Temperament mapping matrix: Difference between revisions

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~~:''This page gives a formal mathematical approach to [[RTT]] mapping. For a page with a simpler introduction, see [[mapping]].''~~

The [[wikipedia: Multiplicative group|multiplicative group]] generated by any finite set of [[wikipedia: Rational number|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

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</math>

for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val| 7 1}} and {{val| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping| 1 2 3 2 4 |0 -3 -5 6 -4 }} as a result again.

for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val| 7 1 }} and {{val| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }} as a result again.

[[Category:Regular temperament theory]]

[[Category:Math]]

[[Category:Mapping]]

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-:''This page gives a formal mathematical approach to [[RTT]] mapping. For a page with a simpler introduction, see [[mapping]].''
+{{Expert|Mapping}}
 The [[wikipedia: Multiplicative group|multiplicative group]] generated by any finite set of [[wikipedia: Rational number|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)
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 </math>
-for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val| 7 1}} and {{val| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping| 1 2 3 2 4 |0 -3 -5 6 -4 }} as a result again.
+for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val| 7 1 }} and {{val| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }} as a result again.
 [[Category:Regular temperament theory]]
 [[Category:Math]]
 [[Category:Mapping]]