Preimage: Difference between revisions

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=== 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 makes to vanish 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.

So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.

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The terms "[[Wikipedia:Image_(mathematics)|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.

== See also ==

* [[Detempering]], replacing tempered pitches with pitches from their preimages

[[Category:Regular temperament theory]]

[[Category:Terms]]

[[Category:Math]]

@@ Line 19: / Line 19: @@
 === 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 makes to vanish 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.
 So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.
@@ Line 32: / Line 32: @@
 The terms "[[Wikipedia:Image_(mathematics)|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.
+== See also ==
+* [[Detempering]], replacing tempered pitches with pitches from their preimages
 [[Category:Regular temperament theory]]
 [[Category:Terms]]
 [[Category:Math]]