Wedgie/Archived version: Difference between revisions
m Simplified some links. TE error is already provided as redirect, makes it much easier to link to it. BTW: @IlL: Please don't use underscores for spaces in internal links! Thanks :) |
→Technical introduction: better link for regular temperaments - at least I hope so |
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We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone. | We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone. | ||
This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [http://en.wikipedia.org/wiki/Greatest_common_divisor GCD] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[ | This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [http://en.wikipedia.org/wiki/Greatest_common_divisor GCD] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s. | ||
These reduced n-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [http://en.wikipedia.org/wiki/Projective_space projective], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments. | These reduced n-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [http://en.wikipedia.org/wiki/Projective_space projective], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments. | ||