# Metallic harmonic series

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

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