Logharmonic series

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Introduction

2-logharmonic series vs. harmonic series

A logharmonic series is a variation of the harmonic series. Like the harmonic series, it is an infinitely long series of pitches from which scales can be drawn. But where the harmonic series is a linear series of pitches, with the formula [math]f(n) = n[/math], the formula for a b-logharmonic series is:

[math]\qquad f(n) = log_b{n} [/math]

Where [math]b \gt 1[/math].

At [math]f(1)[/math], any logharmonic series will be [math]0[/math], which is not useful as a frequency multiplier, since there is no such thing as 0 Hz. So, we ignore the first step of logharmonic series.

If a natural number is chosen as [math]b[/math], the resulting series will be a superset of the harmonic series, inserting extra pitches. For example, the 2-logharmonic series inserts an extra step in between the fundamental and the 2nd harmonic, so that it takes [math]2^1 = 2[/math] steps to reach the 2nd harmonic instead of one. Then it inserts 3 extra steps in between the 2nd harmonic and 3rd harmonic so that it takes [math]2^2 = 4[/math] steps instead of one. Then 7 extra steps before the 4th harmonic so it takes [math]2^3 = 8[/math] steps instead of one.

Frequencies

2-logharmonic series harmonic series
pitch # frequency multiplier (definition) frequency multiplier (decimal) pitch (¢) pitch Δ (¢) octave reduced pitch (¢) pitch # frequency multiplier (definition) frequency multiplier (decimal) pitch (¢) pitch Δ (¢) octave reduced pitch (¢)
2 log22 1.00000000 0.00 - 0.00 1 1 1.000000 0.00 - 0.00
3 log23 1.584962501 797.34 797.34 797.34
4 log24 2.00000000 1200.00 402.66 0.00 2 2 2.000000 1200.00 1200.00 0.00
5 log25 2.321928095 1458.39 258.39 258.39
6 log26 2.584962501 1644.17 185.78 444.17
7 log27 2.807354922 1787.05 142.88 587.05
8 log28 3.00000000 1901.96 114.90 701.96 3 3 3.000000 1901.96 701.96 701.96
9 log29 3.169925001 1997.34 95.38 797.34
10 log210 3.321928095 2078.43 81.09 878.43
11 log211 3.459431619 2148.64 70.22 948.64
12 log212 3.584962501 2210.35 61.71 1010.35
13 log213 3.700439718 2265.24 54.89 1065.24
14 log214 3.807354922 2314.55 49.31 1114.55
15 log215 3.906890596 2359.23 44.68 1159.23
16 log216 4.00000000 2400.00 40.77 0.00 4 4 4.000000 2400.00 498.04 0.00
17 log217 4.087462841 2437.45 37.45 37.45
18 log218 4.169925001 2472.03 34.58 72.03
19 log219 4.247927513 2504.11 32.09 104.11
20 log220 4.321928095 2534.01 29.90 134.01
21 log221 4.392317423 2561.98 27.97 161.98
22 log222 4.459431619 2588.23 26.25 188.23
23 log223 4.523561956 2612.95 24.72 212.95
24 log224 4.584962501 2636.29 23.34 236.29
25 log225 4.64385619 2658.39 22.10 258.39
26 log226 4.700439718 2679.35 20.97 279.35
27 log227 4.754887502 2699.29 19.94 299.29
28 log228 4.807354922 2718.29 19.00 318.29
29 log229 4.857980995 2736.43 18.14 336.43
30 log230 4.906890596 2753.77 17.34 353.77
31 log231 4.95419631 2770.38 16.61 370.38
32 log232 5.00000000 2786.31 15.93 386.31 5 5 5.000000 2786.31 386.31 386.31

A subset of the 2-logharmonic series is explored by Robert P. Schneider in his 2013 paper A Non-Pythagorean Musical Scale Based on Logarithms.

For short, the e-logharmonic series may be simply called the logharmonic series.

Matharmonic series

The logharmonic series can be approximated by pitches taken from the mathematical harmonic series (as opposed to the musical harmonic series):

[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]

We can call this approximating series the matharmonic series.

The difference between pitches of the logharmonic series and the matharmonic series approaches the Euler-Mascheroni constant, [math]≈ 0.5772156649[/math], which represents the difference between the natural logarithm and the mathematical harmonic series.

pitch # logharmonic series matharmonic series difference 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 ln(1) 0 N/A N/A N/A H(1) 1 0.00 701.96 0.00 1
2 ln(2) 0.6931471806 -634.52 - 565.48 H(2) 1.5 701.96 347.41 701.96 0.8068528194
3 ln(3) 1.098612289 162.82 797.34 162.82 H(3) 1.833333333 1049.36 221.31 1049.36 0.7347210447
4 ln(4) 1.386294361 565.48 402.66 565.48 H(4) 2.083333333 1270.67 158.70 70.67 0.6970389722
5 ln(5) 1.609437912 823.87 258.39 823.87 H(5) 2.283333333 1429.37 121.97 229.37 0.6738954209
6 ln(6) 1.791759469 1009.65 185.78 1009.65 H(6) 2.45 1551.34 98.11 351.34 0.6582405308
7 ln(7) 1.945910149 1152.53 142.88 1152.53 H(7) 2.592857143 1649.45 81.51 449.45 0.6469469938
8 ln(8) 2.079441542 1267.44 114.90 67.44 H(8) 2.717857143 1730.96 69.37 530.96 0.6384156012
9 ln(9) 2.197224577 1362.82 95.38 162.82 H(9) 2.828968254 1800.33 60.14 600.33 0.6317436766
10 ln(10) 2.302585093 1443.91 81.09 243.91 H(10) 2.928968254 1860.47 52.92 660.47 0.626383161
11 ln(11) 2.397895273 1514.12 70.22 314.12 H(11) 3.019877345 1913.39 47.13 713.39 0.6219820721
12 ln(12) 2.48490665 1575.83 61.71 375.83 H(12) 3.103210678 1960.51 42.39 760.51 0.6183040284
13 ln(13) 2.564949357 1630.72 54.89 430.72 H(13) 3.180133755 2002.90 38.45 802.90 0.6151843977
14 ln(14) 2.63905733 1680.03 49.31 480.03 H(14) 3.251562327 2041.36 35.14 841.36 0.6125049969
15 ln(15) 2.708050201 1724.71 44.68 524.71 H(15) 3.318228993 2076.50 32.31 876.50 0.6101787921
16 ln(16) 2.772588722 1765.48 40.77 565.48 H(16) 3.380728993 2108.80 29.86 908.80 0.608140271
17 ln(17) 2.833213344 1802.93 37.45 602.93 H(17) 3.439552523 2138.67 27.74 938.67 0.6063391786
18 ln(18) 2.890371758 1837.51 34.58 637.51 H(18) 3.495108078 2166.40 25.88 966.40 0.6047363203
19 ln(19) 2.944438979 1869.59 32.09 669.59 H(19) 3.547739657 2192.28 24.23 992.28 0.603300678
20 ln(20) 2.995732274 1899.49 29.90 699.49 H(20) 3.597739657 2216.51 22.76 1016.51 0.6020073836
21 ln(21) 3.044522438 1927.46 27.97 727.46 H(21) 3.645358705 2239.27 21.45 1039.27 0.600836267
22 ln(22) 3.091042453 1953.71 26.25 753.71 H(22) 3.69081325 2260.73 20.27 1060.73 0.5997707969
23 ln(23) 3.135494216 1978.43 24.72 778.43 H(23) 3.734291511 2281.00 19.21 1081.00 0.5987972952
24 ln(24) 3.17805383 2001.77 23.34 801.77 H(24) 3.775958178 2300.21 18.24 1100.21 0.5979043474
25 ln(25) 3.218875825 2023.87 22.10 823.87 H(25) 3.815958178 2318.45 17.36 1118.45 0.5970823529
26 ln(26) 3.258096538 2044.84 20.97 844.84 H(26) 3.854419716 2335.82 16.56 1135.82 0.5963231782
27 ln(27) 3.295836866 2064.77 19.94 864.77 H(27) 3.891456753 2352.37 15.82 1152.37 0.5956198872
28 ln(28) 3.33220451 2083.77 19.00 883.77 H(28) 3.927171039 2368.19 15.13 1168.19 0.5949665288
29 ln(29) 3.36729583 2101.91 18.14 901.91 H(29) 3.961653798 2383.32 14.51 1183.32 0.5943579676
30 ln(30) 3.401197382 2119.25 17.34 919.25 H(30) 3.994987131 2397.83 13.92 1197.83 0.5937897493 ... -> γ = 0.5772156649

The matharmonic series is to the logharmonic series as an edharmonic series is to a ln-of-a-powharmonic series.

Emulatory matharmonic series

The first two steps of the matharmonic series are 1 and 3/2, which have the same ratio as the second and third steps of the harmonic series, 2 and 3. To make them align the matharmonic series may be rebased onto 2, starting it a step late, inserting a 1 before it starts. This brings it closer in similarity to the harmonic series; 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 matharmonic 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 0.00 - 0.000000 1 1 0.00 0.00 1200.00
2 2.000000 1200.00 1200.00 0.000000 2⋅H(1) 2 1200.00 0.00 701.96
3 3.000000 1901.96 701.96 701.955001 2⋅H(2) 3 1901.96 701.96 347.41
4 4.000000 2400.00 498.04 0.000000 2⋅H(3) 3.666666667 2249.36 1049.36 221.31
5 5.000000 2786.31 386.31 386.313714 2⋅H(4) 4.166666667 2470.67 70.67 158.70
6 6.000000 3101.96 315.64 701.955001 2⋅H(5) 4.566666667 2629.37 229.37 121.97
7 7.000000 3368.83 266.87 968.825906 2⋅H(6) 4.9 2751.34 351.34 98.11
8 8.000000 3600.00 231.17 0.000000 2⋅H(7) 5.185714286 2849.45 449.45 81.51
9 9.000000 3803.91 203.91 203.910002 2⋅H(8) 5.435714286 2930.96 530.96 69.37
10 10.000000 3986.31 182.40 386.313714 2⋅H(9) 5.657936508 3000.33 600.33 60.14
11 11.000000 4151.32 165.00 551.317942 2⋅H(10) 5.857936508 3060.47 660.47 52.92
12 12.000000 4301.96 150.64 701.955001 2⋅H(11) 6.03975469 3113.39 713.39 47.13
13 13.000000 4440.53 138.57 840.527662 2⋅H(12) 6.206421356 3160.51 760.51 42.39
14 14.000000 4568.83 128.30 968.825906 2⋅H(13) 6.36026751 3202.90 802.90 38.45
15 15.000000 4688.27 119.44 1088.268715 2⋅H(14) 6.503124653 3241.36 841.36 35.14
16 16.000000 4800.00 111.73 0.000000 2⋅H(15) 6.636457986 3276.50 876.50 34.44

An analogous emulatory edharmonic series exists.

Sublogharmonic series

Just like the subharmonic series can be found by using one over the frequency of any step of the harmonic series, an equivalent sublogharmonic series may be found by using one over the frequency of any step of a logharmonic series.

Sound of the logharmonic series as a timbre

The usual harmonic series can be thought of as being derived from a periodic timbre, such as a sawtooth wave. In general, we can derive the sawtooth wave as

[math]\sum_{n=1}^\infty \frac{\sin(nt)}{n}[/math]

We can change the "rolloff" on the harmonics to get different derived waveforms which instead have the partials rolling off at a rate of [math]n^s[/math]:

[math]\sum_{n=1}^\infty \frac{\sin(nt)}{n^s}[/math]

As [math]s[/math] increases, so does the rolloff (and hence the brightness of the timbre), so that as we get [math]s \to \infty[/math], our sawtooth wave goes to a sine wave.

If we instead replace our [math]\sin(nt)[/math] term with [math]\sin(\log(n) t)[/math], we instead get the Riemann Zeta function, treated as an audio sound wave:

[math]\sum_{n=1}^\infty \frac{\sin(\log(n) t)}{n^s} = \mathbf{Im}[-\zeta(s+it)][/math]

This is easy to see using the Dirichlet series representation:

[math]\zeta(s+it) = \sum_n \frac{1}{n^{s+it}} = \sum_n \frac{n^{-it}}{n^s} = \sum_n \frac{\exp(-it \log(n))}{n^s}[/math]

And since we have [math]-\sin(t \log(n)) = \exp(-it \log(n))[/math], this completes our proof.

Note that the above only converges if we have [math]s \gt 1[/math], e.g. if the rolloff is greater than that of a sawtooth. However, we can use the analytic continuation of the Riemann Zeta function to procure a function even for [math]s \lt 1[/math]. Here is an example of the critical line of the Riemann zeta function (e.g. [math]s = 0.5[/math] played as a sound wave):

(Mike Battaglia's note: I believe this is technically the real part of the critical line, which sounds basically the same as the imaginary part, only with the phases shifted from sine waves to cosine waves.)

Listening

The Apples in Stereo - C.P.U.

See also

Harmonotonic tunings: logharmonic series are non-arithmetic harmonotonic tunings.

Powharmonic series: another type of non-arithmetic harmonotonic tuning.

Xenharmonic_series

Robert_Schneider - Non-Pythagorean_scale on Wikipedia