Tuning map: Difference between revisions

Line 33:

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 {{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.

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}}}}.

@@ Line 33: / Line 33: @@
 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 {{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.
+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}}}}.