Linear algebra formalism: Difference between revisions

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Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical ~~intervals~~]]) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:

Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:

* Because [[stacking]] corresponds to multiplication of rational numbers:

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Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply.

Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group.

Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors.

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== Vals and tuning maps ==

*

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-Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical intervals]]) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:
+Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:
 * Because [[stacking]] corresponds to multiplication of rational numbers:
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 Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply.
+Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group.
 Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors.
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 == Vals and tuning maps ==
 {{Todo|complete section|inline=1}}
+*