View source for Metallic harmonic series

{{Distinguish|Metallic harmony}}

The sequence of [[wikipedia:Metallic_means|metallic means]] can be used as a variation on the [[harmonic series]].  

{| class="wikitable"
|-
! rowspan="3" | Step
! colspan="4" | Metallic harmonic series
! colspan="3" | Harmonic series
! rowspan="3" | Difference (¢)
|-
! colspan="2" | Frequency multiplier
! colspan="2" | Pitch
! rowspan="2" | Frequency multiplier<br />(decimal)
! colspan="2" | Pitch
|-
! Definition
! Decimaal
! Cents
! Change (¢)
! Cents
! Change (¢)
|-
| '''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
... &rarr; 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 ==
<pre>
! 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
</pre>

<pre>
! 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
</pre>

== Listening ==
[https://soundcloud.com/cmloegcmluin/metallic-harmonic-series-first-four-octaves Metallic Harmonic Series, first four octaves]

[https://soundcloud.com/cmloegcmluin/metallic-harmonic-series-first-four-octaves-octave-reduced Metallic Harmonic Series, first four octaves, octave-reduced]

[[Category:Harmonic series]]
[[Category:Xenharmonic series]]
[[Category:Math]]
[[Category:Tables]]