Temperament addition: Difference between revisions

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= Applications =

[[File:Addability.png|300px|thumb|left|In the first row, we see the sum of two vectors. In the second row, we see how a pair temperaments each defined by 2 vectors may be added as long as the other vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 vectors is able to be added by virtue of all other vectors being the same.]]

The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments.

Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't temper out the meantone comma itself, nor the porcupine comma itself, but instead tempers out whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that tempers out neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot tempers out 81/80 × 250/243, and dicot tempers out 81/80 × 243/250.

Ultimately, this effect is the primary application of temperament arithmetic.

Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe.

Ultimately, this effect is the primary application of temperament arithmetic. With temperament arithmetic, you're essentially never really able to do anything meaningful beyond entry-wise adding a pair of (mono)vectors. What changes from situation to situation is how many other vectors there are alongside, in the vector sets representing the temperament, whether there are 0 other vectors or 2 or 5. As we'll learn later, any other vectors beyond the first ones are always required to be the same between all the summed or differenced temperaments.

== Fokker groups ==

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 = Applications =
-[[File:Addability.png|300px|thumb|left|In the first row, we see the sum of two vectors. In the second row, we see how a pair temperaments each defined by 2 vectors may be added as long as the other vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 vectors is able to be added by virtue of all other vectors being the same.]]
 The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments.
 Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't temper out the meantone comma itself, nor the porcupine comma itself, but instead tempers out whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that tempers out neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot tempers out 81/80 × 250/243, and dicot tempers out 81/80 × 243/250.
+Ultimately, this effect is the primary application of temperament arithmetic.
 Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe.
-Ultimately, this effect is the primary application of temperament arithmetic. With temperament arithmetic, you're essentially never really able to do anything meaningful beyond entry-wise adding a pair of (mono)vectors. What changes from situation to situation is how many other vectors there are alongside, in the vector sets representing the temperament, whether there are 0 other vectors or 2 or 5. As we'll learn later, any other vectors beyond the first ones are always required to be the same between all the summed or differenced temperaments.
 == Fokker groups ==