Metallic harmonic series

The sequence of metallic means can be used as a variation on the harmonic series.

step	metallic harmonic series				harmonic series			difference (¢)
	frequency multiplier (definition)	frequency multiplier (decimal)	pitch (¢)	pitch Δ (¢)	frequency multiplier (decimal)	pitch (¢)	pitch Δ (¢)
0	[math]\displaystyle{ \frac{0 + \sqrt{0^2 + 4}}{2} }[/math]	1	0	-	-	-	-	-
1	[math]\displaystyle{ \frac{1 + \sqrt{1^2 + 4}}{2} }[/math]	1.618033989	833.090296	833.0902964	1.000000	0	-	833.0902964
2	[math]\displaystyle{ \frac{2 + \sqrt{2^2 + 4}}{2} }[/math]	2.414213562	1525.863964	692.7736674	2.000000	1200	1200	325.8639638
3	[math]\displaystyle{ \frac{3 + \sqrt{3^2 + 4}}{2} }[/math]	3.302775638	2068.414762	542.5507985	3.000000	1901.955001	701.9550009	166.4597615
4	[math]\displaystyle{ \frac{4 + \sqrt{4^2 + 4}}{2} }[/math]	4.236067977	2499.270889	430.8561267	4.000000	2400	498.0449991	99.27088907
5	[math]\displaystyle{ \frac{5 + \sqrt{5^2 + 4}}{2} }[/math]	5.192582404	2851.742647	352.4717582	5.000000	2786.313714	386.3137139	65.42893342
6	[math]\displaystyle{ \frac{6 + \sqrt{6^2 + 4}}{2} }[/math]	6.16227766	3148.156427	296.4137793	6.000000	3101.955001	315.6412870	46.20142576
7	[math]\displaystyle{ \frac{7 + \sqrt{7^2 + 4}}{2} }[/math]	7.140054945	3403.122211	254.9657848	7.000000	3368.825906	266.8709056	34.296305
8	[math]\displaystyle{ \frac{8 + \sqrt{8^2 + 4}}{2} }[/math]	8.123105626	3626.437685	223.3154734	8.000000	3600	231.1740935	26.43768483
9	[math]\displaystyle{ \frac{9 + \sqrt{9^2 + 4}}{2} }[/math]	9.109772229	3824.897979	198.4602946	9.000000	3803.910002	203.9100017	20.98797765
10	[math]\displaystyle{ \frac{10 + \sqrt{10^2 + 4}}{2} }[/math]	10.09901951	4003.371993	178.4740134	10.000000	3986.313714	182.4037121	17.05827891
11	[math]\displaystyle{ \frac{11 + \sqrt{11^2 + 4}}{2} }[/math]	11.09016994	4165.451482	162.079489	11.000000	4151.317942	165.0042285	14.13353942
12	[math]\displaystyle{ \frac{12 + \sqrt{12^2 + 4}}{2} }[/math]	12.08276253	4313.854124	148.4026419	12.000000	4301.955001	150.6370585	11.89912281
13	[math]\displaystyle{ \frac{13 + \sqrt{13^2 + 4}}{2} }[/math]	13.07647322	4450.681905	136.8277809	13.000000	4440.527662	138.5726609	10.15424279
14	[math]\displaystyle{ \frac{14 + \sqrt{14^2 + 4}}{2} }[/math]	14.07106781	4577.591891	126.9099868	14.000000	4568.825906	128.2982447	8.76598492
15	[math]\displaystyle{ \frac{15 + \sqrt{15^2 + 4}}{2} }[/math]	15.06637298	4695.912293	118.3204019	15.000000	4688.268715	119.4428083	7.643578517
16	[math]\displaystyle{ \frac{16 + \sqrt{16^2 + 4}}{2} }[/math]	16.06225775	4806.723349	110.8110554	16.000000	4800	111.7312853	6.7233486789

With each successive metallic mean we converge closer to the harmonic series.

Some interesting combination tones may result from this series.

Some Scala Files

! Metallic Harmonic Series - First Four Octaves.scl
! Created using Scale Workshop 1.0.2
!
Metallic Harmonic Series - First Four Octaves
 16
!
 833.0902964
 1525.863964
 2068.414762
 2499.270889
 2851.742647
 3148.156427
 3403.122211
 3626.437685
 3824.897979
 4003.371993
 4165.451482
 4313.854124
 4450.681905
 4577.591891
 4695.912293
 4800.000000

! Metallic Harmonic Series - Octave Reduced.scl
! Created using Scale Workshop 1.0.2
!
Metallic Harmonic Series - First Four Octaves - Octave Reduced
 16
!
 26.43768483
 99.27088907
 224.8979794
 325.8639638
 403.3719928
 451.7426473
 565.4514818
 713.8541237
 748.1564266
 833.0902964
 850.6819046
 868.4147623
 977.5918914
 1003.122211
 1095.912293
 1200.000000

Listening

Metallic Harmonic Series, first four octaves

Metallic Harmonic Series, first four octaves, octave-reduced