User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
→Higher dimensions: Add a note about Tenney-weights w.r.t. the optimal tuning |
Notes about decomposition |
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:<math>e_{12} + 4 e_{13} + 10 e_{14} + 4 e_{23} + 13 e_{24} + 12 e_{34}</math> | :<math>e_{12} + 4 e_{13} + 10 e_{14} + 4 e_{23} + 13 e_{24} + 12 e_{34}</math> | ||
I do not yet know the significance of these numbers, but it is of great practical use that rank-2 temperaments in higher dimensions get unique integral representations (up to scalar multiplication). Because these values can be made canonical they can be used as keys in a database for querying information about temperaments. | I do not yet know the significance of these numbers, but it is of great practical use that rank-2 temperaments in higher dimensions get unique integral representations (up to scalar multiplication). Because these values can be made canonical they can be used as keys in a database for querying information about temperaments. | ||
Observe how vals build temperaments by wedges in increasing rank while pseudovals do increasing damage to just intonation (representable by the pseudoscalar <math>i</math>) and successive vees decrease the rank of the resulting temperament. Pseudovals can be factored into vals | |||
:<math>\overrightarrow{126/125}i = \overleftarrow{12} \wedge \overleftarrow{27} \wedge \overleftarrow{31}</math> | |||
and vice versa | |||
:<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. | |||
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) | ||
:<math>\overleftarrow{TE} = \overleftarrow{JIP} \cdot \mathbf{T} / \mathbf{T}</math> | :<math>\overleftarrow{TE} = \overleftarrow{JIP} \cdot \mathbf{T} / \mathbf{T}</math> | ||
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! | 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 == | |||
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. | |||
Let's return to three dimensions (or two thinking projectively). Meantone can be decomposed into: | |||
:<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>. I think they represent the generators of Meantone 3/1 and 2/1, but I need to some more research. (to be continued...) | |||