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

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Hopefully manipulating these rows like this gives you some sort of feel for how what matters in a temperament mapping is not so much the absolute values but their relationship with each other.

~~Both {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} and {{monzo|{{val|1 1 0}} {{val|0 1 4}}}} are equivalent mappings~~, ~~then. Converting between them we could call~~ a ~~change~~ of ~~basis. This makes more sense,~~ of ~~course, when speaking about converting between equivalent bases; I’ve been cautioned against referring to maps as “bases” despite the label seeming appropriate from an analogy standpoint.~~

To conclude this section, I have a barrage of unrelated points of order:

This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{val|12 19 28}} was simply {{val|5 8 12}} + {{val|7 11 16}}? Well, if {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size<ref>For real numbers <math>p,q</math> we can make the two generators respectively <math>\frac{p}{5p+7q}</math> and <math>\frac{q}{5p+7q}</math> of an octave, e.g. <math>(p,q)=(1,0)</math> for 5-ET, <math>(0,1)</math> for 7-ET, <math>(1,1)</math> for 12-ET, and many other possibilities.</ref>. If they become the same size, then they aren’t truly two separate generators, or at least there’s no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one.

* 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.

* Both {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} and {{monzo|{{val|1 1 0}} {{val|0 1 4}}}} are equivalent mappings, then. Converting between them we could call a change of basis. This makes more sense, of course, when speaking about converting between equivalent bases; I’ve been cautioned against referring to maps as “bases” despite the label seeming appropriate from an analogy standpoint.

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. 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.

* This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{val|12 19 28}} was simply {{val|5 8 12}} + {{val|7 11 16}}? Well, if {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size<ref>For real numbers <math>p,q</math> we can make the two generators respectively <math>\frac{p}{5p+7q}</math> and <math>\frac{q}{5p+7q}</math> of an octave, e.g. <math>(p,q)=(1,0)</math> for 5-ET, <math>(0,1)</math> for 7-ET, <math>(1,1)</math> for 12-ET, and many other possibilities.</ref>. If they become the same size, then they aren’t truly two separate generators, or at least there’s no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one.

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.

* Technically speaking, when we first learned how to map vectors with ETs, we could think of those outputs as vectors too, but they'd be 1-dimensional vectors, i.e. if 12-ET maps 16/15 to 1 step, we could write that as {{val|12 19 28}}{{monzo|4 -1 -1}} = {{monzo|1}}, where writing the answer as {{monzo|1}} expresses that 1 step as 1 of the only generator in this equal temperament.

=== JI as a temperament ===

@@ Line 913: / Line 913: @@
 Hopefully manipulating these rows like this gives you some sort of feel for how what matters in a temperament mapping is not so much the absolute values but their relationship with each other.
-Both {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} and {{monzo|{{val|1 1 0}} {{val|0 1 4}}}} are equivalent mappings, then. Converting between them we could call a change of basis. This makes more sense, of course, when speaking about converting between equivalent bases; I’ve been cautioned against referring to maps as “bases” despite the label seeming appropriate from an analogy standpoint.
+To conclude this section, I have a barrage of unrelated points of order:
-This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{val|12 19 28}} was simply {{val|5 8 12}} + {{val|7 11 16}}? Well, if {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size<ref>For real numbers <span><math>p,q</math></span> we can make the two generators respectively <span><math>\frac{p}{5p+7q}</math></span> and <span><math>\frac{q}{5p+7q}</math></span> of an octave, e.g. <span><math>(p,q)=(1,0)</math></span> for 5-ET, <span><math>(0,1)</math></span> for 7-ET, <span><math>(1,1)</math></span> for 12-ET, and many other possibilities.</ref>. If they become the same size, then they aren’t truly two separate generators, or at least there’s no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one.
+* 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.
+* Both {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} and {{monzo|{{val|1 1 0}} {{val|0 1 4}}}} are equivalent mappings, then. Converting between them we could call a change of basis. This makes more sense, of course, when speaking about converting between equivalent bases; I’ve been cautioned against referring to maps as “bases” despite the label seeming appropriate from an analogy standpoint.
-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. 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.
+* This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{val|12 19 28}} was simply {{val|5 8 12}} + {{val|7 11 16}}? Well, if {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size<ref>For real numbers <span><math>p,q</math></span> we can make the two generators respectively <span><math>\frac{p}{5p+7q}</math></span> and <span><math>\frac{q}{5p+7q}</math></span> of an octave, e.g. <span><math>(p,q)=(1,0)</math></span> for 5-ET, <span><math>(0,1)</math></span> for 7-ET, <span><math>(1,1)</math></span> for 12-ET, and many other possibilities.</ref>. If they become the same size, then they aren’t truly two separate generators, or at least there’s no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one.
-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.
+* Technically speaking, when we first learned how to map vectors with ETs, we could think of those outputs as vectors too, but they'd be 1-dimensional vectors, i.e. if 12-ET maps 16/15 to 1 step, we could write that as {{val|12 19 28}}{{monzo|4 -1 -1}} = {{monzo|1}}, where writing the answer as {{monzo|1}} expresses that 1 step as 1 of the only generator in this equal temperament.
 === JI as a temperament ===