Temperament addition: Difference between revisions

Line 2:

'''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments.

=Introductory examples=

For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}}={{map|(12+7) (19+11) (28+16)}}={{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}}={{map|(12-7) (8-11) (12-16)}}={{map|5 8 12}}.

Line 126:

Line 128:

For <math>g_{\text{min}}>1</math> temperaments, temperament arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_.5Bmath.5D.5Cmin.28g.29.3D1.5B.2Fmath.5D|beyond <math>g_{\text{min}}=1</math> section]] later.

=A note on variance=

For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.

=Visualizing temperament arithmetic=

@@ Line 2: / Line 2: @@
 '''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments.
+=Introductory examples=
 For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}}={{map|(12+7) (19+11) (28+16)}}={{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}}={{map|(12-7) (8-11) (12-16)}}={{map|5 8 12}}.
@@ Line 126: / Line 128: @@
 For <math>g_{\text{min}}>1</math> temperaments, temperament arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_.5Bmath.5D.5Cmin.28g.29.3D1.5B.2Fmath.5D|beyond <math>g_{\text{min}}=1</math> section]] later.
+=A note on variance=
+For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.
 =Visualizing temperament arithmetic=