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

(One intermediate revision by the same user not shown)

Line 65:

<math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> .

Care must be taken when the multiplier of the projective origin is not 1. e.g. Let's consider 1.5 Hz which is a perfect fifth above 1 Hz. It's represented as <math>e_0 - e_2 + e_3</math>.

If some other calculation gave us the result <math>2 e_0 - e_2 + e_3</math> it wouldn't represent 1.5Hz it's

<math>\mathrm{freq}(2 e_0 - e_2 + e_3) = 1.5 Hz^2 \sim \sqrt{\frac{3}{2}} Hz \approx 1.22 Hz</math>

instead.

== Expanding geometry ==

Line 202:

Line 210:

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...

@@ Line 65: / Line 65: @@
 <math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> .
+Care must be taken when the multiplier of the projective origin is not 1. e.g. Let's consider 1.5 Hz which is a perfect fifth above 1 Hz. It's represented as <math>e_0 - e_2 + e_3</math>.
+If some other calculation gave us the result <math>2 e_0 - e_2 + e_3</math> it wouldn't represent 1.5Hz it's
+<math>\mathrm{freq}(2 e_0 - e_2 + e_3) = 1.5 Hz^2 \sim \sqrt{\frac{3}{2}} Hz \approx 1.22 Hz</math>
+instead.
 == Expanding geometry ==
@@ Line 202: / Line 210: @@
 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...