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

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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 {{vector|1 1}}.

If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. 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.

If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator-count vectors or GC-vectors. 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.

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 big part of the power of temperament theory. In this case, a common method for optimizing tunings from temperaments would give these two generators as 73.756¢ and 118.945¢, respectively, which gives the tempered 10/9 as 73.756¢ + 118.945¢ = 192.701¢, which is about 10¢ sharp from its JI cents value of 182.404¢. We’ll learn more about tuning methods later.

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 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 {{vector|1 1}}.
-If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. 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.
+If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator-count vectors or GC-vectors. 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.
 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 big part of the power of temperament theory. In this case, a common method for optimizing tunings from temperaments would give these two generators as 73.756¢ and 118.945¢, respectively, which gives the tempered 10/9 as 73.756¢ + 118.945¢ = 192.701¢, which is about 10¢ sharp from its JI cents value of 182.404¢. We’ll learn more about tuning methods later.