Mapping: Difference between revisions
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=Intro | == Intro to mappings == | ||
''When one val just won't do...'' | ''When one val just won't do...'' | ||
A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated | A [[val]] maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated [[EDO]]s and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough. | ||
== Temperamental rank == | |||
'''A temperament's "rank" denotes how many independent chains of''' '''generators exist within the temperament.''' This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the "dimensionality" of the temperament. | '''A temperament's "rank" denotes how many independent chains of''' '''generators exist within the temperament.''' This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the "dimensionality" of the temperament. | ||
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A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank > 1, we simply need to use more than one val. '''In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.''' | A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank > 1, we simply need to use more than one val. '''In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.''' | ||
=Example= | == Example == | ||
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form: | At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form: | ||
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This is, in fact, the mapping matrix for meantone temperament, which is what we wanted. | This is, in fact, the mapping matrix for meantone temperament, which is what we wanted. | ||
=Change of | == Change of basis == | ||
In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament. | In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament. | ||
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[[Category:Theory]] | |||
[[Category:Math]] | |||
[[Category:todo:increase focus to lemma]] | |||