Temperament addition: Difference between revisions
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=Conditions on temperament arithmetic= | =Conditions on temperament arithmetic= | ||
== | ==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. | 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. | ||