Harmonic template: Difference between revisions
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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 | |||
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: | |||
<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: | ||
< | <math>\displaystyle | ||
2 | \begin{bmatrix} | ||
5 | 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 | 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: | |||
< | <math>\displaystyle | ||
2 | \begin{bmatrix} | ||
5 | 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. | 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> | ||
| < | | <math> | ||
2 | \begin{bmatrix} | ||
5 | 2 & 3 & 4 & 4 & 4 \\ | ||
</ | 5 & 8 & 12 & 15 & 19 \\ | ||
| | \end{bmatrix} | ||
81/80 | </math> | ||
99/98 | | 81/80, <br>99/98, <br>126/125 | ||
126/125 | | [[Meantone family #Undecimal meantone (huygens)|Meantone]] | ||
| 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> | ||
| < | | <math> | ||
2 | \begin{bmatrix} | ||
5 | 2 & 3 & 4 & 4 & 9 \\ | ||
</ | 5 & 8 & 12 & 15 & 16 \\ | ||
| | \end{bmatrix} | ||
81/80 | </math> | ||
126/125 | | 81/80, <br>126/125, <br>385/384 | ||
385/384 | |||
| [[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> | ||
| < | | <math> | ||
2 | \begin{bmatrix} | ||
5 | 2 & 3 & 6 & 8 & 10 \\ | ||
</ | 5 & 8 & 11 & 13 & 16 \\ | ||
| | \end{bmatrix} | ||
100/99 | </math> | ||
225/224 | | 100/99, <br>225/224, <br>245/242 | ||
245/242 | |||
| [[Andromeda]] | | [[Andromeda]] | ||
| Partch<ref name=wilson1975development /> | | Partch<ref name=wilson1975development/> | ||
|- | |- | ||
| <pre> | | <pre> | ||
| Line 144: | Line 138: | ||
. 1/1 . . | . 1/1 . . | ||
</pre> | </pre> | ||
| < | | <math> | ||
2 | \begin{bmatrix} | ||
5 | 2 & 3 & 3 & 6 & 8 \\ | ||
</ | 5 & 8 & 12 & 14 & 17 \\ | ||
| | \end{bmatrix} | ||
55/54 | </math> | ||
64/63 | | 55/54, <br>64/63, <br>99/98 | ||
99/98 | |||
| [[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> | ||
| < | | <math> | ||
4 | \begin{bmatrix} | ||
7 12 17 23 21 28 | 4 & 6 & 9 & 10 & 15 & 14 \\ | ||
</ | 7 & 12 & 17 & 23 & 21 & 28 \\ | ||
| | \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]] | ||