Logharmonic series

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.

Logarithmic intonation is an analogue of just intonation which uses the base-e logharmonic series rather than the harmonic series.

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
pitch #	frequency multiplier (definition)	frequency multiplier (decimal)	pitch (¢)	pitch Δ (¢)	octave reduced pitch (¢)	frequency multiplier (definition)	frequency multiplier (decimal)	pitch (¢)	pitch Δ (¢)	octave reduced pitch (¢)	difference between frequency multipliers
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
pitch #	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.

Logharmonic series

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