Metallic harmonic series

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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