Tuning map: Difference between revisions

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Consider meantone temperament, with the mapping {{mapping| 1 1 0 | 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 tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}.

The tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}. For the error map we use ''J'' = {{val| 1200.000 1901.955 386.314 }} and find ''Ɛ'' = {{val| 0.000 -5.377 0.000 }}, showing us prime 3 is tempered flat by 5.377 cents while primes 2 and 5 are pure.

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. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: 4×0.000 + (-1)×(-5.377) + (-1)×0.000 = +5.377 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 is our answer for TE meantone.

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 }} and error map of {{val| +1.397 -3.509 +1.882 }}. To answer the same questions about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with these different tuning and error maps. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 118.946 cents and 4×1.397 + (-1)×(-3.509) + (-1)×1.882 = +7.215 cents, respectively. And that is our answer for TE meantone.

== Cents versus octaves ==

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 Consider meantone temperament, with the mapping {{mapping| 1 1 0 | 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 tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}.
+The tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}. For the error map we use ''J'' = {{val| 1200.000 1901.955 386.314 }} and find ''Ɛ'' = {{val| 0.000 -5.377 0.000 }}, showing us prime 3 is tempered flat by 5.377 cents while primes 2 and 5 are pure.
-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. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: 4×0.000 + (-1)×(-5.377) + (-1)×0.000 = +5.377 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 is our answer for TE meantone.
+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 }} and error map of {{val| +1.397 -3.509 +1.882 }}. To answer the same questions about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with these different tuning and error maps. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 118.946 cents and 4×1.397 + (-1)×(-3.509) + (-1)×1.882 = +7.215 cents, respectively. And that is our answer for TE meantone.
 == Cents versus octaves ==