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

Line 16:

== Adding Geometry ==

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 ~~orthogonal~~ basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality

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> and enforce orthogonality

<math>e_p \cdot e_q = 0, p \ne q, p, q \in \mathbb{P}</math>

Line 22:

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>

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

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

Line 49:

<math>\mathrm{ratio}(x) = \prod_{p \in \mathbb{P}} p^{x \cdot e^p}</math>

i.e. the superscript vectors act as measuring sticks telling us how much of each prime there is in a vector.

i.e. the superscript vectors act as measuring sticks telling us how much of each prime there is in an arbitrary vector.

Let's ''coin'' a new unit called ''jorp'' (think ''Europe'') <math>€ = ¢^{-1} = 1200 \cdot \overrightarrow{2}^{-1}</math> which measures intervals with a resolution of 1200 ticks per octave.

~~One~~ conceptually important measuring stick is the one constructed from the logarithms of the primes:

A conceptually important measuring stick is the one constructed from the logarithms of the primes:

<math>\mathsf{JIP} = \sum_{p \in \mathbb{P}} e^p \log p</math>

Line 90:

<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>

Note that above I've implicitly used a convenient metric to carry out the calculations which is fine due to the new basis still being orhogonal. Explicitly we'd have

Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have

<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" ~~toward~~ 5/1 or 13/1.

<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1.

== On units ==

Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they ~~are~~ like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This should be enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.

Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This should be enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.

Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.

Line 100:

A reciprocal cent satisfies <math>¢^{-1} \cdot ¢ = 1</math> so as per the usual definition of the geometric inverse of a vector we have <math>¢^{-1} = ¢ / (¢ \cdot ¢) = \frac{1}{1200}e_2 / (\frac{1}{1200}^2 e_2 \cdot e_2) = 1200 w_2 \hat{i} / (w_2^2 \hat{i} \cdot \hat{i}) = \frac{1200}{w_2}\hat{i}</math>.

~~Now~~ we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>. A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.

As we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>; A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.

== Clifford algebra nonsense ==

Line 106:

<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>

Therefore so does any linear combination of them e. g. <math>2 \cdot \overleftarrow{12} + \overleftarrow{7} = \overleftarrow{31} \implies \overleftarrow{31} \cdot \overrightarrow{81/80} = 0</math>

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:

Line 128:

which is is extended linearly to all vectors. It is obvious from the definitions that the Meantone temperament is represented by <math>\overline{\overrightarrow{81/80}}</math>.

~~Defining~~ the vee operator <math>a \vee b := \overline{\overline{b} \wedge \overline{a}}</math>, where the inner overlines are the inverses of the dual (always obvious from context). Wedges combine vals into temperaments that are closer to just intonation while vees do progressive damage to just intonation. As an exercise let's calculate the damage done by combining Meantone with Augmented:

We define the vee operator <math>a \vee b := \overline{\overline{b} \wedge \overline{a}}</math>, where the inner overlines are the inverses of the dual (always obvious from context).

Wedges combine vals into temperaments that are closer to just intonation while vees do progressive damage to just intonation. As an exercise let's calculate the damage done by combining Meantone with Augmented:

<math>

Line 146:

Line 148:

Exterior algebras do have a sense of lying-in-the-plane-of but we need a metric to do projection and tuning which we already touched upon in the units section. As far as data structures go, full Clifford algebras are memory-hungry. Not worth the complication in a general purpose tool like Scale Workshop.

To explore our space a bit longer we can consider <math>e_2</math> as an interval class consiting of all multiples of the octave. After all we have <math>\overrightarrow{8} = 3 \cdot \overrightarrow{2} = 3 e_2</math>. Wedging gives us larger classes. <math>e_2 \wedge e_3</math> can be seen as representing Pythagorean just intonation. Unfortunately the wedges are not ~~powerful~~ enough to distinguish subflavors <math>\overrightarrow{2} \wedge \overrightarrow{9} = 2 \cdot \overrightarrow{2} \wedge \overrightarrow{3} = \overrightarrow{4} \wedge \overrightarrow{3}</math> but at least the "orientation" is correct. Previously we used the term "just intonation" to refer to lack of tempering but it's also used to refer to an interval class such as 5-limit just intonation representable by <math>e_2 \wedge e_3 \wedge e_5</math>.

To explore our space a bit longer we can consider <math>e_2</math> as an interval class consiting of all multiples of the octave. After all we have <math>\overrightarrow{8} = 3 \cdot \overrightarrow{2} = 3 e_2</math>. Wedging gives us larger classes. <math>e_2 \wedge e_3</math> can be seen as representing Pythagorean just intonation. Unfortunately the wedges are not accurate enough to distinguish subflavors <math>\overrightarrow{2} \wedge \overrightarrow{9} = 2 \cdot \overrightarrow{2} \wedge \overrightarrow{3} = \overrightarrow{4} \wedge \overrightarrow{3}</math> but at least the "orientation" is correct. Previously we used the term "just intonation" to refer to lack of tempering but it's also used to refer to an interval class such as 5-limit just intonation representable by <math>e_2 \wedge e_3 \wedge e_5</math>.

One interesting pseudo-object in our space (let's call it <math>\mathbb{S}</math>) is the unison plane:

Line 152:

Line 154:

<math>\{\mathsf{JIP} \cdot x = 0, x \in \mathbb{S}\} = \{\overrightarrow{1}\}</math>

which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so ~~it's informative to look at~~ 3D subspaces of <math>\mathbb{S}</math> ~~facing~~ the unison plane to get a 2D view of all commas of interest. The null-space of all reasonable temperaments are also closely aligned with this plane.

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.

@@ Line 16: / Line 16: @@
 == Adding Geometry ==
-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 orthogonal basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality
+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> and enforce orthogonality
 <math>e_p \cdot e_q = 0, p \ne q, p, q \in \mathbb{P}</math>
@@ Line 22: / Line 22: @@
 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>
+<math>\overrightarrow{2^x 3^y 5^z \ldots} \mapsto x e_2 + y e_3 + z e_5 + \ldots, x, y, z, \ldots \in \mathbb{Q}</math>
 which takes objects from the scalar domain to the geometric pitch domain.
@@ Line 49: / Line 49: @@
 <math>\mathrm{ratio}(x) = \prod_{p \in \mathbb{P}} p^{x \cdot e^p}</math>
-i.e. the superscript vectors act as measuring sticks telling us how much of each prime there is in a vector.
+i.e. the superscript vectors act as measuring sticks telling us how much of each prime there is in an arbitrary vector.
 Let's ''coin'' a new unit called ''jorp'' (think ''Europe'') <math>€ = ¢^{-1} = 1200 \cdot \overrightarrow{2}^{-1}</math> which measures intervals with a resolution of 1200 ticks per octave.
-One conceptually important measuring stick is the one constructed from the logarithms of the primes:
+A conceptually important measuring stick is the one constructed from the logarithms of the primes:
 <math>\mathsf{JIP} = \sum_{p \in \mathbb{P}} e^p \log p</math>
@@ Line 90: / Line 90: @@
 <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>
-Note that above I've implicitly used a convenient metric to carry out the calculations which is fine due to the new basis still being orhogonal. Explicitly we'd have
+Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have
-<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" toward 5/1 or 13/1.
+<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1.
 == On units ==
-Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they are like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This should be enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.
+Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This should be enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.
 Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.
@@ Line 100: / Line 100: @@
 A reciprocal cent satisfies <math>¢^{-1} \cdot ¢ = 1</math> so as per the usual definition of the geometric inverse of a vector we have <math>¢^{-1} = ¢ / (¢ \cdot ¢) = \frac{1}{1200}e_2 / (\frac{1}{1200}^2 e_2 \cdot e_2) = 1200 w_2 \hat{i} / (w_2^2 \hat{i} \cdot \hat{i}) = \frac{1200}{w_2}\hat{i}</math>.
-Now we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>. A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.
+As we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>; A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.
 == Clifford algebra nonsense ==
@@ Line 106: / Line 106: @@
 <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>
+Therefore so does any linear combination of them e. g. <math>2 \cdot \overleftarrow{12} + \overleftarrow{7} = \overleftarrow{31} \implies \overleftarrow{31} \cdot \overrightarrow{81/80} = 0</math>
 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:
@@ Line 128: / Line 128: @@
 which is is extended linearly to all vectors. It is obvious from the definitions that the Meantone temperament is represented by <math>\overline{\overrightarrow{81/80}}</math>.
-Defining the vee operator <math>a \vee b := \overline{\overline{b} \wedge \overline{a}}</math>, where the inner overlines are the inverses of the dual (always obvious from context). Wedges combine vals into temperaments that are closer to just intonation while vees do progressive damage to just intonation. As an exercise let's calculate the damage done by combining Meantone with Augmented:
+We define the vee operator <math>a \vee b := \overline{\overline{b} \wedge \overline{a}}</math>, where the inner overlines are the inverses of the dual (always obvious from context).
+Wedges combine vals into temperaments that are closer to just intonation while vees do progressive damage to just intonation. As an exercise let's calculate the damage done by combining Meantone with Augmented:
 <math>
@@ Line 146: / Line 148: @@
 Exterior algebras do have a sense of lying-in-the-plane-of but we need a metric to do projection and tuning which we already touched upon in the units section. As far as data structures go, full Clifford algebras are memory-hungry. Not worth the complication in a general purpose tool like Scale Workshop.
-To explore our space a bit longer we can consider <math>e_2</math> as an interval class consiting of all multiples of the octave. After all we have <math>\overrightarrow{8} = 3 \cdot \overrightarrow{2} = 3 e_2</math>. Wedging gives us larger classes. <math>e_2 \wedge e_3</math> can be seen as representing Pythagorean just intonation. Unfortunately the wedges are not powerful enough to distinguish subflavors <math>\overrightarrow{2} \wedge \overrightarrow{9} = 2 \cdot \overrightarrow{2} \wedge \overrightarrow{3} = \overrightarrow{4} \wedge \overrightarrow{3}</math> but at least the "orientation" is correct. Previously we used the term "just intonation" to refer to lack of tempering but it's also used to refer to an interval class such as 5-limit just intonation representable by <math>e_2 \wedge e_3 \wedge e_5</math>.
+To explore our space a bit longer we can consider <math>e_2</math> as an interval class consiting of all multiples of the octave. After all we have <math>\overrightarrow{8} = 3 \cdot \overrightarrow{2} = 3 e_2</math>. Wedging gives us larger classes. <math>e_2 \wedge e_3</math> can be seen as representing Pythagorean just intonation. Unfortunately the wedges are not accurate enough to distinguish subflavors <math>\overrightarrow{2} \wedge \overrightarrow{9} = 2 \cdot \overrightarrow{2} \wedge \overrightarrow{3} = \overrightarrow{4} \wedge \overrightarrow{3}</math> but at least the "orientation" is correct. Previously we used the term "just intonation" to refer to lack of tempering but it's also used to refer to an interval class such as 5-limit just intonation representable by <math>e_2 \wedge e_3 \wedge e_5</math>.
 One interesting pseudo-object in our space (let's call it <math>\mathbb{S}</math>) is the unison plane:
@@ Line 152: / Line 154: @@
 <math>\{\mathsf{JIP} \cdot x = 0, x \in \mathbb{S}\} = \{\overrightarrow{1}\}</math>
-which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so it's informative to look at 3D subspaces of <math>\mathbb{S}</math> facing the unison plane to get a 2D view of all commas of interest. The null-space of all reasonable temperaments are also closely aligned with this plane.
+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.