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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>
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</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>
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</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>
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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
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     .    .    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>
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   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>
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   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>
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     .    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>
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   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 ==
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</references>
</references>


[[Category:Mapping]]
[[Category:Terms]]
[[Category:Erv Wilson]]
[[Category:Erv Wilson]]
Retrieved from "https://en.xen.wiki/w/Harmonic_template"