Metallic harmonic series

Not to be confused with Metallic harmony.

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

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

Step	Metallic harmonic series				Harmonic series			Difference (¢)
	Frequency multiplier		Pitch		Frequency multiplier (decimal)	Pitch
	Definition	Decimaal	Cents	Change (¢)		Cents	Change (¢)
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 ... → 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

See also

• Acoustic phi
• Silver third