Tuning map: Difference between revisions
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== Generator tuning map == | == Generator tuning map == | ||
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. | 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. | ||
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== Error map == | == Error map == | ||
An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually JI), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map. If we have an error map '' | An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually JI), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map. If we have an error map ''Ɛ'', tempered tuning map ''T'', and just tuning map ''J'', it follows that | ||
$$\mathcal{E} = T - J$$ | $$\mathcal{E} = T - J$$ | ||
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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 }}. | 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 }}. For the error map we use ''J'' = {{val| 1200.000 1901.955 2786.314 }} and find '' | 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, 4×1200.000 + ( | 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}}}}. | ||
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 | 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 == | == 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 | 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 == | == With respect to linear algebra == | ||
A tuning map can be thought of either as a one-row matrix or as a covector. The same is true of error maps and generator tuning maps. | A tuning map can be thought of either as a one-row matrix or as a covector. The same is true of error maps and generator tuning maps. | ||
[[Category:Regular temperament tuning| ]] <!-- | [[Category:Regular temperament tuning| ]] <!-- Main article --> | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Val]] | [[Category:Val]] | ||