Tuning map: Difference between revisions
Deconfuse generator map and tuning map |
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== Example == | == 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 map of {{map| 1200.000 696.578 }}. | 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 map of {{map| 1200.000 696.578 }}. | ||
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== 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 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 == | == With respect to the JIP == | ||
[[JI]] can be conceptualized as the temperament where nothing is [[tempered out]], and as such, the untempered primes can be thought of as its generators. So, JI subgroups have tuning maps too; they are all subsets of the entries of the [[JIP]]. | [[JI]] can be conceptualized as the temperament where nothing is [[tempered out]], and as such, the untempered primes can be thought of as its generators. So, JI subgroups have tuning maps too; they are all subsets of the entries of the [[JIP]]. | ||
== 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. | |||
[[Category:Regular temperament theory]] | |||
[[Category:Terms]] | |||
[[Category:Math]] | |||
[[Category:Val]] | |||