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

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To make things slightly more formal we define the right facing arrow function

<math>\overrightarrow{2^x 3^y 5^z \ldots} := x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q}</math>

<math>\overrightarrow{2^x 3^y 5^z \ldots} \mapsto x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q}</math>

which takes objects from the scalar domain to the geometric pitch domain.

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Pitch is measured in cents (¢) which we define to be the vector quantity <math>¢ := e_2 / 1200</math> i.e. <math>\mathrm{ratio}(¢) = 2^{\frac{1}{1200}} \approx 1.0005777895</math> .

We also define the backslash function <math>\backslash d \mapsto e_2 / d</math> .

In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> .

== Expanding geometry ==

~~TODO~~

Equal temperaments can be represented by [[Val|vals]] which we define as

<math>\mathrm{val}(n, \overrightarrow{a}, \overrightarrow{b}, \ldots, \overrightarrow{z}) = n \overrightarrow{a} + \lfloor n \log_a(b) + \frac{1}{2}\rfloor \overrightarrow{b} + \ldots + \lfloor n \log_a(z) + \frac{1}{2}\rfloor \overrightarrow{z}</math> .

Usually the basis is obvious from context e.g. <math>a = 2, b = 3, c = 5</math>. In these cases we use a left-facing arrow e.g.

<math>\overleftarrow{12} := \mathrm{val}(12, e_2, e_3, e_5) = 12 e_2 + 19 e_3 + 28 e_5 =: \langle 12, 19, 28 \rbrack</math>

We can use these new objects to calculate how many steps of 12edo a tempered interval spans e.g.

<math>\overleftarrow{12} \cdot \overrightarrow{15/8} = \langle 12, 19, 28 \vert -3, 1, 1 \rangle = 11</math>

The actual pitch is obtained by sandwiching the interval between the val and the step size:

<math>\overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12 = 1100 ¢</math> .

Both 12edo and 7edo temper out the syntonic comma:

<math>\overleftarrow{12} \cdot \overrightarrow{81/80} = 0 = \overleftarrow{7} \cdot \overrightarrow{81/80}</math> .

Therefore so does any linear combination of them e. g. <math>2 \cdot \overleftarrow{12} + \overleftarrow{7} = \overleftarrow{31}</math>

We identify the plane spanned by <math>\overleftarrow{12}</math> and <math>\overleftarrow{7}</math> as the (5-limit) Meantone temperament. We can use wedges to represent it symbolically:

<math>\overleftarrow{12} \wedge \overleftarrow{7} = -4 e_3 \wedge e_5 + 4 e_5 \wedge e_2 - e_2 \wedge e_3</math> ,

where the components are basis planes. E.g. <math>e_3 \wedge e_5</math> is the plane where octaves are tempered out. The wedge of any vector with itself is zero i.e. you can't span a plane with only one direction. The wedge product is also antisymmetric and the planes come with signed weights but we mostly care about the orientation they represent.

The largest possible wedge combines all of the basis vectors and represents just intonation i.e. no tempering whatsoever: <math>e_2 \wedge e_3 \wedge e_5</math>.

@@ Line 20: / Line 20: @@
 To make things slightly more formal we define the right facing arrow function
-<math>\overrightarrow{2^x 3^y 5^z \ldots} := x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q}</math>
+<math>\overrightarrow{2^x 3^y 5^z \ldots} \mapsto x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q}</math>
 which takes objects from the scalar domain to the geometric pitch domain.
@@ Line 30: / Line 30: @@
 Pitch is measured in cents (¢) which we define to be the vector quantity <math>¢ := e_2 / 1200</math> i.e. <math>\mathrm{ratio}(¢) = 2^{\frac{1}{1200}} \approx 1.0005777895</math> .
+We also define the backslash function <math>\backslash d \mapsto e_2 / d</math> .
+In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> .
 == Expanding geometry ==
-TODO
+Equal temperaments can be represented by [[Val|vals]] which we define as
+<math>\mathrm{val}(n, \overrightarrow{a}, \overrightarrow{b}, \ldots, \overrightarrow{z}) = n \overrightarrow{a} + \lfloor n \log_a(b) + \frac{1}{2}\rfloor \overrightarrow{b} + \ldots + \lfloor n \log_a(z) + \frac{1}{2}\rfloor \overrightarrow{z}</math> .
+Usually the basis is obvious from context e.g. <math>a = 2, b = 3, c = 5</math>. In these cases we use a left-facing arrow e.g.
+<math>\overleftarrow{12} := \mathrm{val}(12, e_2, e_3, e_5) = 12 e_2 + 19 e_3 + 28 e_5 =: \langle 12, 19, 28 \rbrack</math>
+We can use these new objects to calculate how many steps of 12edo a tempered interval spans e.g.
+<math>\overleftarrow{12} \cdot \overrightarrow{15/8} = \langle 12, 19, 28 \vert -3, 1, 1 \rangle = 11</math>
+The actual pitch is obtained by sandwiching the interval between the val and the step size:
+<math>\overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12 = 1100 ¢</math> .
+Both 12edo and 7edo temper out the syntonic comma:
+<math>\overleftarrow{12} \cdot \overrightarrow{81/80} = 0 = \overleftarrow{7} \cdot \overrightarrow{81/80}</math> .
+Therefore so does any linear combination of them e. g. <math>2 \cdot \overleftarrow{12} + \overleftarrow{7} = \overleftarrow{31}</math>
+We identify the plane spanned by <math>\overleftarrow{12}</math> and <math>\overleftarrow{7}</math> as the (5-limit) Meantone temperament. We can use wedges to represent it symbolically:
+<math>\overleftarrow{12} \wedge \overleftarrow{7} = -4 e_3 \wedge e_5 + 4 e_5 \wedge e_2 - e_2 \wedge e_3</math> ,
+where the components are basis planes. E.g. <math>e_3 \wedge e_5</math> is the plane where octaves are tempered out. The wedge of any vector with itself is zero i.e. you can't span a plane with only one direction. The wedge product is also antisymmetric and the planes come with signed weights but we mostly care about the orientation they represent.
+The largest possible wedge combines all of the basis vectors and represents just intonation i.e. no tempering whatsoever: <math>e_2 \wedge e_3 \wedge e_5</math>.