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
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 related to the next integer power of [math]\displaystyle{ b }[/math].
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 [math]\displaystyle{ p = log_{3}2 }[/math], will — like the harmonic series — and by virtue of being "of 2" — include every octave of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:
[math]\displaystyle{ 2, 4, 8, 16… }[/math]
they’ll — by virtue of being "base-3" — increase by a factor of 3:
[math]\displaystyle{ 2, 6, 18, 54… }[/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:
- [math]\displaystyle{ \log_{b}x }[/math] gives the power to which [math]\displaystyle{ b }[/math] must be raised to give [math]\displaystyle{ x }[/math]
- 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
- whenever [math]\displaystyle{ \log_{b}x }[/math] is an integer, we raise [math]\displaystyle{ a }[/math] to an integer power
- [math]\displaystyle{ x }[/math], being the pitch # or index, increments linearly by 1
- it takes longer and longer each time for [math]\displaystyle{ x }[/math] to reach the next power of [math]\displaystyle{ b }[/math]
The first period of the series, determined by [math]\displaystyle{ a }[/math], will contain [math]\displaystyle{ b - 1 }[/math] pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain [math]\displaystyle{ 4 - 1 = 3 }[/math] pitches.
ln-of-a-powharmonic series
Irrational values can be used as [math]\displaystyle{ a }[/math] or [math]\displaystyle{ b }[/math].
In particular it may be of interest to use [math]\displaystyle{ e }[/math] as [math]\displaystyle{ b }[/math] — in other words, to use a natural logarithm.
For example, the ln-of-2-powharmonic series fits [math]\displaystyle{ e }[/math] times as many many more pitches into each next octave as the previous octave. Because [math]\displaystyle{ e }[/math] is irrational, however, no integer multiples of the octave will ever be reached.
In fact, this series is equivalent to the example given in the introduction, because [math]\displaystyle{ ln(2) ≈ 0.69314718056 }[/math].
edharmonic series
description
Perhaps even more interestingly, a ln-of-a-powharmonic series can be approximated by moving by steps of increasing equal divisions of [math]\displaystyle{ a }[/math].
For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), 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-2-powharmonic series. We could call this series the 2-edharmonic series.
relation to ln-of-a-powharmonic series
The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches the Euler-Mascheroni constant, which represents the difference between the natural logarithm and the mathematical harmonic series (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of [math]\displaystyle{ a }[/math] is equivalent to a series of pitches [math]\displaystyle{ 2^{H(n)} }[/math] where [math]\displaystyle{ H(n) }[/math] is the [math]\displaystyle{ n^{th} }[/math] harmonic number:
[math]\displaystyle{ \qquad H(1) = 1 \\ \qquad H(2) = \frac{3}{2} = 1 + \frac{1}{2} \\ \qquad H(3) = \frac{11}{6} = 1 + \frac{1}{2} + \frac{1}{3} \\ \qquad H(4) = \frac{25}{12} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ \qquad … }[/math]
In other words, if we have gone by a step of 1ed2, we are at [math]\displaystyle{ 2^1 }[/math]. If we then go by a step of 2ed2, we have gone by [math]\displaystyle{ 2^1 · 2^{\frac12} = 2^{\frac32} }[/math]. And a further step of 3ed2 gets us to [math]\displaystyle{ 2^1 · 2^{\frac12} · 2^{\frac13} = 2^{\frac{11}{6}} }[/math], etc.
| pitch # | ln-of-2-powharmonic series | 2-edharmonic series | ratio between frequency multipliers | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| frequency multiplier (definition) | frequency multiplier (decimal) | pitch (¢) | pitch Δ (¢) | octave reduced pitch (¢) | frequency multiplier (definition) | frequency multiplier (decimal) | pitch (¢) | pitch Δ (¢) | octave reduced pitch (¢) | ||
| 1 | 1ln(2) = 2ln(1) | 1 | 0.00 | - | 0.00 | 2H(1) | 2 | 1200.00 | - | 0.00 | 2 |
| 2 | 2ln(2) = 2ln(2) | 1.616806672 | 831.78 | 831.78 | 831.78 | 2H(2) | 2.828427125 | 1800.00 | 600.00 | 600.00 | 1.749391052 |
| 3 | 3ln(2) = 2ln(3) | 2.141486064 | 1318.33 | 486.56 | 118.33 | 2H(3) | 3.563594873 | 2200.00 | 400.00 | 1000.00 | 1.664075677 |
| 4 | 4ln(2) = 2ln(4) | 2.614063815 | 1663.55 | 345.22 | 463.55 | 2ln(4) | 4.237852377 | 2500.00 | 300.00 | 100.00 | 1.621174033 |
| 5 | 5ln(2) = 2ln(5) | 3.05132936 | 1931.33 | 267.77 | 731.33 | 2ln(5) | 4.868014055 | 2740.00 | 240.00 | 340.00 | 1.595374829 |
| 6 | 6ln(2) = 2ln(6) | 3.462368957 | 2150.11 | 218.79 | 950.11 | 2ln(6) | 5.464161027 | 2940.00 | 200.00 | 540.00 | 1.578156775 |
| 7 | 7ln(2) = 2ln(7) | 3.852807616 | 2335.09 | 184.98 | 1135.09 | 2ln(7) | 6.032922891 | 3111.43 | 171.43 | 711.43 | 1.56585106 |
| 8 | 8ln(2) = 2ln(8) | 4.226435818 | 2495.33 | 160.24 | 95.33 | 2ln(8) | 6.578949063 | 3261.43 | 150.00 | 861.43 | 1.556618708 |
| 9 | 9ln(2) = 2ln(9) | 4.585962562 | 2636.67 | 141.34 | 236.67 | 2ln(9) | 7.105658007 | 3394.76 | 133.33 | 994.76 | 1.549436549 |
| 10 | 10ln(2) = 2ln(10) | 4.933409668 | 2763.10 | 126.43 | 363.10 | 2ln(10) | 7.615655686 | 3514.76 | 120.00 | 1114.76 | 1.543690105 |
| 11 | 11ln(2) = 2ln(11) | 5.270337212 | 2877.47 | 114.37 | 477.47 | 2ln(11) | 8.110986229 | 3623.85 | 109.09 | 23.85 | 1.538988096 |
| 12 | 12ln(2) = 2ln(12) | 5.597981231 | 2981.89 | 104.41 | 581.89 | 2ln(12) | 8.593290568 | 3723.85 | 100.00 | 123.85 | 1.535069557 |
| 13 | 13ln(2) = 2ln(13) | 5.917342318 | 3077.94 | 96.05 | 677.94 | 2ln(13) | 9.063911377 | 3816.16 | 92.31 | 216.16 | 1.531753765 |
| 14 | 14ln(2) = 2ln(14) | 6.22924506 | 3166.87 | 88.93 | 766.87 | 2ln(14) | 9.523965051 | 3901.87 | 85.71 | 301.87 | 1.528911603 |
| 15 | 15ln(2) = 2ln(15) | 6.5343793 | 3249.66 | 82.79 | 849.66 | 2ln(15) | 9.974392624 | 3981.87 | 80.00 | 381.87 | 1.526448369 |
| 16 | 16ln(2) = 2ln(16) | 6.833329631 | 3327.11 | 77.45 | 927.11 | 2ln(16) | 10.41599671 | 4056.87 | 75.00 | 456.87 | 1.524293028 |
naming details
We cross-pollinate the abbreviation for "equal division" with affiliation for the pronunciation of "enharmonic" to get the name "edharmonic series".
Due to the dominance of octave in music, we can actually refer to the 2-edharmonic series simply as the edharmonic series for short.
other examples
As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.
equivalent powharmonic series
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 [math]\displaystyle{ \log_{b}b = 1 }[/math].
Include Jacob chart and point about all harmonic series being the same or per octave per tritave
see also
logharmonic series