class="wikitable"
{| class="wikitable center-2"
! Harmonic template
! Harmonic template
! Mapping
! Mapping
Line 84: Line 84:
     .    .    1/1    .    .
     .    .    1/1    .    .
</pre>
</pre>
| <pre>
| <math>
2 3 4 4 4
\begin{bmatrix}
5 8 12 15 19
2 & 3 & 4 & 4 & 4 \\
</pre>
5 & 8 & 12 & 15 & 19 \\
| <pre>
\end{bmatrix}
81/80
</math>
99/98
| 81/80, <br>99/98, <br>126/125
126/125
| [[Meantone family #Undecimal meantone (huygens)|Meantone]]
</pre>
| D'alessandro<ref name=wilson1989dalessandro/>
| [[Meantone family#Undecimal meantone (huygens)|Meantone]]
| D'alessandro<ref name=wilson1989dalessandro />
|-
|-
| <pre>
| <pre>
Line 104: Line 102:
   1/1    .    .    .
   1/1    .    .    .
</pre>
</pre>
| <pre>
| <math>
2 3 4 4 9
\begin{bmatrix}
5 8 12 15 16
2 & 3 & 4 & 4 & 9 \\
</pre>
5 & 8 & 12 & 15 & 16 \\
| <pre>
\end{bmatrix}
81/80
</math>
126/125
| 81/80, <br>126/125, <br>385/384
385/384
</pre>
| [[Meanpop]]
| [[Meanpop]]
| Inverted D'alessandro<ref name=wilson1989dalessandro />
| Inverted D'alessandro<ref name=wilson1989dalessandro/>
|-
|-
| <pre>
| <pre>
Line 124: Line 120:
   1/1    .    .    .    .
   1/1    .    .    .    .
</pre>
</pre>
| <pre>
| <math>
2 3 6 8 10
\begin{bmatrix}
5 8 11 13 16
2 & 3 & 6 & 8 & 10 \\
</pre>
5 & 8 & 11 & 13 & 16 \\
| <pre>
\end{bmatrix}
100/99
</math>
225/224
| 100/99, <br>225/224, <br>245/242
245/242
</pre>
| [[Andromeda]]
| [[Andromeda]]
| Partch<ref name=wilson1975development />
| Partch<ref name=wilson1975development/>
|-
|-
| <pre>
| <pre>
Line 144: Line 138:
     .    1/1    .    .
     .    1/1    .    .
</pre>
</pre>
| <pre>
| <math>
2 3 3 6 8
\begin{bmatrix}
5 8 12 14 17
2 & 3 & 3 & 6 & 8 \\
</pre>
5 & 8 & 12 & 14 & 17 \\
| <pre>
\end{bmatrix}
55/54
</math>
64/63
| 55/54, <br>64/63, <br>99/98
99/98
</pre>
| [[Suprapyth]]
| [[Suprapyth]]
| Pascal<ref name=wilson1989dalessandro />
| Pascal<ref name=wilson1989dalessandro/>
|-
|-
| <pre>
| <pre>
Line 168: Line 160:
   1/1    .    .  11/8    .
   1/1    .    .  11/8    .
</pre>
</pre>
| <pre>
| <math>
4 6 9 10 15 14
\begin{bmatrix}
7 12 17 23 21 28
4 & 6 & 9 & 10 & 15 & 14 \\
</pre>
7 & 12 & 17 & 23 & 21 & 28 \\
| <pre>
\end{bmatrix}
169/168
</math>
225/224
| 169/168, <br>225/224, <br>325/324, <br>385/384
325/324
385/384
</pre>
</pre>
| [[Catakleismic]]
| [[Catakleismic]]
| Hebdomekontany<ref name=wilsonhebdomekontany />
| Hebdomekontany<ref name=wilsonhebdomekontany/>
|}
|}


== References ==
== References ==
Line 202: Line 191:
</references>
</references>


[[Category:Mapping]]
[[Category:Terms]]
[[Category:Erv Wilson]]
[[Category:Erv Wilson]]

Latest revision as of 10:53, 16 June 2026

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

  1. ↑ Terumi Narushima, Microtonality and the Tuning Systems of Erv Wilson, Routledge (2017)
  2. ↑ Naren Ratan, Another look at Wilson's keyboard mapping system, Xenharmonikon Online (2026)
  3. ↑ 3.0 3.1 Erv Wilson, On the development of intonational systems by extended linear mapping, Xenharmonikon 3 (1975)
  4. ↑ 4.0 4.1 4.2 Erv Wilson, D'alessandro, like a Hurricane, Xenharmonikon 12 (1989)
  5. ↑ Erv Wilson, Hebdomekontany Notes
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