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

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And somehow… from this… we can generate meantone?! This is true, but it’s not immediately easy to see how that would happen.

First we should show how to actually use rank-2 mappings. It’s actually not that complicated. It’s just like using a rank-1 mapping, except you have to ~~use both~~ of them. Let’s ~~check~~ out 10/9, or {{monzo|1 -2 1}}. In {{val|5 8 12}} ~~that’s~~ 5 - 16 + 12 = 1 ~~step. In~~ {{val|7 11 16}} ~~that’s~~ 7 - 22 + 16 = 1 ~~step.~~ So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{monzo|1 1}}.

First we should show how to actually use rank-2 mappings. It’s actually not that complicated. It’s just like using a rank-1 mapping, except you have to find each of them separately, and then put them back together at the end. Let’s see how this plays out for 10/9, or {{monzo|1 -2 1}}.

'''{{val|5 8 12}}:'''

* {{val|5 8 12}}{{monzo|1 -2 1}}

* 5×1 + 8×-2 + 12×1

* 5 + -16 + 12

* 1

'''{{val|7 11 16}}:'''

* {{val|7 11 16}}{{monzo|1 -2 1}}

* 7×1 + 11×-2 + 16×1

* 7 + -22 + 16

* 1

So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{monzo|1 1}}.

If the familiar usage of vectors has been as prime count lists, we can now generalize that definition to things like this {{monzo|1 1}}: generator count lists. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.

@@ Line 841: / Line 841: @@
 And somehow… from this… we can generate meantone?! This is true, but it’s not immediately easy to see how that would happen.
-First we should show how to actually use rank-2 mappings. It’s actually not that complicated. It’s just like using a rank-1 mapping, except you have to use both of them. Let’s check out 10/9, or {{monzo|1 -2 1}}. In {{val|5 8 12}} that’s 5 - 16 + 12 = 1 step. In {{val|7 11 16}} that’s 7 - 22 + 16 = 1 step. So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{monzo|1 1}}.
+First we should show how to actually use rank-2 mappings. It’s actually not that complicated. It’s just like using a rank-1 mapping, except you have to find each of them separately, and then put them back together at the end. Let’s see how this plays out for 10/9, or {{monzo|1 -2 1}}.
+'''{{val|5 8 12}}:'''
+* {{val|5 8 12}}{{monzo|1 -2 1}}
+* 5×1 + 8×-2 + 12×1
+* 5 + -16 + 12
+* 1
+'''{{val|7 11 16}}:'''
+* {{val|7 11 16}}{{monzo|1 -2 1}}
+* 7×1 + 11×-2 + 16×1
+* 7 + -22 + 16
+* 1
+So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{monzo|1 1}}.
 If the familiar usage of vectors has been as prime count lists, we can now generalize that definition to things like this {{monzo|1 1}}: generator count lists. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.