Tuning map: Difference between revisions

Line 14:

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

Consider meantone temperament, with the mapping {{rket|{{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 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 }}.

The tuning map from <math>𝒈</math> = {{map|1200.000 696.578}} and <math>M</math> = {{rket|{{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.

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.

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.

== 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 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 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 the JIP ==

@@ Line 14: / Line 14: @@
 == 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 generator tuning map of {{map| 1200.000 696.578 }}.
+Consider meantone temperament, with the mapping {{rket|{{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 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 }}.
+The tuning map from <math>𝒈</math> = {{map|1200.000 696.578}} and <math>M</math> = {{rket|{{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.
+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.
+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.
 == 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 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 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 the JIP ==