Temperament addition: Difference between revisions

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Like [[temperament merging]], temperament arithmetic takes temperaments as inputs and finds a new temperament sharing properties of the inputs. And they both can be understood as, in some sense, ''adding'' these input temperaments together.

But there is a big difference between temperament arithmetic and merging. Temperament arithmetic is done using ''entry-wise'' addition (or subtraction), whereas merging is done using ''concatenation''. So the temperament sum of mappings with two rows each is a new mapping that still has exactly two rows, while the other hand, the merging of mappings with two rows each is a new mapping that has a total of four rows<ref>At least, this mapping would have a total of four rows before it is ~~reduced~~. After ~~reduction~~, it may end up with only three (or two if you map-merged a temperament with itself for some reason).</ref>.

But there is a big difference between temperament arithmetic and merging. Temperament arithmetic is done using ''entry-wise'' addition (or subtraction), whereas merging is done using ''concatenation''. So the temperament sum of mappings with two rows each is a new mapping that still has exactly two rows, while the other hand, the merging of mappings with two rows each is a new mapping that has a total of four rows<ref>At least, this mapping would have a total of four rows before it is canonicalized. After canonicalization, it may end up with only three (or two if you map-merged a temperament with itself for some reason).</ref>.

==The linear dependence connection==

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Temperament arithmetic, as stated earlier, always requires addability, which is a more complex property involving linear dependence.

Merging does not ''necessarily'' involve linear dependence. Linear dependence only matters for merging when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and ~~reduce~~. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

Merging does not ''necessarily'' involve linear dependence. Linear dependence only matters for merging when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and canonicalize. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

=<math>g_{\text{min}}=1</math>=

@@ Line 195: / Line 195: @@
 Like [[temperament merging]], temperament arithmetic takes temperaments as inputs and finds a new temperament sharing properties of the inputs. And they both can be understood as, in some sense, ''adding'' these input temperaments together.
-But there is a big difference between temperament arithmetic and merging. Temperament arithmetic is done using ''entry-wise'' addition (or subtraction), whereas merging is done using ''concatenation''. So the temperament sum of mappings with two rows each is a new mapping that still has exactly two rows, while the other hand, the merging of mappings with two rows each is a new mapping that has a total of four rows<ref>At least, this mapping would have a total of four rows before it is reduced. After reduction, it may end up with only three (or two if you map-merged a temperament with itself for some reason).</ref>.
+But there is a big difference between temperament arithmetic and merging. Temperament arithmetic is done using ''entry-wise'' addition (or subtraction), whereas merging is done using ''concatenation''. So the temperament sum of mappings with two rows each is a new mapping that still has exactly two rows, while the other hand, the merging of mappings with two rows each is a new mapping that has a total of four rows<ref>At least, this mapping would have a total of four rows before it is canonicalized. After canonicalization, it may end up with only three (or two if you map-merged a temperament with itself for some reason).</ref>.
 ==The linear dependence connection==
@@ Line 203: / Line 203: @@
 Temperament arithmetic, as stated earlier, always requires addability, which is a more complex property involving linear dependence.
-Merging does not ''necessarily'' involve linear dependence. Linear dependence only matters for merging when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].
+Merging does not ''necessarily'' involve linear dependence. Linear dependence only matters for merging when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and canonicalize. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].
 =<math>g_{\text{min}}=1</math>=