Temperament addition: Difference between revisions

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.

[[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.]]