User:Frostburn/Theory From First Principles: Difference between revisions

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When we take the logarithm of a positive rational scalar its factors separate into a sum e.g. <math>\log(15/8) = \log(3) + \log(5) - 3\log(2)</math>.

== Absolute Pitch Domain ==

The way the logarithm converts multiplicative quantities into linear representation motivates the designation of intermediary domain where the units are kept intact inside the logarithm e.g.

<math>

\log(440 Hz) = \log(Hz \cdot 2^3 \cdot 5 \cdot 11) = \log(Hz) + 3 \log(2) + \log(5) + \log(11) .

</math>

To make use of this representation we would like to add quantities to each other so let's see how that works:

<math>

\log(361 Hz) + \log(529 Hz) = \log(Hz) + \log(19^2) + \log(Hz) + \log(23^2) = 2 \log(Hz) + 2 \log(19) + 2 \log(23) = 2 \log(437 Hz) ,

</math>

which looks perfectly reasonable besides that extra factor of 2 in the front. We choose to ignore it and interprete the sum of absolute pitch quantities as their [[Wikipedia:Geometric_mean|geometric mean]].

In the non-logarithmic domain this simply means that we have to keep track of the exponent of Hz and take the appropriate root. This has the added benefit of unifying the time and frequency domains. e.g. 1 millisecond is simply identified as the period of oscillation and corresponds to a frequency of 1 kilohertz.

== Adding Geometry ==

By the [[Wikipedia:Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] logarithms of primes are linearly independent over <math>\mathbb{Q}</math>, so we can interprete <math>\log(2), \log(3), \ldots</math> as basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality

@@ Line 15: / Line 15: @@
 When we take the logarithm of a positive rational scalar its factors separate into a sum e.g. <math>\log(15/8) = \log(3) + \log(5) - 3\log(2)</math>.
+== Absolute Pitch Domain ==
+The way the logarithm converts multiplicative quantities into linear representation motivates the designation of intermediary domain where the units are kept intact inside the logarithm e.g.
+<math>
+\log(440 Hz) = \log(Hz \cdot 2^3 \cdot 5 \cdot 11) = \log(Hz) + 3 \log(2) + \log(5) + \log(11) .
+</math>
+To make use of this representation we would like to add quantities to each other so let's see how that works:
+<math>
+\log(361 Hz) + \log(529 Hz) = \log(Hz) + \log(19^2) + \log(Hz) + \log(23^2) = 2 \log(Hz) + 2 \log(19) + 2 \log(23) = 2 \log(437 Hz) ,
+</math>
+which looks perfectly reasonable besides that extra factor of 2 in the front. We choose to ignore it and interprete the sum of absolute pitch quantities as their [[Wikipedia:Geometric_mean|geometric mean]].
+In the non-logarithmic domain this simply means that we have to keep track of the exponent of Hz and take the appropriate root. This has the added benefit of unifying the time and frequency domains. e.g. 1 millisecond is simply identified as the period of oscillation and corresponds to a frequency of 1 kilohertz.
 == Adding Geometry ==
 By the [[Wikipedia:Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] logarithms of primes are linearly independent over <math>\mathbb{Q}</math>, so we can interprete <math>\log(2), \log(3), \ldots</math> as basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality