Douglas Blumeyer's RTT How-To: Difference between revisions

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We haven’t specified the size of either of these generators, but that’s not important here. These mappings are just like a set of requirements for any pair of generators that might implement this temperament. This is as good a time as any to emphasize the fact that temperaments are abstract; they are not ready-to-go tunings, but more like instructions for a tuning to follow. This can sometimes feel frustrating or hard to understand, but ultimately it’s a bit part of the power of temperament theory.

The critical thing here is that if {{monzo|-4 4 -1}} is mapped to 0 steps by {{val|5 8 12}} individually and to 0 steps by {{val|7 11 16}} individually, then in total it comes out to 0 steps in the temperament, and thus is tempered out, or has vector {{monzo|0 0~~}}, or {{monzo|~~}}.

The critical thing here is that if {{monzo|-4 4 -1}} is mapped to 0 steps by {{val|5 8 12}} individually and to 0 steps by {{val|7 11 16}} individually, then in total it comes out to 0 steps in the temperament, and thus is tempered out, or has vector {{monzo|0 0}}.

And how about something silly like {{monzo|1}}, the octave? Well, that maps to {{monzo|5 7}}. If the first generator was the exact same size as 5-ET’s generator, and the second generator was the exact same size as 7-ET’s generator, then 5 of the first and 7 of the second wouldn’t take us to the octave, it would take us to the double octave, 4/1. So maybe the first generator is the size of half a step of 5-ET, maybe? And the second generator is the size as half a step of 7-ET, maybe? I guess that would work out, since 120¢ + 85.714¢ = 205.714¢ and 10/9 is 182.404¢; I mean, it’s not super close, but it’s in the ballpark at least.

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Note well: this is not to say that {{val|1 1 0}} or {{val|0 1 4}} are the generators for meantone. These are generator mappings. They describe behavior of the generators. But they are not themselves the generators.

=== JI as a temperament ===

@@ Line 786: / Line 786: @@
 We haven’t specified the size of either of these generators, but that’s not important here. These mappings are just like a set of requirements for any pair of generators that might implement this temperament. This is as good a time as any to emphasize the fact that temperaments are abstract; they are not ready-to-go tunings, but more like instructions for a tuning to follow. This can sometimes feel frustrating or hard to understand, but ultimately it’s a bit part of the power of temperament theory.
-The critical thing here is that if {{monzo|-4 4 -1}} is mapped to 0 steps by {{val|5 8 12}} individually and to 0 steps by {{val|7 11 16}} individually, then in total it comes out to 0 steps in the temperament, and thus is tempered out, or has vector {{monzo|0 0}}, or {{monzo|}}.
+The critical thing here is that if {{monzo|-4 4 -1}} is mapped to 0 steps by {{val|5 8 12}} individually and to 0 steps by {{val|7 11 16}} individually, then in total it comes out to 0 steps in the temperament, and thus is tempered out, or has vector {{monzo|0 0}}.
 And how about something silly like {{monzo|1}}, the octave? Well, that maps to {{monzo|5 7}}. If the first generator was the exact same size as 5-ET’s generator, and the second generator was the exact same size as 7-ET’s generator, then 5 of the first and 7 of the second wouldn’t take us to the octave, it would take us to the double octave, 4/1. So maybe the first generator is the size of half a step of 5-ET, maybe? And the second generator is the size as half a step of 7-ET, maybe? I guess that would work out, since 120¢ + 85.714¢ = 205.714¢ and 10/9 is 182.404¢; I mean, it’s not super close, but it’s in the ballpark at least.
@@ Line 847: / Line 847: @@
 Note well: this is not to say that {{val|1 1 0}} or {{val|0 1 4}} are the generators for meantone. These are generator mappings. They describe behavior of the generators. But they are not themselves the generators.
 === JI as a temperament ===