User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
→Geometric interpretation of temperaments: Add POTE mapping for meantone. Move (pseudo)scalar multiples to the right side of the vector. |
→Higher dimensions: Add a note about Tenney-weights w.r.t. the optimal tuning |
||
| Line 50: | Line 50: | ||
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. | ||
The projection formula for calculating the optimal tuning for a temperament | The projection formula for calculating the optimal tuning for a temperament (in Tenney-weighted coordinates) | ||
:<math>\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! | ||