Powharmonic series: Difference between revisions

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[[File:Ln-of-2-powharmonic series.png|thumb|

~~natural~~ powharmonic series

ln-of-2-powharmonic series

]]

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In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)|<math>e</math>]] as <math>b</math> — in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].

For example, the ln-of-2-~~powharmonic series, or ''natural~~ powharmonic series'', fits <math>e</math> times as many many more pitches into each next octave as the previous octave. Because <math>e</math> is irrational, however, no integer multiples of the octave will ever be reached.

For example, the ''ln-of-2-powharmonic series'' fits <math>e</math> times as many many more pitches into each next octave as the previous octave. Because <math>e</math> is irrational, however, no integer multiples of the octave will ever be reached.

In fact, this series is equivalent to the example given in the introduction, because <math>ln(2) ≈ 0.69314718056</math>, and if any powharmonic series were to qualify to be referred to for short as "the" powharmonic series, this would be the one.

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Perhaps even more interestingly, a ln-of-a-powharmonic series can be approximated by moving by steps of increasing equal divisions of <math>a</math>.

For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ~~natural~~ powharmonic series. We could call this series the 2-edharmonic series.

For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.

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

The ratio between pitches of the ~~natural~~ 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]]:

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>

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|+

! rowspan="2" |pitch #

! colspan="5" |~~natural~~ powharmonic series

! colspan="5" |ln-of-2-powharmonic series

! colspan="5" |2-edharmonic series

! rowspan="2" |ratio between frequency multipliers

@@ Line 193: / Line 193: @@
 [[File:Ln-of-2-powharmonic series.png|thumb|
-natural powharmonic series
+ln-of-2-powharmonic series
 ]]
@@ Line 200: / Line 200: @@
 In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)|<span><math>e</math></span>]] as <span><math>b</math></span> — in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].
-For example, the ln-of-2-powharmonic series, or ''natural powharmonic series'', fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.
+For example, the ''ln-of-2-powharmonic series'' fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.
 In fact, this series is equivalent to the example given in the introduction, because <span><math>ln(2) ≈ 0.69314718056</math></span>, and if any powharmonic series were to qualify to be referred to for short as "the" powharmonic series, this would be the one.
@@ Line 208: / Line 208: @@
 Perhaps even more interestingly, a ln-of-a-powharmonic series can be approximated by moving by steps of increasing equal divisions of <span><math>a</math></span>.
-For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the natural powharmonic series. We could call this series the 2-edharmonic series.
+For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.
 === Relation to ln-of-a-powharmonic series ===
-The ratio between pitches of the natural 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]]:
+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>
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 |+
 ! rowspan="2" |pitch #
-! colspan="5" |natural powharmonic series
+! colspan="5" |ln-of-2-powharmonic series
 ! colspan="5" |2-edharmonic series
 ! rowspan="2" |ratio between frequency multipliers