User:Frostburn/Theory From First Principles: Difference between revisions
Designate absolute pitch domain. |
→Adding Geometry: Expand geometry with a projective origin based on logarithmic frequency. |
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In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> . | In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> . | ||
Absolute pitch can be incorporated when we introduce a basis vector <math>e_0</math> and associate it with logarithmic frequency | |||
<math>\overrightarrow{Hz} \mapsto e_0</math> | |||
In represents a single point of origin in projective geometry (remember how we chose to ignore that extra scalar in front when summing absolute pitches). | |||
In practice it's handy to have a reference frequency that humans can hear so we might use the shifted origin | |||
<math>\tilde{e_0} := \overrightarrow{440Hz} = e_0 + 3 e_2 + e_5 + e_{11}</math>. | |||
When working with absolute pitch, the inverse of the arrow function is called <math>\mathrm{freq}</math> instead of <math>\mathrm{ratio}</math> | |||
<math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> . | |||
== Expanding geometry == | == Expanding geometry == | ||
It's instructive to construct <math>\mathrm{ratio}</math> more explicitly by considering the inverses of the basis vectors | It's instructive to construct <math>\mathrm{ratio}</math> more explicitly by considering the inverses of the basis vectors | ||