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

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We’ve made it to a critical point here: we are now able to explain why RTT is called “regular” temperament theory. Regular here is a mathematical term, and I don’t have a straightforward definition of it for you, but it apparently refers to the fact that all intervals in the tuning are combinations of only these specified generators. So there you go.

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. It can ~~get~~ confusing, because ~~it's certainly tempting~~ to ~~say~~ {{~~val~~|1 1 0}} ~~"is" the octave or the period and that~~ {{val|0 ~~1 4~~}} ~~"is"~~ the ~~fifth~~ or the ~~generator~~, because {{monzo|1 0 0}} maps to {{monzo|1 0~~}} and {{monzo|-~~1 ~~1 0~~}} ~~maps~~ to {{~~monzo~~|0 1}}~~. And maybe~~ when the context is clear it ~~may even be acceptable~~ to ~~say~~ that~~. I suppose if~~ "~~2/1 maps to one~~ of the ~~first~~ generator (and ~~zero of~~ the second generator)" ~~it may be to some extent acceptable to say that~~ the first generator ~~"is 2/~~1". ~~And~~ if ~~we're looking at this~~ mapping ~~and you ask me, "which generator is the octave~~?~~" I~~'d ~~be hard-pressed to find a better answer than "it's~~ the {{val|1 1 0}} ~~one"~~, but ~~some folks may~~ be ~~less tolerant of such stretches~~ of the ~~terminology~~.

Note well: this is not to say that {{val|1 1 0}} or {{val|0 1 4}} ''are'' the generators for meantone. They are generator ''mappings'': when assembled together, they collectively describe behavior of the generators, but they are ''not'' themselves the generators. This situation can be confusing; it confused me for many weeks. I thought of it this way: because the first generator is 2/1 — i.e. {{monzo|1 0 0}} maps to {{monzo|1 0}} — referring to {{val|1 1 0}} as the octave or period seems reasonable and is effective when the context is clear. And similarly, because the second generator is 3/2 — i.e. {{monzo|-1 1 0}} maps to {{monzo|0 1}} — referring to {{val|0 1 4}} as the fifth or the generator seems reasonable as is effective when the context is clear. But it's critical to understand that the first generator "being" the octave here is ''contingent upon the definition of the second generator'', and vice versa, the second generator "being" the fifth here is ''contingent upon the definition of the first generator''. Considering {{val|1 1 0}} or {{val|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{monzo|{{val|0 1 4}} {{val|1 2 4}}} instead? We'd still have the first generator mapping as {{val|1 1 0}}, but now that the second generator mapping is {{val|1 2 4}}, the two generators must be the fourth and the fifth. In summary, neither mapping row describes a generator in a vacuum, but does so in the context of all the other mapping rows.

=== JI as a temperament ===

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 We’ve made it to a critical point here: we are now able to explain why RTT is called “regular” temperament theory. Regular here is a mathematical term, and I don’t have a straightforward definition of it for you, but it apparently refers to the fact that all intervals in the tuning are combinations of only these specified generators. So there you go.
-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. It can get confusing, because it's certainly tempting to say {{val|1 1 0}} "is" the octave or the period and that {{val|0 1 4}} "is" the fifth or the generator, because {{monzo|1 0 0}} maps to {{monzo|1 0}} and {{monzo|-1 1 0}} maps to {{monzo|0 1}}. And maybe when the context is clear it may even be acceptable to say that. I suppose if "2/1 maps to one of the first generator (and zero of the second generator)" it may be to some extent acceptable to say that the first generator "is 2/1". And if we're looking at this mapping and you ask me, "which generator is the octave?" I'd be hard-pressed to find a better answer than "it's the {{val|1 1 0}} one", but some folks may be less tolerant of such stretches of the terminology.
+Note well: this is not to say that {{val|1 1 0}} or {{val|0 1 4}} ''are'' the generators for meantone. They are generator ''mappings'': when assembled together, they collectively describe behavior of the generators, but they are ''not'' themselves the generators. This situation can be confusing; it confused me for many weeks. I thought of it this way: because the first generator is 2/1 — i.e. {{monzo|1 0 0}} maps to {{monzo|1 0}} — referring to {{val|1 1 0}} as the octave or period seems reasonable and is effective when the context is clear. And similarly, because the second generator is 3/2 — i.e. {{monzo|-1 1 0}} maps to {{monzo|0 1}} — referring to {{val|0 1 4}} as the fifth or the generator seems reasonable as is effective when the context is clear. But it's critical to understand that the first generator "being" the octave here is ''contingent upon the definition of the second generator'', and vice versa, the second generator "being" the fifth here is ''contingent upon the definition of the first generator''. Considering {{val|1 1 0}} or {{val|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{monzo|{{val|0 1 4}} {{val|1 2 4}}} instead? We'd still have the first generator mapping as {{val|1 1 0}}, but now that the second generator mapping is {{val|1 2 4}}, the two generators must be the fourth and the fifth. In summary, neither mapping row describes a generator in a vacuum, but does so in the context of all the other mapping rows.
 === JI as a temperament ===