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>.

 

 

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.

=== 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]]:

{| class="wikitable center-all"

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

! rowspan="3" | Pitch #

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

! colspan="5" | 2-Edharmonic series

! rowspan="3" | Ratio between frequency multipliers

! colspan="2" | Frequency multiplier

! colspan="3" | Pitch

! colspan="2" | Frequency multiplier

! colspan="3" | Pitch

! Change (cents)

! Octave-reduced (cents)

! Change (cents)

! Octave-reduced (cents)

| 1

| 1^ln(2) = 2^ln(1)

| 1

| 0.00

| &mdash;

| 0.00

| 2^H(1) = 2¹

| 2

| 1200.00

| &mdash;

| 0.00

| 2

| 2

| 2^ln(2) = 2^ln(2)

| 1.616806672

| 831.78

| 831.78

| 831.78

| 2^H(2) = 2^3/2

| 2.828427125

| 1800.00

| 600.00

| 600.00

| 1.749391052

| 3

| 3^ln(2) = 2^ln(3)

| 2.141486064

| 1318.33

| 486.56

| 118.33

| 2^H(3) = 2^11/6

| 3.563594873

| 2200.00

| 400.00

| 1000.00

| 1.664075677

| 4

| 4^ln(2) = 2^ln(4)

| 2.614063815

| 1663.55

| 345.22

| 463.55

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

| 4.237852377

| 2500.00

| 300.00

| 100.00

| 1.621174033

| 5

| 5^ln(2) = 2^ln(5)

| 3.05132936

| 1931.33

| 267.77

| 731.33

| 2^H(5) = 2^137/60

| 4.868014055

| 2740.00

| 240.00

| 340.00

| 1.595374829

| 6

| 6^ln(2) = 2^ln(6)

| 3.462368957

| 2150.11

| 218.79

| 950.11

| 2^H(6) = 2^49/20

| 5.464161027

| 2940.00

| 200.00

| 540.00

| 1.578156775

| 7

| 7^ln(2) = 2^ln(7)

| 3.852807616

| 2335.09

| 184.98

| 1135.09

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

| 6.032922891

| 3111.43

| 171.43

| 711.43

| 1.56585106

| 8

| 8^ln(2) = 2^ln(8)

| 4.226435818

| 2495.33

| 160.24

| 95.33

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

| 6.578949063

| 3261.43

| 150.00

| 861.43

| 1.556618708

| 9

| 9^ln(2) = 2^ln(9)

| 4.585962562

| 2636.67

| 141.34

| 236.67

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

| 7.105658007

| 3394.76

| 133.33

| 994.76

| 1.549436549

| 10

| 10^ln(2) = 2^ln(10)

| 4.933409668

| 2763.10

| 126.43

| 363.10

| 2^H(10)

| 7.615655686

| 3514.76

| 120.00

| 1114.76

| 1.543690105

| 11

| 11^ln(2) = 2^ln(11)

| 5.270337212

| 2877.47

| 114.37

| 477.47

| 2^H(11)

| 8.110986229

| 3623.85

| 109.09

| 23.85

| 1.538988096

| 12

| 12^ln(2) = 2^ln(12)

| 5.597981231

| 2981.89

| 104.41

| 581.89

| 2^H(12)

| 8.593290568

| 3723.85

| 100.00

| 123.85

| 1.535069557

| 13

| 13^ln(2) = 2^ln(13)

| 5.917342318

| 3077.94

| 96.05

| 677.94

| 2^H(13)

| 9.063911377

| 3816.16

| 92.31

| 216.16

| 1.531753765

| 14

| 14^ln(2) = 2^ln(14)

| 6.22924506

| 3166.87

| 88.93

| 766.87

| 2^H(14)

| 9.523965051

| 3901.87

| 85.71

| 301.87

| 1.528911603

| 16

| 16^ln(2) = 2^ln(16)

| 6.833329631

| 3327.11

| 77.45

| 927.11

| 2^H(16)

| 10.41599671

| 4056.87

| 75.00

| 456.87

| 1.524293028 ... &rarr; 2^&gamma; = 1.49196704047

=== Other examples ===

 

 

In accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <span><math>a</math></span>.

 

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.

 

 

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]]