Preimage: Difference between revisions
Cmloegcmluin (talk | contribs) Created page with "For a regular temperament, the '''preimage''' of a mapped interval is the set of all (typically justly intoned) intervals that map to it. For any interval that's..." |
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For any interval that's a member of such a preimage, altering it by any one of the [[temper out|tempered-out comma]]s of the temperament finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest. | For any interval that's a member of such a preimage, altering it by any one of the [[temper out|tempered-out comma]]s of the temperament finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest. | ||
== | == Examples == | ||
This section uses curly brackets for [[tmonzos and tvals|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals. | This section uses curly brackets for [[tmonzos and tvals|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals. | ||
== A temperament with only one comma == | === A temperament with only one comma === | ||
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}}. | 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}}. | ||
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We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament. | We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament. | ||
== A temperament with multiple commas == | === A temperament with multiple commas === | ||
We could repeat this experiment but with septimal meantone, {{ket|{{val|1 0 -4 -13}} {{val|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it tempers out is [[126/125]] {{vector|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector|1 -2 1 0}} and 9/8 {{vector|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector|1 -2 1 0}} + {{vector|1 2 -3 1}} = {{vector|2 0 -2 1}}, which is 28/25, or {{vector|-3 2 0 0}} - {{vector|1 2 -3 1}} = {{vector|-4 0 3 -1}}, which is 125/112. | We could repeat this experiment but with septimal meantone, {{ket|{{val|1 0 -4 -13}} {{val|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it tempers out is [[126/125]] {{vector|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector|1 -2 1 0}} and 9/8 {{vector|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector|1 -2 1 0}} + {{vector|1 2 -3 1}} = {{vector|2 0 -2 1}}, which is 28/25, or {{vector|-3 2 0 0}} - {{vector|1 2 -3 1}} = {{vector|-4 0 3 -1}}, which is 125/112. | ||