Temperament addition: Difference between revisions

Line 177:

Temperament addition 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 addition 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~~==

==The temperaments share the same domain basis==

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 addition.

If you're unfamiliar with [[domain basis|domain bases]], 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 addition.

==The temperaments are addable==

@@ Line 177: / Line 177: @@
 Temperament addition 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 addition 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==
+==The temperaments share the same domain basis==
-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 addition.
+If you're unfamiliar with [[domain basis|domain bases]], 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 addition.
 ==The temperaments are addable==