Tuning map: Difference between revisions
Error map |
This means just tuning map must be introduced earlier |
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From the generator tuning map <math>π</math> and the mapping <math>M</math>, we can obtain the tuning map <math>π</math> as <math>πM</math>. To go the other wayβββthat is, to find the generator tuning map from the (primes) tuning mapβββwe can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] <math>M^{+}</math>, as in <math>π = πM^{+}</math>. For more information, see the explanation [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_in_nonstandard_domains#9._Find_pseudoinverse|here]]. | From the generator tuning map <math>π</math> and the mapping <math>M</math>, we can obtain the tuning map <math>π</math> as <math>πM</math>. To go the other wayβββthat is, to find the generator tuning map from the (primes) tuning mapβββwe can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] <math>M^{+}</math>, as in <math>π = πM^{+}</math>. For more information, see the explanation [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_in_nonstandard_domains#9._Find_pseudoinverse|here]]. | ||
== With respect to JIP == | |||
{{Main| JIP }} | |||
[[JI]] can be conceptualized as the temperament where no intervals are made to [[vanish]], and as such, the untempered primes can be thought of as its generators, or of course its basis elements. So, JI subgroups have generator tuning maps and tuning maps too; the generator tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of the [[JIP]]. | |||
== 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 | 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. Β | ||
== Example == | == Example == | ||
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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 4Γ1 + (-1)Γ1.580 + (-1)Γ2.322 = 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 4Γ1 + (-1)Γ1.580 + (-1)Γ2.322 = 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 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| ]] <!-- main article --> | [[Category:Regular temperament tuning| ]] <!-- main article --> | ||