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]

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 (¢)	octave reduced pitch (¢)	pitch Δ (¢)	pitch #	frequency multiplier (definition)	frequency multiplier (decimal)	pitch (¢)	octave reduced pitch (¢)	pitch Δ (¢)
2	log22	1.00000000	0.00	0.00	-	1	1	1.000000	0.00	0.00	1200.00
3	log23	1.584962501	797.34	797.34	797.34
4	log24	2.00000000	1200.00	0.00	402.66	2	2	2.000000	1200.00	0.00	701.96
5	log25	2.321928095	1458.39	258.39	258.39
6	log26	2.584962501	1644.17	444.17	185.78
7	log27	2.807354922	1787.05	587.05	142.88
8	log28	3.00000000	1901.96	701.96	114.90	3	3	3.000000	1901.96	701.96	498.04
9	log29	3.169925001	1997.34	797.34	95.38
1	log210	3.321928095	2078.43	878.43	81.09
11	log211	3.459431619	2148.64	948.64	70.22
12	log212	3.584962501	2210.35	1010.35	61.71
13	log213	3.700439718	2265.24	1065.24	54.89
14	log214	3.807354922	2314.55	1114.55	49.31
15	log215	3.906890596	2359.23	1159.23	44.68
16	log216	4.00000000	2400.00	0.00	40.77	4	4	4.000000	2400.00	0.00	386.31
17	log217	4.087462841	2437.45	37.45	37.45
18	log218	4.169925001	2472.03	72.03	34.58
19	log219	4.247927513	2504.11	104.11	32.09
20	log220	4.321928095	2534.01	134.01	29.90
21	log221	4.392317423	2561.98	161.98	27.97
22	log222	4.459431619	2588.23	188.23	26.25
23	log223	4.523561956	2612.95	212.95	24.72
24	log224	4.584962501	2636.29	236.29	23.34
25	log225	4.64385619	2658.39	258.39	22.10
26	log226	4.700439718	2679.35	279.35	20.97
27	log227	4.754887502	2699.29	299.29	19.94
28	log228	4.807354922	2718.29	318.29	19.00
29	log229	4.857980995	2736.43	336.43	18.14
30	log230	4.906890596	2753.77	353.77	17.34
31	log231	4.95419631	2770.38	370.38	16.61
32	log232	5.00000000	2786.31	386.31	15.93	5	5	5.000000	2786.31	386.31	315.64

matharmonic series

The e-logharmonic series can be approximated by pitches taken from the mathematical harmonic series:

1/1, 3/2, 11/6, 25/12, 137/60, 49/20, ...