Powharmonic series: Difference between revisions

== Introduction ==

A powerharmonic series, like the [[wikipedia:Harmonic_series_(music)|harmonic series]], is an infinitely ascending set of pitches from which scales can be drawn.

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:

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)]].

 

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 ==

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'''.

For example, the log-base-3-of-2-powharmonic series, where <span><math>p = log_{3}2</math></span>, will&mdash;like the harmonic series and by virtue of being "of 2"&mdash;include every octave (multiple of 2) of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:

they will&mdash;by virtue of being "base-3"&mdash;increase by a factor of 3:

<math>\qquad n^{\log_{b}a} = a^{log_{b}n}

# <span><math>\log_{b}n</math></span> gives the power to which <span><math>b</math></span> must be raised to give <span><math>n</math></span>

# whenever <span><math>n</math></span> is an integer power (squared, cubed, etc.) of <span><math>b</math></span>, <span><math>\log_{b}n</math></span> will be an integer

# whenever <span><math>\log_{b}n</math></span> is an integer, we raise <span><math>a</math></span> to an integer power

# <span><math>n</math></span>, being the pitch # or index, increments linearly by 1

# it takes longer and longer each time for <span><math>n</math></span> to reach the next power of <span><math>b</math></span>

=== Equivalences ===

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>.

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>&mdash;in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].

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.

=== 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]]:

|+ style="font-size: 105%;" | Comparison of ln-of-2-Powharmonic with 2-Edharmonic series+

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

! colspan="5" | 2-Edharmonic series

| 2^H(4) = 2^25/12

 

<math> \qquad f(n) = a^{H(n)}

=== Other examples ===

 

 

The 0^th harmonic number is not defined, however, if it were, it seems reasonable to assume it would be defined as 0; in other words, the first step of the harmonic series would be to add <span><math>\frac11</math></span> to 0.

 

 

 

| 2^H(7) = 2^363/140

| 2^H(8) = 2^761/280

| 2^H(9) = 2^7129/2520

An analogous [https://en.xen.wiki/w/Logharmonic_series#Emulatory_matharmonic_series emulatory matharmonic series] exists.

== See also ==

[[Harmonotonic tunings]]: powharmonic series are non-[[Arithmetic tunings|arithmetic]] harmonotonic tunings.

[[Logharmonic series]]: another type of non-arithmetic harmonotonic tuning.

[[Category:Xenharmonic series]]