User:Frostburn/Geometric algebra for regular temperaments - Revision history

Frostburn: /* Mapping */ Remove the requirement for matrix inversion when calculating the mapping.

2022-06-12T06:35:03Z

Mapping: Remove the requirement for matrix inversion when calculating the mapping.

@@ Line 83: / Line 83: @@
 With a temperament of rank 2 the period <math>\overrightarrow{p}</math> and the generator <math>\overrightarrow{g}</math> together with n - 2 commas <math>\overrightarrow{c}_i</math> that are tempered out form a basis for the whole n-dimensional JI subgroup and can thus be inverted to produce a [[mapping]] for the temperament.
-Let's choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. In this case there's only a single comma <math>\overrightarrow{c} = \overrightarrow{81/80} = [-4, 4, 1 ></math>. Inverting the matrix <math>[\overrightarrow{p}, \overrightarrow{g}, \overrightarrow{c}]</math> gives us the mapping:
+Any interval within the JI subgroup can be written as
 :<math>\begin{align}
 \overleftarrow{p} &= < 1, 1, 0 ] \\
@@ Line 93: / Line 118: @@
 For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>.
-If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', I'm pretty sure the non-integral components of the matrix inverse will fall onto the comma mapping vals so they don't matter in the end. I also believe that its always possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically.
+If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', the fractions in the mapping cluster on the commas so they don't matter in the end. The period and generator always get integral mappings. It should also always be possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically.
 == Optimizing database keys ==

Frostburn: /* Geometric interpretation */

2022-05-24T12:13:31Z

Geometric interpretation

← Older revision		Revision as of 12:13, 24 May 2022
Line 10:		Line 10:

	3D lines that pass through the origin find representation in the [[Wikipedia:Geometric Algebra\|Geometric Algebra]] <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space.		3D lines that pass through the origin find representation in the [[Wikipedia:Geometric Algebra\|Geometric Algebra]] <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space.
	:<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math>		:<math> \langle 5, 8, 12 \rbrack \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math>

	I will use the term pseudoscalar to refer to the combination of all basis vectors and denote it with ''i''.		I will use the term pseudoscalar to refer to the combination of all basis vectors and denote it with ''i''.

Frostburn: /* Improving the notation */ Think about units of measurement.

2022-05-24T09:18:34Z

Improving the notation: Think about units of measurement.

← Older revision		Revision as of 09:18, 24 May 2022
Line 112:		Line 112:
	:<math>\begin{align}		:<math>\begin{align}
	\overrightarrow{e_i} &:= e_i \log(p_i) \\		\overrightarrow{e_i} &:= e_i \log(p_i) \\
	\overleftarrow{e_i} &:= e_i / \log(p_i)		\overleftarrow{e_i} &:= e_i / \log(p_i) \\
	\end{align}</math>		\end{align}
			\implies \overleftarrow{e_i}\overrightarrow{e_i} = 1</math>
	where ''p''<sub>i</sub> are the [[Just_intonation_subgroup\|formal primes]] of the fractional just intonation subgroup.		where ''p''<sub>i</sub> are the [[Just_intonation_subgroup\|formal primes]] of the fractional just intonation subgroup.
	It should be noted that the pseudoscalar relevant for tempering is <math>\prod_i \overleftarrow{e_i}</math>.		It should be noted that the pseudoscalar relevant for tempering is <math>\overleftarrow{i} := \prod_i \overleftarrow{e_i}</math>.

	This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>.		This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>.

	I will look into it more before turning this draft into a page in the Wiki proper.		I will look into it more before turning this draft into a page in the Wiki proper.

			It could also be nice to use units that make sure that all of the formulae make sense. Starting with sound frequency in Hz we identify ratios of frequencies as musically interesting such as the fifth = 660 Hz / 440 Hz. To make working with ratios easier we take the logarithm and measure pitch in cents: <math>\mathrm{fifth}_{cents} = \log_2(\frac{3}{2}) * 1200 ¢ = \log_2(3) * 1200 ¢ - \log_2(2) * 1200 ¢ = \overleftarrow{JIP} \cdot \overrightarrow{3/2}</math>. If we leave the just intonation point unit-less then monzo components will be measured in cents and val components in reciprocal cents <math>¢^{-1}</math>.

Frostburn: Add notes about improving the notation so that Tenney weights are implied. Typeset some val brackets better.

2022-05-24T06:39:28Z

Add notes about improving the notation so that Tenney weights are implied. Typeset some val brackets better.

← Older revision		Revision as of 06:39, 24 May 2022
Line 1:		Line 1:
	This is a work-in-progress for my (Frostburn) thoughts about using geometric algebra to work with regular temperaments. I'm trying to write this in such a way that both geometers and musicians can follow the line of thought.		This is a work-in-progress for my (Frostburn) thoughts about using geometric algebra to work with regular temperaments. I'm trying to write this in such a way that both geometers and musicians can follow the line of thought.
	== Vals ==		== Vals ==
	The simplest kinds of tunings only have a finite set of notes like our western [[12edo\|12-tone equal temperament]]. To understand such scales harmonically we need to know how to translate musically meaningful fractions such as [[3/2]] or [[5/3]] into numbers of steps in the scale of interest. Let's start with 5 equal divisions of the octave (the octave is the fraction 2/1). How should we represent the next prime 3/1? If 5 steps is the octave then the closest we can get to 3/1 is with 8 steps. The next prime 5/1 takes 12 steps of our 5-tone scale. These step counts are collected into a vector <math>< 5, 8, 12 ]</math> known as a [[Val\|val]] where the prime factorization of intervals of interest are collected into vectors known as [[Monzo\|monzos]]: <math>3/2 \mapsto [ -1, 1, 0 ></math> and <math>5/3 \mapsto [ 0, -1, 1 ></math>. We can now use the dot product to work out how many steps we need to represent 3/2 (known as the fifth) <math>< 5, 8, 12 \| -1, 1, 0 > = 8 - 5 = 3</math> steps of 5-tone equal temperament. Similarly 5/3 (known as the major sixth) equals <math>< 5, 8, 12 \| 0, -1, 1 > = 12 - 8 = 4</math> steps.		The simplest kinds of tunings only have a finite set of notes like our western [[12edo\|12-tone equal temperament]]. To understand such scales harmonically we need to know how to translate musically meaningful fractions such as [[3/2]] or [[5/3]] into numbers of steps in the scale of interest. Let's start with 5 equal divisions of the octave (the octave is the fraction 2/1). How should we represent the next prime 3/1? If 5 steps is the octave then the closest we can get to 3/1 is with 8 steps. The next prime 5/1 takes 12 steps of our 5-tone scale. These step counts are collected into a vector <math>\langle 5, 8, 12 \rbrack</math> known as a [[Val\|val]] where the prime factorization of intervals of interest are collected into vectors known as [[Monzo\|monzos]]: <math>3/2 \mapsto \lbrack -1, 1, 0 \rangle</math> and <math>5/3 \mapsto \lbrack 0, -1, 1 \rangle</math>. We can now use the dot product to work out how many steps we need to represent 3/2 (known as the fifth) <math>\langle 5, 8, 12 \| -1, 1, 0 \rangle = 8 - 5 = 3</math> steps of 5-tone equal temperament. Similarly 5/3 (known as the major sixth) equals <math>\langle 5, 8, 12 \| 0, -1, 1 \rangle = 12 - 8 = 4</math> steps.

	=== Combining vals ===		=== Combining vals ===
	Notice that it doesn't matter if we scale our val. <math>< 10, 16, 24 ]</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>< 7, 11, 16 ]</math> and add it to <math>< 5, 8, 12 ]</math>. The result is <math>< 12, 19, 28 ]</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>< 17, 27, 40 ]</math> which is different from the optimal ([[Patent val\|patent]]) val for 17-tone equal temperament <math>< 17, 27, \mathbf{39} ]</math>. By insisting that the first non-zero component of a val is positive and that all of the components are in lowest terms (GCD = 1) we get 1-[[Wedgies and multivals\|wedgies]] which are unique identifiers of tunings.		Notice that it doesn't matter if we scale our val. <math>\langle 10, 16, 24 \rbrack</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>\langle 7, 11, 16 \rbrack</math> and add it to <math>\langle 5, 8, 12 \rbrack</math>. The result is <math>\langle 12, 19, 28 \rbrack</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>\langle 17, 27, 40 \rbrack</math> which is different from the optimal ([[Patent val\|patent]]) val for 17-tone equal temperament <math>\langle 17, 27, \mathbf{39} \rbrack</math>. By insisting that the first non-zero component of a val is positive and that all of the components are in lowest terms (GCD = 1) we get 1-[[Wedgies and multivals\|wedgies]] which are unique identifiers of tunings.

	=== Geometric interpretation ===		=== Geometric interpretation ===
Line 107:		Line 107:

	As an example [[Marvel_family#Marvel\|Marvel]] ~ [1, 2, -2] which is just the non-two monzo components of the [[225/224\|marvel comma]] reversed and with alternating signs. This makes sense: a comma's two's component can be deduced from the other components by assuming that the comma has a positive size less than the octave.		As an example [[Marvel_family#Marvel\|Marvel]] ~ [1, 2, -2] which is just the non-two monzo components of the [[225/224\|marvel comma]] reversed and with alternating signs. This makes sense: a comma's two's component can be deduced from the other components by assuming that the comma has a positive size less than the octave.

			== Improving the notation ==
			It might be possible to consistently define everything so that the Tenney weighted metric is incorporated in the basis vectors. If ''e''<sub>i</sub> are the GA basis vectors that square to 1 we can define basis vectors
			:<math>\begin{align}
			\overrightarrow{e_i} &:= e_i \log(p_i) \\
			\overleftarrow{e_i} &:= e_i / \log(p_i)
			\end{align}</math>
			where ''p''<sub>i</sub> are the [[Just_intonation_subgroup\|formal primes]] of the fractional just intonation subgroup.
			It should be noted that the pseudoscalar relevant for tempering is <math>\prod_i \overleftarrow{e_i}</math>.

			This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>.

			I will look into it more before turning this draft into a page in the Wiki proper.

Frostburn: /* Optimizing database keys */ Add notes about generalizing to higher ranks.

2022-05-24T03:36:09Z

Optimizing database keys: Add notes about generalizing to higher ranks.

← Older revision		Revision as of 03:36, 24 May 2022
Line 97:		Line 97:
	== Optimizing database keys ==		== Optimizing database keys ==
	As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies\|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>2</sub>.''p''<sub>3</sub>.(…).''p''<sub>n</sub> calculate		As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies\|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>2</sub>.''p''<sub>3</sub>.(…).''p''<sub>n</sub> calculate
	:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math>		:<math>\mathbf{T}' = \langle 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) \rbrack \wedge \langle 0, W_{12}, W_{13}, \ldots, W_{1n} \rbrack </math>
	If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.		If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.

	Examples in 2.3.5 include [[Meantone_family#Meantone\|Meantone]] ~ [1, 4], [[Augmented_family#Augmented\|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29\|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore\|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle\|Miracle]] ~ [6, -7, -2] in 2.3.5.7.		Examples in 2.3.5 include [[Meantone_family#Meantone\|Meantone]] ~ [1, 4], [[Augmented_family#Augmented\|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29\|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore\|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle\|Miracle]] ~ [6, -7, -2] in 2.3.5.7.

			Note that this generalizes to arbitrary rank. For example in rank 3:
			:<math>\mathbf{T}' = \langle 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) \rbrack \wedge \langle \langle 0, \ldots, 0, W_{123}, W_{124}, \ldots, W_{1(n-1)n} \rbrack \rbrack</math>
			Where <math>W_{1ij}</math> are the 3-wedgie components associated with ''e<sub>1</sub>'' of a temperament <math>\mathbf{T}</math> and are back-filled into a 2-val. If <math>\mathbf{T}</math> is sufficiently good we have <math>\mathbf{T} = \mathrm{round}(\mathbf{T}')</math>.

			As an example [[Marvel_family#Marvel\|Marvel]] ~ [1, 2, -2] which is just the non-two monzo components of the [[225/224\|marvel comma]] reversed and with alternating signs. This makes sense: a comma's two's component can be deduced from the other components by assuming that the comma has a positive size less than the octave.

Frostburn: /* Mapping */ Comments about non-integral period monzos.

2022-05-21T13:45:11Z

Mapping: Comments about non-integral period monzos.

← Older revision		Revision as of 13:45, 21 May 2022
Line 92:		Line 92:

	For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>.		For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>.

			If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', I'm pretty sure the non-integral components of the matrix inverse will fall onto the comma mapping vals so they don't matter in the end. I also believe that its always possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically.

	== Optimizing database keys ==		== Optimizing database keys ==

Frostburn: /* Mapping */ Claim that the mapping can be calculated using matrix inversion.

2022-05-21T13:25:20Z

Mapping: Claim that the mapping can be calculated using matrix inversion.

← Older revision		Revision as of 13:25, 21 May 2022
Line 81:		Line 81:
	An expression such as <math>\mathbf{T} = \overleftarrow{12} \wedge \overleftarrow{19}</math> is nice for getting the feel of a temperament: Meantone feels like what [[12edo]] and [[19edo]] have in common. However it's hard to figure out how a given monzo should be mapped to periods and generators.		An expression such as <math>\mathbf{T} = \overleftarrow{12} \wedge \overleftarrow{19}</math> is nice for getting the feel of a temperament: Meantone feels like what [[12edo]] and [[19edo]] have in common. However it's hard to figure out how a given monzo should be mapped to periods and generators.

	~~We can invert~~ <math>\overrightarrow{p}</math> and <math>\overrightarrow{g}</math> ~~''(TODO: How?)''~~ to produce a [[mapping]] for the temperament. Let's choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. ~~The~~ mapping ~~vals will be~~:		With a temperament of rank 2 the period <math>\overrightarrow{p}</math> and the generator <math>\overrightarrow{g}</math> together with n - 2 commas <math>\overrightarrow{c}_i</math> that are tempered out form a basis for the whole n-dimensional JI subgroup and can thus be inverted to produce a [[mapping]] for the temperament.

			Let's choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. In this case there's only a single comma <math>\overrightarrow{c} = \overrightarrow{81/80} = [-4, 4, 1 ></math>. Inverting the matrix <math>[\overrightarrow{p}, \overrightarrow{g}, \overrightarrow{c}]</math> gives us the mapping:
	:<math>\begin{align}		:<math>\begin{align}
	\overleftarrow{p} &= < 1, 1, 0 ] \\		\overleftarrow{p} &= < 1, 1, 0 ] \\
	\overleftarrow{g} &= < 0, 1, 4 ]		\overleftarrow{g} &= < 0, 1, 4 ] \\
			\overleftarrow{c} &= < 0, 0, -1]
	\end{align}</math>		\end{align}</math>
	Now <math>\mathbf{T} = \overleftarrow{p} \wedge \overleftarrow{g}</math> and for any monzo <math>\overrightarrow{m}</math> we can get the number of periods as <math>\overleftarrow{p} \cdot \overrightarrow{m}</math> and the number of generators as <math>\overleftarrow{g} \cdot \overrightarrow{m}</math>.		Now <math>\mathbf{T} = \overleftarrow{p} \wedge \overleftarrow{g}</math> and for any monzo <math>\overrightarrow{m}</math> we can get the number of periods as <math>\overleftarrow{p} \cdot \overrightarrow{m}</math> and the number of generators as <math>\overleftarrow{g} \cdot \overrightarrow{m}</math>. (We don't care about how many commas are needed because they're tempered out anyway.)

	For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone ~~because~~ <math>\~~frac~~{5}{3}\frac{81}{80} = \frac{27}{16}~~</math> and the comma <math>~~\frac{81}{80}</math> ~~vanishes in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave~~.		For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>.

	== Optimizing database keys ==		== Optimizing database keys ==

Frostburn: Add notes about mapping

2022-05-21T05:43:37Z

Add notes about mapping

@@ Line 77: / Line 77: @@
 :<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math>
 If the temperament is not expressed in lowest terms the number of divisions will miscompensate and you will end up with the wrong equave.
 == Optimizing database keys ==

Frostburn: /* Optimizing database keys */ Fix subgroup prime indexing

2022-05-20T14:34:39Z

Optimizing database keys: Fix subgroup prime indexing

← Older revision		Revision as of 14:34, 20 May 2022
Line 79:		Line 79:

	== Optimizing database keys ==		== Optimizing database keys ==
	As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies\|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>1</sub>.''p''<sub>2</sub>.(…).''p''<sub>n</sub> calculate		As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies\|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>2</sub>.''p''<sub>3</sub>.(…).''p''<sub>n</sub> calculate
	:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math>		:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math>
	If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.		If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.

	Examples in 2.3.5 include [[Meantone_family#Meantone\|Meantone]] ~ [1, 4], [[Augmented_family#Augmented\|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29\|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore\|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle\|Miracle]] ~ [6, -7, -2] in 2.3.5.7.		Examples in 2.3.5 include [[Meantone_family#Meantone\|Meantone]] ~ [1, 4], [[Augmented_family#Augmented\|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29\|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore\|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle\|Miracle]] ~ [6, -7, -2] in 2.3.5.7.

Frostburn: Add notes about optimizing database lookup. Remove boldface from scalars.

2022-05-20T14:33:17Z

Add notes about optimizing database lookup. Remove boldface from scalars.

← Older revision		Revision as of 14:33, 20 May 2022
Line 54:		Line 54:
	and vice versa		and vice versa
	:<math>\overleftarrow{12} = \overrightarrow{64/63}i \vee \overrightarrow{81/80}i \vee \overrightarrow{128/125}i</math>		:<math>\overleftarrow{12} = \overrightarrow{64/63}i \vee \overrightarrow{81/80}i \vee \overrightarrow{128/125}i</math>
	In higher dimensions with algebra <math>\mathcal{G}(n,0)</math> pseudovals can be ~~decomposed~~ into a wedge of <math>(n-1)</math> vals and vals into a vee of <math>(n-1)</math> pseudovals.		In higher dimensions with algebra <math>\mathcal{G}(n,0)</math> pseudovals can be factored into a wedge of <math>(n-1)</math> vals and vals into a vee of <math>(n-1)</math> pseudovals.

	The projection formula for calculating the optimal tuning for a temperament (in Tenney-weighted coordinates)		The projection formula for calculating the optimal tuning for a temperament (in Tenney-weighted coordinates)
Line 65:		Line 65:
	Let's translate the [[Wedgies_and_multivals#The_procedure\|procedure]] introduced in the wedgie page to the notation I'm using.		Let's translate the [[Wedgies_and_multivals#The_procedure\|procedure]] introduced in the wedgie page to the notation I'm using.

	To find the '''period''' (in n dimensions). Calculate <math>d = gcd(~~\mathbf{W}_~~{12}, ~~\mathbf{W}_~~{13}, \ldots, ~~\mathbf{W}_~~{1n})</math>, where <math>~~\mathbf{W}_~~{1n}</math> are all of the bivector components associated with <math>e_1</math> (i.e. with prime 2) of the 2-wedgie for the temperament. Your period will be 1\d in backslash notation i.e. one step of d equal divisions of the octave.		To find the '''period''' (in n dimensions). Calculate <math>d = gcd(W_{12}, W_{13}, \ldots, W_{1n})</math>, where <math>W_{1n}</math> are all of the bivector components associated with <math>e_1</math> (i.e. with prime 2) of the 2-wedgie for the temperament. Your period will be 1\d in backslash notation i.e. one step of d equal divisions of the octave.

	To find the '''generator'''. Solve the equation <math>c_2~~\mathbf{W}_~~{12} + c_3~~\mathbf{W}_~~{13} + \ldots + c_n~~\mathbf{W}_~~{1n} = d</math> for the coefficients <math>c_2,\ldots,c_n</math> (using some algorithm such as the [[Wikipedia:extended Euclidean algorithm\|extended Euclidean algorithm]]). Then one valid generator for the temperament is <math>\overrightarrow{g}=c_2e_2 + c_3e_3 + \ldots + c_ne_n</math>. Thus (the tempered version of) the generator represents <math>q_2^{c_2} \cdots q_n^{c_n}</math> where 2.''q''<sub>2</sub>.(…).''q''<sub>n</sub> is the [[JI subgroup]].		To find the '''generator'''. Solve the equation <math>c_2 W_{12} + c_3 W_{13} + \ldots + c_n W_{1n} = d</math> for the coefficients <math>c_2,\ldots,c_n</math> (using some algorithm such as the [[Wikipedia:extended Euclidean algorithm\|extended Euclidean algorithm]]). Then one valid generator for the temperament is <math>\overrightarrow{g}=c_2e_2 + c_3e_3 + \ldots + c_ne_n</math>. Thus (the tempered version of) the generator represents <math>q_2^{c_2} \cdots q_n^{c_n}</math> where 2.''q''<sub>2</sub>.(…).''q''<sub>n</sub> is the [[JI subgroup]].

	This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament.		This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament.
Line 77:		Line 77:
	:<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math>		:<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math>
	If the temperament is not expressed in lowest terms the number of divisions will miscompensate and you will end up with the wrong equave.		If the temperament is not expressed in lowest terms the number of divisions will miscompensate and you will end up with the wrong equave.

			== Optimizing database keys ==
			As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies\|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>1</sub>.''p''<sub>2</sub>.(…).''p''<sub>n</sub> calculate
			:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math>
			If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.

			Examples in 2.3.5 include [[Meantone_family#Meantone\|Meantone]] ~ [1, 4], [[Augmented_family#Augmented\|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29\|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore\|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle\|Miracle]] ~ [6, -7, -2] in 2.3.5.7.