Metallic harmonic series

Not to be confused with Metallic Harmony.

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

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