User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
→Combining vals: Add a note about wedgies |
→Higher dimensions: Add notes about the numbers being a wedgie |
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where the overline represents the Hodge dual. | where the overline represents the Hodge dual. | ||
To give an example we have Septimal Meantone <math>= \overleftarrow{19} \wedge \overleftarrow{12} = \overrightarrow{81/80}i \vee \overrightarrow{126/125}i</math> with the numerical value: | To give an example we have Septimal Meantone <math>= \overleftarrow{19} \wedge \overleftarrow{12} = \overrightarrow{81/80}i \vee \overrightarrow{126/125}i</math> which is a bival with the numerical value: | ||
:<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> | ||
It's already positive and in lowest terms so it's a 2-wedgie. It is of great practical use that rank-2 temperaments in higher dimensions get unique integral representations. For example 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 | 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 | ||