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

Line 10:

== Scalar Domain ==

Frequencies are scalar multiples of each other and especially the positive rational scalars are of ~~special~~ interest in music.

Frequencies are scalar multiples of each other and especially the positive rational scalars are of particular interest in music.

== Pitch Domain ==

Line 18:

By the [[Wikipedia:Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] logarithms of primes are linearly independent over <math>\mathbb{Q}</math>, so we can interprete <math>\log(2), \log(3), \ldots</math> as basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math>.

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

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

<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>

Line 36:

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> .

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

where the negative superscripts are geometric inverses i.e. <math>\overrightarrow{v}^{-1} \cdot \overrightarrow{v} = 1</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>

<math>\overleftarrow{12} := \mathrm{val}(12; 2, 3, 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.

Line 50:

Line 52:

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

The geometric inverses are mainly relevant for subgroup temperaments. Consider [[The_Archipelago#Subgroup_temperaments|Barbados]]:

<math>\overleftarrow{5} := \mathrm{val}(5; 2, 3, 13/5) = 5 \cdot \overrightarrow{2}^{-1} + 8 \cdot \overrightarrow{3}^{-1} + 7 \cdot \overrightarrow{13/5}^{-1} = 5 e_2 + 8 e_3 - \frac{7}{2}e_5 + \frac{7}{2}e_{13}</math>

We can verify that the comma 676/675 indeed vanishes using this val:

<math>\overleftarrow{5} \cdot \overrightarrow{676/675} = \langle 5, 8, -\frac{7}{2}, 0, 0, \frac{7}{2} \vert 2, -3, -2, 0, 0, 2 \rangle = 0</math>

== Exterior algebra nonsense ==

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

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

Line 55:

Line 66:

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:

We can 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> ,

Line 62:

Line 73:

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>.

Exterior algebras do have a sense of orthogonality but we need a metric to do projection and tuning which we already defined implicitly by giving numerical values to geometric inverses and dot products. As far as data structures go, full Clifford algebras are memory-hungry. Not worth the complication in a general purpose tool like Scale Workshop.

@@ Line 10: / Line 10: @@
 == Scalar Domain ==
-Frequencies are scalar multiples of each other and especially the positive rational scalars are of special interest in music.
+Frequencies are scalar multiples of each other and especially the positive rational scalars are of particular interest in music.
 == Pitch Domain ==
@@ Line 18: / Line 18: @@
 By the [[Wikipedia:Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] logarithms of primes are linearly independent over <math>\mathbb{Q}</math>, so we can interprete <math>\log(2), \log(3), \ldots</math> as basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math>.
-To make things slightly more formal we define the right facing arrow function
+To make things slightly more formal we define the right-facing arrow function
 <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>
@@ Line 36: / Line 36: @@
 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> .
+<math>\mathrm{val}(n; a, b, \ldots, z) := n \overrightarrow{a}^{-1} + \lfloor n \log_a(b) + \frac{1}{2}\rfloor \overrightarrow{b}^{-1} + \ldots + \lfloor n \log_a(z) + \frac{1}{2}\rfloor \overrightarrow{z}^{-1}</math> ,
+where the negative superscripts are geometric inverses i.e. <math>\overrightarrow{v}^{-1} \cdot \overrightarrow{v} = 1</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>
+<math>\overleftarrow{12} := \mathrm{val}(12; 2, 3, 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.
@@ Line 50: / Line 52: @@
 <math>\overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12 = 1100 ¢</math> .
+The geometric inverses are mainly relevant for subgroup temperaments. Consider [[The_Archipelago#Subgroup_temperaments|Barbados]]:
+<math>\overleftarrow{5} := \mathrm{val}(5; 2, 3, 13/5) = 5 \cdot \overrightarrow{2}^{-1} + 8 \cdot \overrightarrow{3}^{-1} + 7 \cdot \overrightarrow{13/5}^{-1} = 5 e_2 + 8 e_3 - \frac{7}{2}e_5 + \frac{7}{2}e_{13}</math>
+We can verify that the comma 676/675 indeed vanishes using this val:
+<math>\overleftarrow{5} \cdot \overrightarrow{676/675} = \langle 5, 8, -\frac{7}{2}, 0, 0, \frac{7}{2} \vert 2, -3, -2, 0, 0, 2 \rangle = 0</math>
+== Exterior algebra nonsense ==
 Both 12edo and 7edo temper out the syntonic comma:
 <math>\overleftarrow{12} \cdot \overrightarrow{81/80} = 0 = \overleftarrow{7} \cdot \overrightarrow{81/80}</math> .
@@ Line 55: / Line 66: @@
 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:
+We can 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> ,
@@ Line 62: / Line 73: @@
 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>.
+Exterior algebras do have a sense of orthogonality but we need a metric to do projection and tuning which we already defined implicitly by giving numerical values to geometric inverses and dot products.  As far as data structures go, full Clifford algebras are memory-hungry. Not worth the complication in a general purpose tool like Scale Workshop.