Temperament addition: Difference between revisions

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As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the trickier cases of <math>g_{\text{min}}>1</math>.

Throughout this section, we will be using a green color on linearly dependent objects and values, and a red color on linearly independent objects and values, to help differentiate between the two.

==Addability==

In order to understand how to do temperament arithmetic on <math>g_{\text{min}}>1</math> temperaments, we must first understand addability.

===Verbal explanation===

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==Multivector approach==

The simplest approach is to use multivectors. This is discussed in more detail here: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament arithmetic]].

The simplest approach to <math>g_{\text{min}}>1</math> temperament arithmetic is to use multivectors. This is discussed in more detail here: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament arithmetic]].

==Matrix approach==

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. These vectors are called the linear-dependence basis, or <math>L_{\text{dep}}</math>. But it is not as simple as determining <math>L_{\text{dep}}</math> (using the technique described [[Linear_dependence#For_a_given_set_of_matrices.2C_how_to_compute_a_basis_for_their_linearly_dependent_vectors|here]]) and then supplying the remaining vectors necessary to match the grade of the original matrix, because the results may then be [[enfactored]]. And defactoring them without compromising the explicit <math>L_{\text{dep}}</math> cannot be done using existing [[defactoring algorithms]]; it's a tricky process, or at least computationally intensive.

Temperament arithmetic for <math>g_{\text{min}}>1</math> temperaments (again, that's 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. These vectors are called the linear-dependence basis, or <math>L_{\text{dep}}</math>. But it is not as simple as determining <math>L_{\text{dep}}</math> (using the technique described [[Linear_dependence#For_a_given_set_of_matrices.2C_how_to_compute_a_basis_for_their_linearly_dependent_vectors|here]]) and then supplying the remaining vectors necessary to match the grade of the original matrix, because the results may then be [[enfactored]]. And defactoring them without compromising the explicit <math>L_{\text{dep}}</math> cannot be done using existing [[defactoring algorithms]]; it's a tricky process, or at least computationally intensive. This is called '''addabilization defactoring''' and is worked through by example below.

~~Throughout this section, we will be using a green color on linearly dependent objects~~ and ~~values, and a red color on linearly independent objects and values, to help differentiate between the two~~.

===Negation===

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(Examples WIP)

===Proof that addabilization defactoring is always possible===

This conjecture has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] suggest this is true, though.

= References =

@@ Line 226: / Line 226: @@
 As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the trickier cases of <math>g_{\text{min}}>1</math>.
+Throughout this section, we will be using <span style="color: #3C8031;">a green color on linearly dependent objects and values</span>, and <span style="color: #B6321C;">a red color on linearly independent objects and values</span>, to help differentiate between the two.
 ==Addability==
+In order to understand how to do temperament arithmetic on <math>g_{\text{min}}>1</math> temperaments, we must first understand addability.
 ===Verbal explanation===
@@ Line 684: / Line 688: @@
 ==Multivector approach==
-The simplest approach is to use multivectors. This is discussed in more detail here: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament arithmetic]].
+The simplest approach to <math>g_{\text{min}}>1</math> temperament arithmetic is to use multivectors. This is discussed in more detail here: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament arithmetic]].
 ==Matrix approach==
-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 <span style="color: #3C8031;">the basis for the linearly dependent vectors shared</span> 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. These vectors are called the <span style="color: #3C8031;">linear-dependence basis, or <math>L_{\text{dep}}</math></span>. But it is not as simple as determining <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> (using the technique described [[Linear_dependence#For_a_given_set_of_matrices.2C_how_to_compute_a_basis_for_their_linearly_dependent_vectors|here]]) and then supplying the remaining vectors necessary to match the grade of the original matrix, because the results may then be [[enfactored]]. And defactoring them without compromising the explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> cannot be done using existing [[defactoring algorithms]]; it's a tricky process, or at least computationally intensive.
+Temperament arithmetic for <math>g_{\text{min}}>1</math> temperaments (again, that's 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 <span style="color: #3C8031;">the basis for the linearly dependent vectors shared</span> 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. These vectors are called the <span style="color: #3C8031;">linear-dependence basis, or <math>L_{\text{dep}}</math></span>. But it is not as simple as determining <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> (using the technique described [[Linear_dependence#For_a_given_set_of_matrices.2C_how_to_compute_a_basis_for_their_linearly_dependent_vectors|here]]) and then supplying the remaining vectors necessary to match the grade of the original matrix, because the results may then be [[enfactored]]. And defactoring them without compromising the explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> cannot be done using existing [[defactoring algorithms]]; it's a tricky process, or at least computationally intensive. This is called '''addabilization defactoring''' and is worked through by example below.
-Throughout this section, we will be using <span style="color: #3C8031;">a green color on linearly dependent objects and values</span>, and <span style="color: #B6321C;">a red color on linearly independent objects and values</span>, to help differentiate between the two.
 ===Negation===
@@ Line 852: / Line 854: @@
 (Examples WIP)
+===Proof that addabilization defactoring is always possible===
+This conjecture has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] suggest this is true, though.
 = References =