User:Frostburn/Theory From First Principles: Difference between revisions
→Adding Geometry: Expand geometry with a projective origin based on logarithmic frequency. |
→Expanding geometry: Make the projective origin null and ground an example. |
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Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have | Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have | ||
<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1. | <math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1. | ||
We make the projective origin <math>e_0</math> non-invertible by enforcing a null metric weight <math>e_0 \cdot e_0 = 0</math> which can be handy in some calculations and often does the right thing. | |||
Most of the theory developed here deals with relative pitch. It's always possible to ground a result on an origin e.g. | |||
<math> | |||
\mathrm{freq}(\tilde{e_0} + \overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12) = \mathrm{freq}(e_0 + \frac{47}{12} e_2 + e_5 + e_{11}) \approx 830.6 Hz | |||
</math> | |||
== On units == | == On units == | ||