Logharmonic series: Difference between revisions
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<math>\qquad f(n) = log_b{n} | <math>\qquad f(n) = log_b{n} | ||
</math> | </math> | ||
At <span><math>f(1)</math></span>, any logharmonic series will be <span><math>0</math></span>, which is not useful as a frequency multiplier, since there is no such thing as 0 Hz. So, we ignore the first step of logharmonic series. | |||
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 one. | 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 one. | ||
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|386.31 | |386.31 | ||
|} | |} | ||
For short, the e-logharmonic series may be simply called the logharmonic series. | |||
= matharmonic series = | = matharmonic series = | ||
The | The logharmonic series can be approximated by pitches taken from the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series): | ||
<math> | <math> | ||
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We can call this approximating series the ''matharmonic series''. | We can call this approximating series the ''matharmonic series''. | ||
The difference between pitches of the | The difference between pitches of the logharmonic series and the matharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the mathematical harmonic series. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| rowspan="2" |'''pitch #''' | | rowspan="2" |'''pitch #''' | ||
| colspan="5" |''' | | colspan="5" |'''logharmonic series''' | ||
| colspan="5" |'''matharmonic series''' | | colspan="5" |'''matharmonic series''' | ||
| rowspan="2" |'''difference between frequency multipliers''' | | rowspan="2" |'''difference between frequency multipliers''' | ||
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|0.5937897493 ... -> ''γ ='' 0.5772156649 | |0.5937897493 ... -> ''γ ='' 0.5772156649 | ||
|} | |} | ||