Powharmonic series

introduction

A powerharmonic series, like the harmonic series, is an infinitely ascending set of pitches from which scales can be drawn.

A powharmonic series can be built on any number [math]\displaystyle{ p }[/math], whether it is rational or irrational, positive or negative. The formula for a p-powharmonic series is simply:

[math]\displaystyle{ \qquad f(n) = n^p }[/math]

For example, the 0.69314718056-powharmonic series looks like this:

pitch #	frequency multiplier (definition)	frequency multiplier (decimal)	pitch (¢)	pitch Δ (¢)	octave reduced pitch (¢)
1	10.69314718056	1	0.00	-	0.00
2	20.69314718056	1.616806672	831.78	831.78	831.78
3	30.69314718056	2.141486064	1318.33	486.56	118.33
4	40.69314718056	2.614063815	1663.55	345.22	463.55
5	50.69314718056	3.05132936	1931.33	267.77	731.33
6	60.69314718056	3.462368957	2150.11	218.79	950.11
7	70.69314718056	3.852807616	2335.09	184.98	1135.09
8	80.69314718056	4.226435818	2495.33	160.24	95.33
9	90.69314718056	4.585962562	2636.67	141.34	236.67
10	100.69314718056	4.933409668	2763.10	126.43	363.10
11	110.69314718056	5.270337212	2877.47	114.37	477.47
12	120.69314718056	5.597981231	2981.89	104.41	581.89
13	130.69314718056	5.917342318	3077.94	96.05	677.94
14	140.69314718056	6.22924506	3166.87	88.93	766.87
15	150.69314718056	6.5343793	3249.66	82.79	849.66
16	160.69314718056	6.833329631	3327.11	77.45	927.11

The harmonic series is technically a powharmonic series: the 1-powharmonic series.

log-base-b-of-a-powharmonic series

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

When we choose a [math]\displaystyle{ p }[/math] of the form [math]\displaystyle{ \log_{b}a }[/math], the resulting scale will include every integer power of [math]\displaystyle{ a }[/math], and the count of steps between each power of [math]\displaystyle{ a }[/math] will be equal to the next integer power of [math]\displaystyle{ b }[/math].

By extension of 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 [math]\displaystyle{ p = log_{3}2 }[/math], 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]\displaystyle{ 2, 4, 8, 16… }[/math]

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

[math]\displaystyle{ 3, 9, 27, 81… }[/math]

An equality involving exponents and logarithms helps us understand why:

[math]\displaystyle{ \qquad x^{\log_{b}a} = a^{log_{b}x} }[/math]

Breaking this down step by step:

1. [math]\displaystyle{ \log_{b}x }[/math] gives the power to which [math]\displaystyle{ b }[/math] must be raised to give [math]\displaystyle{ x }[/math]
2. whenever [math]\displaystyle{ x }[/math] is an integer power (squared, cubed, etc.) of [math]\displaystyle{ b }[/math], [math]\displaystyle{ \log_{b}x }[/math] will be an integer
3. whenever [math]\displaystyle{ \log_{b}x }[/math] is an integer, we raise [math]\displaystyle{ a }[/math] to an integer power
4. [math]\displaystyle{ x }[/math] increments linearly by 1
5. it takes longer and longer each time for [math]\displaystyle{ x }[/math] to reach the next power of [math]\displaystyle{ b }[/math]

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

Include Jacob chart and point about all harmonic series being the same or per octave per tritave

see also

logharmonic series