Temperament addition: Difference between revisions

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==The temperaments have the same dimensions==

Temperament arithmetic is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament arithmetic will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments.

==The temperaments share the same interval subspace==

If you're unfamiliar with [[interval basis|interval subspaces]], then you can probably safely assume your temperaments are in the same subspace, because they should be in the default, standard, [[prime limit|prime-limit]] interval subspace. If they're not, change them to be on the same interval subspace if you can, and then come back to temperament arithmetic.

==The temperaments are addable==

[[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 of temperaments each defined by 2 basis vectors may be added as long as the other basis vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 basis vectors is able to be added by virtue of all other basis vectors being the same.]]

@@ Line 174: / Line 174: @@
 ==The temperaments have the same dimensions==
 Temperament arithmetic is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament arithmetic will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments.
+==The temperaments share the same interval subspace==
+If you're unfamiliar with [[interval basis|interval subspaces]], then you can probably safely assume your temperaments are in the same subspace, because they should be in the default, standard, [[prime limit|prime-limit]] interval subspace. If they're not, change them to be on the same interval subspace if you can, and then come back to temperament arithmetic.
 ==The temperaments are addable==
 [[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 of temperaments each defined by 2 basis vectors may be added as long as the other basis vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 basis vectors is able to be added by virtue of all other basis vectors being the same.]]