Tuning map: Difference between revisions

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A '''tuning map''' represents the tuning of a [[regular temperament]]. It can take a vector representation of an interval ([[monzo]]) as input and outputs its pitch, usually measured in cents or octaves. A tuning map has one entry for each [[formal prime]] of the temperament, giving its size in cents or octaves.

== ~~Example~~ ==

== Generator tuning map ==

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 '''generator tuning map''' ~~of {{map| 1200.000 696.578 }}. The generator tuning map~~ is like a tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.

A '''generator tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.

From the generator tuning map G and the ~~temperament~~ mapping V, we can obtain the tuning map T:

From the generator 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>.

<~~math~~>

== Example ==

~~T = GV~~

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 generator tuning map of {{map| 1200.000 696.578 }}.

</~~math~~>

The tuning map from G = {{map| 1200.000 696.578 }} and V = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is {{map| 1200.000 1896.578 2786.314 }}.

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 }}.

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.

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 A '''tuning map''' represents the tuning of a [[regular temperament]]. It can take a vector representation of an interval ([[monzo]]) as input and outputs its pitch, usually measured in cents or octaves. A tuning map has one entry for each [[formal prime]] of the temperament, giving its size in cents or octaves.
-== Example ==
+== Generator tuning map ==
-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 '''generator tuning map''' of {{map| 1200.000 696.578 }}. The generator tuning map is like a tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.
+A '''generator tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.
-From the generator tuning map G and the temperament mapping V, we can obtain the tuning map T:
+From the generator 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>.
-<math>
+== Example ==
-T = GV
+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 generator tuning map of {{map| 1200.000 696.578 }}.
-</math>
-The tuning map from G = {{map| 1200.000 696.578 }} and V = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is {{map| 1200.000 1896.578 2786.314 }}.
+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 }}.
 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.