User:Frostburn/Theory From First Principles: Difference between revisions

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which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so in 3D subspaces of <math>\mathbb{S}</math> it's informative to face the unison plane head-on to get a 2D view of all commas of interest. The null-space of all reasonable temperaments (i.e. the set of all commas and their products that vanish in the temperament) are also closely aligned with this plane.

== The nature of 1 and improving representation ==

We've purposefully avoided using <math>e_1</math> i.e. <math>\overrightarrow{1} = 0</math>. This is because it introduces ambiguity into the logarithm formula. e.g.

<math>\log(6) = \log(2\cdot 3) = \log(1 \cdot 1 \cdot 2 \cdot 3) = \log(1) + \log(1) + \log(2) + log(3) \mapsto^? 2 e_1 + e_2 + e_3</math>.

If we stipulate that the logarithm of unity is only added once and that <math>e_1 \cdot e_1 = 0</math> we might get a useful bookkeeping tool.

The right-facing arrow function should be redefined:

<math>\overrightarrow{1\cdot 2^x\cdot 3^y\cdot\ldots} \mapsto e_1 + x e_2 + y e_3 + \ldots</math>

The point is to differentiate intervals from interval classes using projective geometric algebra. A ratio and it's square are different intervals, but any rational multiple of a val represents the same equal temperament and the same goes for temperaments (wedges). I've forgotten how to do the projective stuff to Make Things Work™, so I'm just leaving this chapter here to remind myself to read up on the relevant literature...

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 which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so in 3D subspaces of <math>\mathbb{S}</math> it's informative to face the unison plane head-on to get a 2D view of all commas of interest. The null-space of all reasonable temperaments (i.e. the set of all commas and their products that vanish in the temperament) are also closely aligned with this plane.
+== The nature of 1 and improving representation ==
+We've purposefully avoided using <math>e_1</math> i.e. <math>\overrightarrow{1} = 0</math>. This is because it introduces ambiguity into the logarithm formula. e.g.
+<math>\log(6) = \log(2\cdot 3) = \log(1 \cdot 1 \cdot 2 \cdot 3) = \log(1) + \log(1) + \log(2) + log(3) \mapsto^? 2 e_1 + e_2 + e_3</math>.
+If we stipulate that the logarithm of unity is only added once and that <math>e_1 \cdot e_1 = 0</math> we might get a useful bookkeeping tool.
+The right-facing arrow function should be redefined:
+<math>\overrightarrow{1\cdot 2^x\cdot 3^y\cdot\ldots} \mapsto e_1 + x e_2 + y e_3 + \ldots</math>
+The point is to differentiate intervals from interval classes using projective geometric algebra. A ratio and it's square are different intervals, but any rational multiple of a val represents the same equal temperament and the same goes for temperaments (wedges). I've forgotten how to do the projective stuff to Make Things Work™, so I'm just leaving this chapter here to remind myself to read up on the relevant literature...