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:

For example, the 0.69314718056-powharmonic series gives the series of pitches:

{| class="wikitable"

|'''frequency multiplier definition'''

|'''frequency multiplier (decimal)'''

|'''pitch (¢)'''

|'''pitch Δ (¢)'''

|'''octave reduced pitch (¢)'''

|1

|1^{0.69314718056}

|1

|0.00

| -

|0.00

|2

|2^{0.69314718056}

|1.616806672

|831.78

|831.78

|831.78

|3

|3^{0.69314718056}

|2.141486064

|1318.33

|486.56

|118.33

|4

|4^{0.69314718056}

|2.614063815

|1663.55

|345.22

|463.55

|5

|5^{0.69314718056}

|3.05132936

|1931.33

|267.77

|731.33

|6

|6^{0.69314718056}

|3.462368957

|2150.11

|218.79

|950.11

|7

|7^{0.69314718056}

|3.852807616

|2335.09

|184.98

|1135.09

|8

|8^{0.69314718056}

|4.226435818

|2495.33

|160.24

|95.33

|9

|9^{0.69314718056}

|4.585962562

|2636.67

|141.34

|236.67

|10

|10^{0.69314718056}

|4.933409668

|2763.10

|126.43

|363.10

|11

|11^{0.69314718056}

|5.270337212

|2877.47

|114.37

|477.47

|12

|12^{0.69314718056}

|5.597981231

|2981.89

|104.41

|581.89

|13

|13^{0.69314718056}

|5.917342318

|3077.94

|96.05

|677.94

|14

|14^{0.69314718056}

|6.22924506

|3166.87

|88.93

|766.87

|15

|15^{0.69314718056}

|6.5343793

|3249.66

|82.79

|849.66

|16

|16^{0.69314718056}

|6.833329631

|3327.11

|77.45

|927.11

For example, the log-base-3-of-2-powharmonic series, where <span><math>p = log_{3}2</math></span>, will — like the harmonic series — include every octave of the fundamental. However, instead of the octaves containing counts of pitches in increasing powers of 2

<math>2, 4, 8, 16…

they’ll contain counts of pitches in increasing powers of 3

<math>3, 9, 27, 81…

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

so if log b of x is what power b must be raised to to give x, then as you increment x linearly by 1, whenever you reach a power of b, b will be a natural number, which when you raise a to that, you'll get a "natural"/pure/whatever power of a.

== ln-of-a-powharmonic series ==

Irrational values can be used as a or b. In particular it may be of interest to use e as b, i.e. a natural logarithm, such as in ln-of-2-powharmonic series. In fact, this is the example given earlier, since ln(2) = 0.69314718056.

== edharmonic series ==

 

Perhaps even more interestingly, ln-of-a powharmonic series can be approximated by series constructed by moving by steps of increasing equal divisions of a. For example, if we first move by a step of 1ed2, then by 2ed2, then 3ed2, 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-a series. The difference between the frequency multipliers on our fundamental will approach the Euler-Masceroni constant, which represents the difference between the natural logarithm and the mathematical harmonic series. This is because moving by steps of increasing equal divisions of a is equivalent to a series of pitches 2^H(n) where H(n) is the nth harmonic number.

(insert chart with edharmonic series, and maybe a few columns comparing it with ln-of-2 powharmonic series)

We can refer to the 2-edharmonic series as the edharmonic series for short. The 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.

== equivalent powharmonic series ==

logharmonic series