Temperament addition: Difference between revisions

'''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.

Matching rank and dimensionality is only the first of two conditions on the possibility of temperament arithmetic. The second condition is that the temperaments must all be '''addable'''. This condition is trickier, though, and so a detailed discussion of it will be deferred to a later section (here: [[Temperament arithmetic#Addability]]). But we can at least say here that any set of min-grade-1 temperaments are addable<ref>or they are all the same temperament, in which case they share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</ref>, fortunately, so we don't need to worry about it in that case.

===The linear dependence connection===

Another connection between temperament arithmetic and meet/join is that they ''may'' involve checks for linear dependence.  

Temperament arithmetic, as stated earlier, always requires addability, which is a more complex property involving linear dependence.

Meet and join does not ''necessarily'' involve linear dependence. Linear dependence only matters for meet and join when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

Temperament arithmetic for temperaments with both <math>r>1</math> and <math>n>1</math> can also be done using matrices, but it's significantly more involved than it is with multivectors. It works in essentially the same way — entry-wise addition or subtraction — but for matrices, it is necessary to make explicit the basis for the linearly dependent vectors shared between the involved matrices before performing the arithmetic. In other words, any vectors that can be found through linear combinations of any of the involved matrices' basis vectors must appear explicitly and in the same position of each matrix before the sum or difference is taken. But it is not as simple as determining the basis for these linearly dependent vectors and pasting them over the vectors as you found them, because the results may then be [[enfactored]]. And defactoring them without compromising the explicit linearly dependent basis vectors cannot be done using existing [[defactoring algorithms]]; it's a tricky process, or at least computationally intensive.

==Addability==

In order to understand addability, we must work up to it, understanding these concepts in this order:

#linear dependence

#linear dependence ''between temperaments''  

#linear ''in''dependence between temperaments

#linear independence between temperaments by only one basis vector (that's addability)

====1. Linear dependence====

This is explained here: [[linear dependence]].

====2. Linear dependence between temperaments====

Linear dependence has been defined for the matrices and multivectors that represent temperaments, but it can also be defined for temperaments themselves. The conditions of temperament arithmetic motivate a definition of linear dependence for temperaments whereby temperaments are considered linearly dependent if ''either of their mappings or their comma bases are linearly dependent''<ref>or — equivalently, in EA — either their multimaps or their multicommas are linearly dependent</ref>.  

For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{map|7 11 16}}}} may at first seem to be linearly independent, because the basis vectors visible in their mappings are clearly linearly independent (when comparing two vectors, the only way they could be linearly dependent is if they are multiples of each other, as discussed [[Linear dependence#Linear dependence between individual vectors|here]]). And indeed their ''mappings'' are linearly independent. But these two ''temperaments'' are linearly ''de''pendent, because if we consider their corresponding comma bases, we will find that they share the basis vector of the meantone comma {{vector|4 -4 1}}.

====3. Linear independence between temperaments====

Linear dependence may be considered as a boolean (yes/no, linearly dependent/independent) or it may be considered as an integer count of linearly dependent basis vectors (e.g. 5-ET and 7-ET, per the example in the previous section, are linear-dependence-1 temperaments).  

It does not make sense to speak of the linear dependence between temperaments in this integer count sense. Here's an example that illustrates why. Consider two different 11-limit rank-2 temperaments. Both their mappings and comma bases are linearly dependent, but their mappings have linear-dependence of 1, while their comma bases have linear-dependence of 2. So what could the linear-dependence of this temperament be? We could, of course, define "min-linear-dependence" and "max-linear-dependence", as we defined "min-grade" and "max-grade", but this does not turn out to be helpful.  

However, it turns out that it does make sense to speak of the ''linear-independence'' of the temperament as an integer count. This is because the count of linearly independent basis vectors of two temperaments' mappings and the count of linearly independent basis vectors of their comma bases will always be the same. So the temperament linear-independence is simply this number. In the 11-limit rank-2 example from the previous paragraph, these would be linear-independence-1 temperaments.

A proof of this is given [[Temperament arithmetic#Sintel's proof of the linear independence conjecture|here]].

 

Two temperaments are addable if they are linear-independence-1. In other words, both their mappings and their comma bases share all but one basis vector.