Logharmonic series: Difference between revisions
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== introduction == | == introduction == | ||
[[File:2-logharmonic vs harmonic.png|thumb| | |||
2-logharmonic series vs. harmonic series | |||
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A logharmonic series is a variation of the harmonic series. Like the harmonic series, it is an infinitely long series of pitches from which scales can be drawn. But where the harmonic series is a linear series of pitches, with the formula f(n) = n, the formula for a b-logharmonic series is: | A logharmonic series is a variation of the harmonic series. Like the harmonic series, it is an infinitely long series of pitches from which scales can be drawn. But where the harmonic series is a linear series of pitches, with the formula f(n) = n, the formula for a b-logharmonic series is: | ||
Revision as of 01:47, 3 February 2020
introduction
A logharmonic series is a variation of the harmonic series. Like the harmonic series, it is an infinitely long series of pitches from which scales can be drawn. But where the harmonic series is a linear series of pitches, with the formula f(n) = n, the formula for a b-logharmonic series is:
[math]\displaystyle{ \qquad f(n) = log_b{n} }[/math]
If a natural number is chosen as b, the resulting series will be a superset of the harmonic series, inserting extra pitches. For example, the 2-logharmonic series inserts an extra step in between the fundamental and the 2nd harmonic, so that it takes two steps to reach the 2nd harmonic instead of one. Then it inserts 3 extra steps in between the 2nd harmonic and 3rd harmonic so that it takes four steps instead of one. Then 7 extra steps before the 4th harmonic so it takes 2^3 = 8 steps instead of 1.
matharmonic series
The e-logharmonic series can be approximated by pitches taken from the mathematical harmonic series:
1/1, 3/2, 11/6, 25/12, 137/60, 49/20, ...