User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
Add notes about improving the notation so that Tenney weights are implied. Typeset some val brackets better. |
→Improving the notation: Think about units of measurement. |
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:<math>\begin{align} | :<math>\begin{align} | ||
\overrightarrow{e_i} &:= e_i \log(p_i) \\ | \overrightarrow{e_i} &:= e_i \log(p_i) \\ | ||
\overleftarrow{e_i} &:= e_i / \log(p_i) | \overleftarrow{e_i} &:= e_i / \log(p_i) \\ | ||
\end{align}</math> | \end{align} | ||
\implies \overleftarrow{e_i}\overrightarrow{e_i} = 1</math> | |||
where ''p''<sub>i</sub> are the [[Just_intonation_subgroup|formal primes]] of the fractional just intonation subgroup. | where ''p''<sub>i</sub> are the [[Just_intonation_subgroup|formal primes]] of the fractional just intonation subgroup. | ||
It should be noted that the pseudoscalar relevant for tempering is <math>\prod_i \overleftarrow{e_i}</math>. | It should be noted that the pseudoscalar relevant for tempering is <math>\overleftarrow{i} := \prod_i \overleftarrow{e_i}</math>. | ||
This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>. | This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>. | ||
I will look into it more before turning this draft into a page in the Wiki proper. | I will look into it more before turning this draft into a page in the Wiki proper. | ||
It could also be nice to use units that make sure that all of the formulae make sense. Starting with sound frequency in Hz we identify ratios of frequencies as musically interesting such as the fifth = 660 Hz / 440 Hz. To make working with ratios easier we take the logarithm and measure pitch in cents: <math>\mathrm{fifth}_{cents} = \log_2(\frac{3}{2}) * 1200 ¢ = \log_2(3) * 1200 ¢ - \log_2(2) * 1200 ¢ = \overleftarrow{JIP} \cdot \overrightarrow{3/2}</math>. If we leave the just intonation point unit-less then monzo components will be measured in cents and val components in reciprocal cents <math>¢^{-1}</math>. | |||