Powharmonic series

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Introduction

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

p-powharmonic series

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

[math]\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. [math]p[/math] closer to 1 give powharmonic 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.

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

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

Description

When we choose a [math]p[/math] of the form [math]\log_{b}a[/math], the resulting scale will include every integer power of [math]a[/math], and the count of steps between each power of [math]a[/math] will increase by a factor of [math]b[/math].

Extending the naming scheme p-powharmonic series, we call this a log-base-b-of-a-powharmonic series.

Pitches per period

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

[math]2, 4, 8, 16… [/math]

they’ll — by virtue of being "base-3" — increase by a factor of 3:

[math]2, 6, 18, 54… [/math]

Equality explanation

An equality involving exponents and logarithms helps us understand why:

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

Breaking this down step by step:

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

Initial count

The first period of the series, determined by [math]a[/math], will contain [math]b - 1[/math] pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain [math]4 - 1 = 3[/math] pitches.

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 [math]\log_{b}b = 1[/math].

Any powharmonic series has infinite equivalent ways of being expressed. We can visualize the equivalences with the following coloration of powharmonic space:

Powharmonic space.png

a-edharmonic series

Prerequisite: ln-of-a-powharmonic series

ln-of-2-powharmonic series

Irrational values can be used as [math]a[/math] or [math]b[/math].

In particular it may be of interest to use [math]e[/math] as [math]b[/math] — in other words, to use a natural logarithm.

For example, the ln-of-2-powharmonic series fits [math]e[/math] times as many many more pitches into each next octave as the previous octave. Because [math]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]ln(2) ≈ 0.69314718056[/math], 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 [math]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 [math]2^γ ≈ 1.49196704047[/math], where [math]γ[/math] is the Euler-Mascheroni constant, [math]≈ 0.5772156649[/math], 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]a[/math] is equivalent to a series of pitches [math]2^{H(n)}[/math] where [math]H(n)[/math] is the [math]n^{th}[/math] harmonic number:

[math] \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]2^1[/math]. If we then go by a step of 2ed2, we have gone by [math]2^1 · 2^{\frac12} = 2^{\frac32}[/math]. And a further step of 3ed2 gets us to [math]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) = 21 2 1200.00 - 0.00 2
2 2ln(2) = 2ln(2) 1.616806672 831.78 831.78 831.78 2H(2) = 23/2 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) = 211/6 3.563594873 2200.00 400.00 1000.00 1.664075677
4 4ln(2) = 2ln(4) 2.614063815 1663.55 345.22 463.55 2H(4) = 225/12 4.237852377 2500.00 300.00 100.00 1.621174033
5 5ln(2) = 2ln(5) 3.05132936 1931.33 267.77 731.33 2H(5) = 2137/60 4.868014055 2740.00 240.00 340.00 1.595374829
6 6ln(2) = 2ln(6) 3.462368957 2150.11 218.79 950.11 2H(6) = 249/20 5.464161027 2940.00 200.00 540.00 1.578156775
7 7ln(2) = 2ln(7) 3.852807616 2335.09 184.98 1135.09 2H(7) = 2363/140 6.032922891 3111.43 171.43 711.43 1.56585106
8 8ln(2) = 2ln(8) 4.226435818 2495.33 160.24 95.33 2H(8) = 2761/280 6.578949063 3261.43 150.00 861.43 1.556618708
9 9ln(2) = 2ln(9) 4.585962562 2636.67 141.34 236.67 2H(9) = 27129/2520 7.105658007 3394.76 133.33 994.76 1.549436549
10 10ln(2) = 2ln(10) 4.933409668 2763.10 126.43 363.10 2H(10) 7.615655686 3514.76 120.00 1114.76 1.543690105
11 11ln(2) = 2ln(11) 5.270337212 2877.47 114.37 477.47 2H(11) 8.110986229 3623.85 109.09 23.85 1.538988096
12 12ln(2) = 2ln(12) 5.597981231 2981.89 104.41 581.89 2H(12) 8.593290568 3723.85 100.00 123.85 1.535069557
13 13ln(2) = 2ln(13) 5.917342318 3077.94 96.05 677.94 2H(13) 9.063911377 3816.16 92.31 216.16 1.531753765
14 14ln(2) = 2ln(14) 6.22924506 3166.87 88.93 766.87 2H(14) 9.523965051 3901.87 85.71 301.87 1.528911603
15 15ln(2) = 2ln(15) 6.5343793 3249.66 82.79 849.66 2H(15) 9.974392624 3981.87 80.00 381.87 1.526448369
16 16ln(2) = 2ln(16) 6.833329631 3327.11 77.45 927.11 2H(16) 10.41599671 4056.87 75.00 456.87 1.524293028 ... -> 2γ = 1.49196704047

In yet other words, the definition of an a-edharmonic series is:

[math] \qquad f(n) = a^{H(n)} [/math]

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.

Analogy with matharmonic series

Edharmonic series are to powharmonic series as the matharmonic series is to the logharmonic series.

Emulatory edharmonic series

The 0th 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 [math]\frac11[/math] to 0.

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 [math]a[/math].

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.

pitch # harmonic series emulatory edharmonic series
frequency multiplier (decimal) pitch (¢) pitch Δ (¢) octave reduced pitch (¢) frequency multiplier (definition) frequency multiplier (decimal) pitch (¢) pitch Δ (¢) octave reduced pitch (¢)
1 1.000000 0 - 0 2H(0) = 20 1.000000000 0 - 0
2 2.000000 1200 1200 0 2H(1) = 21 2.000000000 1200.00 1200.00 0.00
3 3.000000 1901.955001 701.955001 701.955001 2H(2) = 23/2 2.828427125 1800.00 600.00 600.00
4 4.000000 2400 498.044999 0 2H(3) = 211/6 3.563594873 2200.00 400.00 1000.00
5 5.000000 2786.313714 386.313714 386.313714 2H(4) = 225/12 4.237852377 2500.00 300.00 100.00
6 6.000000 3101.955001 315.6412870 701.955001 2H(5) = 2137/60 4.868014055 2740.00 240.00 340.00
7 7.000000 3368.825906 266.8709056 968.825906 2H(6) = 249/20 5.464161027 2940.00 200.00 540.00
8 8.000000 3600 231.1740935 0 2H(7) = 2363/140 6.032922891 3111.43 171.43 711.43
9 9.000000 3803.910002 203.9100017 203.910002 2H(8) = 2761/280 6.578949063 3261.43 150.00 861.43
10 10.000000 3986.313714 182.4037121 386.313714 2H(9) = 27129/2520 7.105658007 3394.76 133.33 994.76
11 11.000000 4151.317942 165.0042285 551.317942 2H(10) 7.615655686 3514.76 120.00 1114.76
12 12.000000 4301.955001 150.6370585 701.955001 2H(11) 8.110986229 3623.85 109.09 23.85
13 13.000000 4440.527662 138.5726609 840.527662 2H(12) 8.593290568 3723.85 100.00 123.85
14 14.000000 4568.825906 128.2982447 968.825906 2H(13) 9.063911377 3816.16 92.31 216.16
15 15.000000 4688.268715 119.4428083 1088.268715 2H(14) 9.523965051 3901.87 85.71 301.87
16 16.000000 4800 111.7312853 0 2H(15) 9.974392624 3981.87 80.00 381.87

An analogous emulatory matharmonic series exists.

See also

Logharmonic series