Preimage: Difference between revisions

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For the meantone mapping {{ket|{{map|1 1 0}} {{map|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector|1 -2 1}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector|-3 2 0}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector|-4 4 -1}}) apart: {{vector|1 -2 1}} + {{vector|-4 4 -1}} = {{vector|-3 2 0}}.

The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma-basis]] or ~~null-space~~ contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

First we find {{vector|-3 2 0}} + {{vector|-4 4 -1}} = {{vector|-7 6 -1}} which is 729/640; then from there we find {{vector|-7 6 -1}} + {{vector|-4 4 -1}} = {{vector|-11 10 -2}} = 59049/51200; then we find {{vector|-11 10 -2}} + {{vector|-4 4 -1}} = {{vector|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.

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 For the meantone mapping {{ket|{{map|1 1 0}} {{map|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector|1 -2 1}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector|-3 2 0}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector|-4 4 -1}}) apart: {{vector|1 -2 1}} + {{vector|-4 4 -1}} = {{vector|-3 2 0}}.
-The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma-basis]] or null-space contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.
+The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.
 First we find {{vector|-3 2 0}} + {{vector|-4 4 -1}} = {{vector|-7 6 -1}} which is 729/640; then from there we find {{vector|-7 6 -1}} + {{vector|-4 4 -1}} = {{vector|-11 10 -2}} = 59049/51200; then we find {{vector|-11 10 -2}} + {{vector|-4 4 -1}} = {{vector|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.