Powharmonic series: Difference between revisions
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== ''p''-Powharmonic series == | == ''p''-Powharmonic series == | ||
A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a ''p-powharmonic series'' is simply: | A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a '''''p''-powharmonic series''' is simply: | ||
<math>\qquad f(n) = n^p | <math>\qquad f(n) = n^p | ||
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The harmonic series is technically a powharmonic series, but it is the trivial case, with the exponent equal to 1. <span><math>p</math><span> closer to 1 give series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics. | The harmonic series is technically a powharmonic series, but it is the trivial case, with the exponent equal to 1. <span><math>p</math><span> closer to 1 give series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics. | ||
Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS|AFS (arithmetic frequency sequence)]]. | Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant ''c''. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS|AFS (arithmetic frequency sequence)]]. | ||
Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series. | Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series. | ||
== log-base-b-of-a-Powharmonic series == | == log-base-''b''-of-''a''-Powharmonic series == | ||
[[File:Log-base-3-of-2-powharmonic series.png|thumb| | [[File:Log-base-3-of-2-powharmonic series.png|thumb| | ||
log-base-3-of-2-powharmonic series | log-base-3-of-2-powharmonic series | ||
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When we choose a <span><math>p</math></span> of the form <span><math>\log_{b}a</math></span>, the resulting scale will include every integer power of <span><math>a</math></span>, and the count of steps between each power of <span><math>a</math></span> will increase by a factor of <span><math>b</math></span>. | When we choose a <span><math>p</math></span> of the form <span><math>\log_{b}a</math></span>, the resulting scale will include every integer power of <span><math>a</math></span>, and the count of steps between each power of <span><math>a</math></span> will increase by a factor of <span><math>b</math></span>. | ||
Extending the naming scheme ''p-powharmonic series'', we call this a ''log-base-b-of-a-powharmonic series''. | Extending the naming scheme '''''p''-powharmonic series''', we call this a '''log-base-''b''-of-''a''-powharmonic series'''. | ||
=== Pitches per period === | === Pitches per period === | ||
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== ''a''-Edharmonic series == | == ''a''-Edharmonic series == | ||
=== Prerequisite: ln-of-a-powharmonic series === | === Prerequisite: ln-of-''a''-powharmonic series === | ||
[[File:Ln-of-2-powharmonic series.png|thumb| | [[File:Ln-of-2-powharmonic series.png|thumb| | ||
ln-of-2-powharmonic series | ln-of-2-powharmonic series | ||
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Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>. | Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>. | ||
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 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'' 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. | ||
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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. | 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 === | === Relation to ln-of-''a''-powharmonic series === | ||
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]]: | 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]]: | ||
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In yet other words, the definition of an a-edharmonic series is: | In yet other words, the definition of an ''a''-edharmonic series is: | ||
<math> \qquad f(n) = a^{H(n)} | <math> \qquad f(n) = a^{H(n)} | ||