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

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 works for temperament <math>\mathbf{T}</math> of any rank in just intonation subgroups of any size and most notably without using a single matrix operation!
-== Decomposition ==
+== How the period and generator falls out of a rank-2 wedgie  ==
-In equal temperament it is obvious that the building block of any musical interval is a single scale step, it's not immediately obvious what intervals in higher rank temperaments are made out of. We can wedge two vals together but the factorization is not unique. What we need are two vals that are as simple as possible. '''TODO''': Scrap this and translate the algorithm from [[Wedgies and multivals]] to Geometric algebra.
+In equal temperament it is obvious that the building block of any musical interval is a single scale step. In rank-2 there are two independent building blocks. The one associated with the octave is called the period and the other the generator.
-Let's return to three dimensions (or two thinking projectively). Meantone can be decomposed into:
+Let's translate the [[Wedgies_and_multivals#The_procedure|procedure]] introduced in the wedgie page to the notation I'm using.
-:<math>< 0, 1, 4 ] \wedge <1, 0, -4]</math>
-As tunings these would be a division of 5/1 into 4 equal parts each representing a 3/1, while the other is the octave "divided" into a single unit. Let's call these vals <math>\overleftarrow{v_0}</math> and <math>\overleftarrow{v_1}</math>. Any val <math>\overleftarrow{v}</math> that supports Meantone can be expressed as a linear combination of these two vals and thus the number of scale steps for a given monzo is
-:<math>\begin{align}
-n &= \overleftarrow{v} \cdot \overrightarrow{m} \\
-  &= (p\overleftarrow{v_0} + q\overleftarrow{v_1}) \cdot \overrightarrow{m} \\
-  &= p(\overleftarrow{v_0} \cdot \overrightarrow{m}) + q(\overleftarrow{v_1} \cdot \overrightarrow{m})
-\end{align}
-</math>
-With a lot of hand waving we can say that 3/1 and 2/1 generate the Meantone temperament. Any frequency/pitch from an interval tuned to a specific version of meantone has a counterpart composed of only 3/1 and 2/1 in the same tuning with the same frequency/pitch.
-Another example might be Blackwood <math>= e_3 \wedge \overleftarrow{5}</math> where the generators are 5/1 (based on <math>e_2</math>) and one step of the octave divided into 5 equal parts (denoted in backslash notation as 1\5). It is impossible to find an integral val with a first component larger than zero but smaller than 5.
+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.
-A similar thing happens with Augmented <math>= e_2 \wedge \overleftarrow{3}</math>, but now the generators are 3/1 and 1\3.
+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 [https://en.wikipedia.org/wiki/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]].
-Something without a split octave and a 5/1 generator would be Dicot <math>= < 0, 2, 1] \wedge < 1, 1, 2]</math>.
+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.
-There are more musically useful ways to express these generators (the one related to the octave is usually called the period), but observations like this should be enough to make software that finds some generators with brute force and uses those to build [[MOS]] scales for rank 2 temperaments which is what the user most likely cares about.