Tuning map: Difference between revisions

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The tuning map from {{nowrap|''G'' {{=}} {{map| 1200.000 696.578 }}}} and {{nowrap|''M'' {{=}} {{mapping| 1 1 0 | 0 1 4 }}}} is {{nowrap|''T'' {{=}} {{map| 1200.000 1896.578 2786.314 }}}}. For the error map we use {{nowrap|''J'' {{=}} {{val| 1200.000 1901.955 2786.314 }}}} and find {{nowrap|''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, {{nowrap|4×1200.000 + (~~−1~~) × 1896.578 + (~~−1~~) × 2786.314 {{=}} 117.108{{c}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap|~~4×0~~.000 + (~~−1~~) × (-5.377) + (~~−1~~) × 0.000 {{=}} +5.377{{c}}}}.

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, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap|4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}}}}.

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 {{nowrap|4 × 1201.397 + (~~−1~~) × 1898.446 + (~~−1~~) × 2788.196 {{=}} 118.946{{c}}}} and {{nowrap|4 × 1.397 + (~~−1~~) × (~~−3~~.509) + (~~−1~~) × 1.882 {{=}} +7.215{{c}}}}, respectively. 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 {{nowrap|4 × 1201.397 + (−1) × 1898.446 + (−1) × 2788.196 {{=}} 118.946{{c}}}} and {{nowrap|4 × 1.397 + (−1) × (−3.509) + (−1) × 1.882 {{=}} +7.215{{c}}}}, respectively. And that is our answer for TE meantone.

== Cents versus octaves ==

Sometimes you will see tuning maps given in octaves instead of cents. They work the same exact way. The only difference is that these octave-based tuning maps have each entry divided by 1200. For example, the quarter-comma meantone tuning map, in octaves, would be {{map|1200 1896.578 2786.314}}/1200 = {{map|1 1.580 2.322}}. If we dot product {{vector|4 -1 -1}} with that, we get {{nowrap|4 × 1 + (~~−1~~) × 1.580 + (~~−1~~) × 2.322 {{=}} 0.098}}, which tells us that 16/15 is a little less than 1/10 of an octave here.

Sometimes you will see tuning maps given in octaves instead of cents. They work the same exact way. The only difference is that these octave-based tuning maps have each entry divided by 1200. For example, the quarter-comma meantone tuning map, in octaves, would be {{map|1200 1896.578 2786.314}}/1200 = {{map|1 1.580 2.322}}. If we dot product {{vector|4 -1 -1}} with that, we get {{nowrap|4 × 1 + (−1) × 1.580 + (−1) × 2.322 {{=}} 0.098}}, which tells us that 16/15 is a little less than 1/10 of an octave here.

== With respect to linear algebra ==

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 The tuning map from {{nowrap|''G'' {{=}} {{map| 1200.000 696.578 }}}} and {{nowrap|''M'' {{=}} {{mapping| 1 1 0 | 0 1 4 }}}} is {{nowrap|''T'' {{=}} {{map| 1200.000 1896.578 2786.314 }}}}. For the error map we use {{nowrap|''J'' {{=}} {{val| 1200.000 1901.955 2786.314 }}}} and find {{nowrap|''&#400;'' {{=}} {{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, {{nowrap|4×1200.000 + (&minus;1) × 1896.578 + (&minus;1) × 2786.314 {{=}} 117.108{{c}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap|4×0.000 + (&minus;1) × (-5.377) + (&minus;1) × 0.000 {{=}} +5.377{{c}}}}.
+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, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap|4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}}}}.
-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 {{nowrap|4 × 1201.397 + (&minus;1) × 1898.446 + (&minus;1) × 2788.196 {{=}} 118.946{{c}}}} and {{nowrap|4 × 1.397 + (&minus;1) × (&minus;3.509) + (&minus;1) × 1.882 {{=}} +7.215{{c}}}}, respectively. 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 {{nowrap|4 × 1201.397 + (−1) × 1898.446 + (−1) × 2788.196 {{=}} 118.946{{c}}}} and {{nowrap|4 × 1.397 + (−1) × (−3.509) + (−1) × 1.882 {{=}} +7.215{{c}}}}, respectively. And that is our answer for TE meantone.
 == Cents versus octaves ==
-Sometimes you will see tuning maps given in octaves instead of cents. They work the same exact way. The only difference is that these octave-based tuning maps have each entry divided by 1200. For example, the quarter-comma meantone tuning map, in octaves, would be {{map|1200 1896.578 2786.314}}/1200 = {{map|1 1.580 2.322}}. If we dot product {{vector|4 -1 -1}} with that, we get {{nowrap|4 × 1 + (&minus;1) × 1.580 + (&minus;1) × 2.322 {{=}} 0.098}}, which tells us that 16/15 is a little less than 1/10 of an octave here.
+Sometimes you will see tuning maps given in octaves instead of cents. They work the same exact way. The only difference is that these octave-based tuning maps have each entry divided by 1200. For example, the quarter-comma meantone tuning map, in octaves, would be {{map|1200 1896.578 2786.314}}/1200 = {{map|1 1.580 2.322}}. If we dot product {{vector|4 -1 -1}} with that, we get {{nowrap|4 × 1 + (−1) × 1.580 + (−1) × 2.322 {{=}} 0.098}}, which tells us that 16/15 is a little less than 1/10 of an octave here.
 == With respect to linear algebra ==