Powharmonic series: Difference between revisions

Line 190:

=== relation to ln-of-a-powharmonic series ===

The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <math>a</math> is equivalent to a series of pitches <math>2^{H(n)}</math> where <math>H(n)</math> is the <math>n^{th}</math> [[wikipedia:Harmonic_number|harmonic number]]:

The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <math>2^γ ≈ 1.49196704047</math>, where <math>γ</math> is the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <math>≈ 0.5772156649</math>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <math>a</math> is equivalent to a series of pitches <math>2^{H(n)}</math> where <math>H(n)</math> is the <math>n^{th}</math> [[wikipedia:Harmonic_number|harmonic number]]:

<math>

@@ Line 190: / Line 190: @@
 === relation to ln-of-a-powharmonic series ===
-The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number|harmonic number]]:
+The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <span><math>2^γ ≈ 1.49196704047</math><span>, where <span><math>γ</math></span> is the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number|harmonic number]]:
 <math>