Logharmonic series: Difference between revisions
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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 [[wikipedia:Harmonic_series_(music)|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 <span><math>f(n) = n</math></span>, the formula for a ''b-logharmonic series'' is: | ||
<math>\qquad f(n) = log_b{n} | <math>\qquad f(n) = log_b{n} | ||
</math> | </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 | If a natural number is chosen as <span><math>b</math></span>, 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 <span><math>2^1 = 2</math></span> 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 <span><math>2^2 = 4</math></span> steps instead of one. Then 7 extra steps before the 4th harmonic so it takes <span><math>2^3 = 8</math></span> steps instead of 1. | ||
== matharmonic series == | == matharmonic series == | ||
Revision as of 01:50, 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 [math]\displaystyle{ f(n) = n }[/math], the formula for a b-logharmonic series is:
[math]\displaystyle{ \qquad f(n) = log_b{n} }[/math]
If a natural number is chosen as [math]\displaystyle{ b }[/math], 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 [math]\displaystyle{ 2^1 = 2 }[/math] 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 [math]\displaystyle{ 2^2 = 4 }[/math] steps instead of one. Then 7 extra steps before the 4th harmonic so it takes [math]\displaystyle{ 2^3 = 8 }[/math] 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, ...