User:Frostburn/Theory From First Principles - Revision history

Frostburn: /* Adding Geometry */ Elucidate the behaviour of scalar multiples of the projective origin.

2023-11-27T07:34:51Z

Adding Geometry: Elucidate the behaviour of scalar multiples of the projective origin.

← Older revision		Revision as of 07:34, 27 November 2023
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	<math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> .		<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 ==		== Expanding geometry ==

Frostburn: Remove personal reminder.

2023-11-27T07:23:00Z

Remove personal reminder.

← Older revision		Revision as of 07:23, 27 November 2023
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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.		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...

Frostburn: /* On units */

2023-11-27T07:21:48Z

On units

← Older revision		Revision as of 07:21, 27 November 2023
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	== On units ==		== On units ==
	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.		Scalars do not have units. That's what makes them scalars. Do relative 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 is 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>.		Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals such as the ''jorp'' (€) 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>.

	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>.		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>.

Frostburn: /* Adding Geometry */

2023-11-27T07:17:05Z

Adding Geometry

← Older revision		Revision as of 07:17, 27 November 2023
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	<math>\overrightarrow{Hz} \mapsto e_0</math>		<math>\overrightarrow{Hz} \mapsto e_0</math>

	In represents a single point of origin in projective geometry (remember how we chose to ignore that extra scalar in front when summing absolute pitches).		It represents a single point of origin in projective geometry (remember how we chose to ignore that extra scalar in front when summing absolute pitches).

	In practice it's handy to have a reference frequency that humans can hear so we might use the shifted origin		In practice it's handy to have a reference frequency that humans can hear so we might use the shifted origin

Frostburn: /* Expanding geometry */ Make the projective origin null and ground an example.

2023-11-27T07:15:24Z

Expanding geometry: Make the projective origin null and ground an example.

← Older revision		Revision as of 07:15, 27 November 2023
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	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" towards 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.

			We make the projective origin <math>e_0</math> non-invertible by enforcing a null metric weight <math>e_0 \cdot e_0 = 0</math> which can be handy in some calculations and often does the right thing.

			Most of the theory developed here deals with relative pitch. It's always possible to ground a result on an origin e.g.

			<math>
			\mathrm{freq}(\tilde{e_0} + \overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12) = \mathrm{freq}(e_0 + \frac{47}{12} e_2 + e_5 + e_{11}) \approx 830.6 Hz
			</math>

	== On units ==		== On units ==

Frostburn: /* Adding Geometry */ Expand geometry with a projective origin based on logarithmic frequency.

2023-11-27T06:59:42Z

Adding Geometry: Expand geometry with a projective origin based on logarithmic frequency.

@@ Line 51: / Line 51: @@
 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 ==
 It's instructive to construct <math>\mathrm{ratio}</math> more explicitly by considering the inverses of the basis vectors

Frostburn: Designate absolute pitch domain.

2023-11-27T06:34:32Z

Designate absolute pitch domain.

@@ Line 15: / Line 15: @@
 When we take the logarithm of a positive rational scalar its factors separate into a sum e.g. <math>\log(15/8) = \log(3) + \log(5) - 3\log(2)</math>.
 == 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 basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality

Frostburn: Add a reminder to self.

2023-11-26T09:58:11Z

Add a reminder to self.

← Older revision		Revision as of 09:58, 26 November 2023
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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.		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...

Frostburn: Be more accurate by calling the "val" function "gpv" instead.

2023-11-26T08:47:38Z

Be more accurate by calling the "val" function "gpv" instead.

← Older revision		Revision as of 08:47, 26 November 2023
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	</math>		</math>

	However rational approximations to the <math>\mathsf{JIP}</math> do belong in our space. Equal temperaments can be represented by such approximations called [[~~Val~~\|vals]] which we define as		However rational approximations to the <math>\mathsf{JIP}</math> do belong in our space. Equal temperaments can be represented by such approximations called [[Patent_val#Generalized_patent_val\|generalized patent vals]] which we define as

	<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> ,		<math>\mathrm{gpv}(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> ,

			which belong to a class of [[Val\|vals]] complementary to [[Monzo\|monzos]].

	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.		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; 2, 3, 5) = 12 e^2 + 19 e^3 + 28 e^5 =: \langle 12, 19, 28 \rbrack</math>		<math>\overleftarrow{12} := \mathrm{gpv}(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.		We can use these new objects to calculate how many steps of 12edo a tempered interval spans e.g.
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	The geometric inverses are mainly relevant for subgroup temperaments. Consider [[The_Archipelago#Subgroup_temperaments\|Barbados]]:		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>		<math>\overleftarrow{5} := \mathrm{gpv}(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:		We can verify that the comma 676/675 indeed vanishes using this val:

Frostburn: /* On units */ Make it clearer that there's no "cent" for the tritave in this system.

2023-11-26T08:38:06Z

On units: Make it clearer that there's no "cent" for the tritave in this system.

← Older revision		Revision as of 08:38, 26 November 2023
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	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.		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.

			To be perfectly clear about the nature of these objects: There is no "cent" associated with primes besides 2. The closest we can get is:

			<math>\overrightarrow{3} \cdot \mathsf{JIP} / \log(2) * 1200 ¢ \approx 1901.955 ¢</math>

			Where we had to ''a)'' violate linear indepence over <math>\mathbb{Q}</math> ''b)'' remove "twoness" by dividing by the logarithm ''c)'' explicitly add the units to "twist" the answer into the requisitioned direction.

	== Clifford algebra nonsense ==		== Clifford algebra nonsense ==