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

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* Both {{vector|{{map|5 8 12}} {{map|7 11 16}}}} and {{vector|{{map|1 1 0}} {{map|0 1 4}}}} are equivalent mappings, then. In other words, they are both mapping-row bases of the same mapping row-space. Converting between them we could call a change of basis.

* Note well: this is not to say that {{map|1 1 0}} or {{map|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. {{vector|1 0 0}} maps to {{vector|1 0}} — referring to {{map|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. {{vector|-1 1 0}} maps to {{vector|0 1}} — referring to {{map|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 {{map|1 1 0}} or {{map|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{vector|{{map|0 1 4}} {{map|1 2 4}}} instead? We'd still have the first generator mapping as {{map|1 1 0}}, but now that the second generator mapping is {{map|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 {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{vector|{{map|5 8 12}} {{map|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.

* This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{vector|{{map|5 8 12}} {{map|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. You could imagine gradually increasing the size of one generator and decreasing the size of the other until they were both 100¢. As long as you maintain the correct proportion, you'll stay along the meantone line..

* 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 {{map|12 19 28}}{{vector|4 -1 -1}} = {{vector|1}}, where writing the answer as {{vector|1}} expresses that 1 step as 1 of the only generator in this equal temperament.

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 * Both {{vector|{{map|5 8 12}} {{map|7 11 16}}}} and {{vector|{{map|1 1 0}} {{map|0 1 4}}}} are equivalent mappings, then. In other words, they are both mapping-row bases of the same mapping row-space. Converting between them we could call a change of basis.
 * Note well: this is not to say that {{map|1 1 0}} or {{map|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. {{vector|1 0 0}} maps to {{vector|1 0}} — referring to {{map|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. {{vector|-1 1 0}} maps to {{vector|0 1}} — referring to {{map|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 {{map|1 1 0}} or {{map|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{vector|{{map|0 1 4}} {{map|1 2 4}}} instead? We'd still have the first generator mapping as {{map|1 1 0}}, but now that the second generator mapping is {{map|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 {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{vector|{{map|5 8 12}} {{map|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.
+* This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{vector|{{map|5 8 12}} {{map|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. You could imagine gradually increasing the size of one generator and decreasing the size of the other until they were both 100¢. As long as you maintain the correct proportion, you'll stay along the meantone line..
 * 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 {{map|12 19 28}}{{vector|4 -1 -1}} = {{vector|1}}, where writing the answer as {{vector|1}} expresses that 1 step as 1 of the only generator in this equal temperament.