User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ 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.
 == 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 ===
-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 ===
@@ 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.
+== 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.