Tuning map: Difference between revisions

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It may be helpful, then, to think of the units of each entry of a generators tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.

From the generators tuning map <math>~~\textbf{g}~~</math> and the mapping <math>M</math>, we can obtain the tuning map <math>~~\textbf{t}~~</math> as <math>~~\textbf{g}~~.M</math>.

From the generators tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.

== Example ==

Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }}. Temperaments, as represented by mappings, remain abstract; while this mapping does convey that the generators are ~2/1 and ~3/2, it does not specify exact tunings for those approximations. One example tuning would be quarter-comma meantone, where the octave is pure and the perfect fifth is 5<sup>1/4</sup>; this gives a generators tuning map of {{map| 1200.000 696.578 }}.

The tuning map from <math>~~\textbf{g}~~</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>~~\textbf{t}~~</math> = {{map| 1200.000 1896.578 2786.314 }}.

The tuning map from <math>𝒈</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>𝒕</math> = {{map| 1200.000 1896.578 2786.314 }}.

So, to answer the question, "how many cents is the approximation of the interval 16/15 in quarter-comma meantone?" we use the dot product to map 16/15's [[prime-count vector]] {{vector| 4 -1 -1 }} via the tuning map given above, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.

@@ Line 11: / Line 11: @@
 It may be helpful, then, to think of the units of each entry of a generators tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.
-From the generators tuning map <math>\textbf{g}</math> and the mapping <math>M</math>, we can obtain the tuning map <math>\textbf{t}</math> as <math>\textbf{g}.M</math>.
+From the generators tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.
 == Example ==
 Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }}. Temperaments, as represented by mappings, remain abstract; while this mapping does convey that the generators are ~2/1 and ~3/2, it does not specify exact tunings for those approximations. One example tuning would be quarter-comma meantone, where the octave is pure and the perfect fifth is 5<sup>1/4</sup>; this gives a generators tuning map of {{map| 1200.000 696.578 }}.
-The tuning map from <math>\textbf{g}</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>\textbf{t}</math> = {{map| 1200.000 1896.578 2786.314 }}.
+The tuning map from <math>𝒈</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>𝒕</math> = {{map| 1200.000 1896.578 2786.314 }}.
 So, to answer the question, "how many cents is the approximation of the interval 16/15 in quarter-comma meantone?" we use the dot product to map 16/15's [[prime-count vector]] {{vector| 4 -1 -1 }} via the tuning map given above, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.