Temperament addition: Difference between revisions

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For single vectors (and multivectors), negation is as simple as changing the sign of every entry.

Suppose you have a matrix representing temperament <math>~~T_1~~</math> and another matrix representing <math>~~T_2~~</math>. If you want to find both their sum and difference, you can calculate both <math>~~T_1~~ + ~~T_2~~</math> and <math>~~T_1~~ + -~~T_2~~</math>. There's no need to also find <math>-~~T_1~~ + ~~T_2~~</math>; this will merely give the negation of <math>~~T_1~~ + -~~T_2~~</math>. The same goes for <math>-~~T_1~~ + -~~T_2~~</math>, which is the negation of <math>~~T_1~~ + ~~T_2~~</math>.

Suppose you have a matrix representing temperament <math>𝓣_1</math> and another matrix representing <math>𝓣_2</math>. If you want to find both their sum and difference, you can calculate both <math>𝓣_1 + 𝓣_2</math> and <math>𝓣_1 + -𝓣_2</math>. There's no need to also find <math>-𝓣_1 + 𝓣_2</math>; this will merely give the negation of <math>𝓣_1 + -𝓣_2</math>. The same goes for <math>-𝓣_1 + -𝓣_2</math>, which is the negation of <math>𝓣_1 + 𝓣_2</math>.

But a question remains: which result between <math>~~T_1~~ + ~~T_2~~</math> and <math>~~T_1~~ + -~~T_2~~</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>~~T_1~~</math> or <math>~~T_2~~</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.

But a question remains: which result between <math>𝓣_1 + 𝓣_2</math> and <math>𝓣_1 + -𝓣_2</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>𝓣_1</math> or <math>𝓣_2</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.

The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.

@@ Line 225: / Line 225: @@
 For single vectors (and multivectors), negation is as simple as changing the sign of every entry.
-Suppose you have a matrix representing temperament <math>T_1</math> and another matrix representing <math>T_2</math>. If you want to find both their sum and difference, you can calculate both <math>T_1 + T_2</math> and <math>T_1 + -T_2</math>. There's no need to also find <math>-T_1 + T_2</math>; this will merely give the negation of <math>T_1 + -T_2</math>. The same goes for <math>-T_1 + -T_2</math>, which is the negation of <math>T_1 + T_2</math>.
+Suppose you have a matrix representing temperament <math>𝓣_1</math> and another matrix representing <math>𝓣_2</math>. If you want to find both their sum and difference, you can calculate both <math>𝓣_1 + 𝓣_2</math> and <math>𝓣_1 + -𝓣_2</math>. There's no need to also find <math>-𝓣_1 + 𝓣_2</math>; this will merely give the negation of <math>𝓣_1 + -𝓣_2</math>. The same goes for <math>-𝓣_1 + -𝓣_2</math>, which is the negation of <math>𝓣_1 + 𝓣_2</math>.
-But a question remains: which result between <math>T_1 + T_2</math> and <math>T_1 + -T_2</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>T_1</math> or <math>T_2</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.
+But a question remains: which result between <math>𝓣_1 + 𝓣_2</math> and <math>𝓣_1 + -𝓣_2</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>𝓣_1</math> or <math>𝓣_2</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.
 The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.