Powharmonic series: Difference between revisions

{{powharmonic series|12|expo=0.69314718056}}

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.

[[File:Log-base-3-of-2-powharmonic series.png|thumb|

log-base-3-of-2-powharmonic series

=== Description ===

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>.

=== Pitches per period ===

<math>2, 4, 8, 16…

</math>

<math>2, 6, 18, 54…

=== Equality explanation ===

An equality involving exponents and logarithms helps us understand why:

=== Initial count ===

The first period of the series, determined by <span><math>a</math></span>, will contain <span><math>b - 1</math></span> pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain <span><math>4 - 1 = 3</math></span> pitches.

The harmonic series features counts of pitches of increasing powers of 2 in each next octave, but it also contains counts of pitches of increasing powers of 3 in each next tritave, and counts of pitches in increasing powers of 5 in each next 5/1 interval, and so forth. This is because the harmonic series is equivalent to the log-base-2-of-2-powharmonic series, the log-base-3-of-3-powharmonic series, the log-base-5-of-5-powharmonic series, and so forth (the log-base-b-of-b-powharmonic series). This because any <span><math>\log_{b}b = 1</math></span>.

[[File:Powharmonic space.png|378x378px]]

=== Prerequisite: ln-of-a-powharmonic series ===

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

ln-of-2-powharmonic series

=== Description ===

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>.

=== 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]]:

In other words, if we have gone by a step of 1ed2, we are at <span><math>2^1</math></span>. If we then go by a step of 2ed2, we have gone by <span><math>2^1 · 2^{\frac12} = 2^{\frac32}</math></span>. And a further step of 3ed2 gets us to <span><math>2^1 · 2^{\frac12} · 2^{\frac13} = 2^{\frac{11}{6}}</math></span>, etc.

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In the case of the (2-)edharmonic series, doing so brings it closer in similarity to the (musical) harmonic series; the first step is exactly an octave, the second step a fifth (701.96¢ vs 600.00¢), the third step a fourth (498.04¢ vs 400.00¢), the fourth step a third, (386.31¢ vs 300¢), etc. This similarity could be useful when using the entire series as a scale rather than drawing scales from it. We therefore propose referring to this variation as the "emulatory edharmonic series", because it emulates the harmonic series.

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