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 (¢)	frequency multiplier (decimal)	pitch (¢)
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	1.000000	0	833.0902964
2	[math]\displaystyle{ \frac{2 + \sqrt{2^2 + 4}}{2} }[/math]	2.414213562	1525.863964	2.000000	1200	325.8639638
3	[math]\displaystyle{ \frac{3 + \sqrt{3^2 + 4}}{2} }[/math]	3.302775638	2068.414762	3.000000	1901.955001	166.4597615
4	[math]\displaystyle{ \frac{4 + \sqrt{4^2 + 4}}{2} }[/math]	4.236067977	2499.270889	4.000000	2400	99.27088907
5	[math]\displaystyle{ \frac{5 + \sqrt{5^2 + 4}}{2} }[/math]	5.192582404	2851.742647	5.000000	2786.313714	65.42893342
6	[math]\displaystyle{ \frac{6 + \sqrt{6^2 + 4}}{2} }[/math]	6.16227766	3148.156427	6.000000	3101.955001	46.20142576
7	[math]\displaystyle{ \frac{7 + \sqrt{7^2 + 4}}{2} }[/math]	7.140054945	3403.122211	7.000000	3368.825906	34.296305
8	[math]\displaystyle{ \frac{8 + \sqrt{8^2 + 4}}{2} }[/math]	8.123105626	3626.437685	8.000000	3600	26.43768483
9	[math]\displaystyle{ \frac{9 + \sqrt{9^2 + 4}}{2} }[/math]	9.109772229	3824.897979	9.000000	3803.910002	20.98797765
10	[math]\displaystyle{ \frac{10 + \sqrt{10^2 + 4}}{2} }[/math]	10.09901951	4003.371993	10.000000	3986.313714	17.05827891
11	[math]\displaystyle{ \frac{11 + \sqrt{11^2 + 4}}{2} }[/math]	11.09016994	4165.451482	11.000000	4151.317942	14.13353942
12	[math]\displaystyle{ \frac{12 + \sqrt{12^2 + 4}}{2} }[/math]	12.08276253	4313.854124	12.000000	4301.955001	11.89912281
13	[math]\displaystyle{ \frac{13 + \sqrt{13^2 + 4}}{2} }[/math]	13.07647322	4450.681905	13.000000	4440.527662	10.15424279
14	[math]\displaystyle{ \frac{14 + \sqrt{14^2 + 4}}{2} }[/math]	14.07106781	4577.591891	14.000000	4568.825906	8.76598492
15	[math]\displaystyle{ \frac{15 + \sqrt{15^2 + 4}}{2} }[/math]	15.06637298	4695.912293	15.000000	4688.268715	7.643578517
16	[math]\displaystyle{ \frac{16 + \sqrt{16^2 + 4}}{2} }[/math]	16.06225775	4806.723349	16.000000	4800	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