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]\displaystyle{ f(n) = n }[/math], the formula for a b-logharmonic series is:

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

Where [math]\displaystyle{ b > 1 }[/math].

At [math]\displaystyle{ f(1) }[/math], any logharmonic series will be [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 2^2 = 4 }[/math] steps instead of one. Then 7 extra steps before the 4th harmonic so it takes [math]\displaystyle{ 2^3 = 8 }[/math] steps instead of one.

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
1	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]\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]

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]\displaystyle{ ≈ 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.

An analogous [emulatory edharmonic series] exists.

Listening

The Apples in Stereo - C.P.U.

See also

Powharmonic series