Tuning map: Difference between revisions

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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.

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.

Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that's our answer for TE meantone.

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 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.
+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.
 Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that's our answer for TE meantone.