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Metallic harmonic series: Difference between revisions

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


{| class="wikitable"
{| class="wikitable"
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|-
|-
|'''0'''
|'''0'''
|<nowiki><math>\frac{0+sqrt{0^2 + 4}}{2}</math></nowiki>
|<math>\frac{0 + \sqrt{0^2 + 4}}{2}</math>
|1
|1
|0
|0
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|-
|-
|'''1'''
|'''1'''
|
|<math>\frac{1 + \sqrt{1^2 + 4}}{2}</math>
|1.618033989
|1.618033989
|833.090296
|833.090296
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|-
|-
|'''2'''
|'''2'''
|
|<math>\frac{2 + \sqrt{2^2 + 4}}{2}</math>
|2.414213562
|2.414213562
|1525.863964
|1525.863964
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|-
|-
|'''3'''
|'''3'''
|
|<math>\frac{3 + \sqrt{3^2 + 4}}{2}</math>
|3.302775638
|3.302775638
|2068.414762
|2068.414762
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|-
|-
|'''4'''
|'''4'''
|
|<math>\frac{4 + \sqrt{4^2 + 4}}{2}</math>
|4.236067977
|4.236067977
|2499.270889
|2499.270889
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|-
|-
|'''5'''
|'''5'''
|
|<math>\frac{5 + \sqrt{5^2 + 4}}{2}</math>
|5.192582404
|5.192582404
|2851.742647
|2851.742647
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|-
|-
|'''6'''
|'''6'''
|
|<math>\frac{6 + \sqrt{6^2 + 4}}{2}</math>
|6.16227766
|6.16227766
|3148.156427
|3148.156427
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|-
|-
|'''7'''
|'''7'''
|
|<math>\frac{7 + \sqrt{7^2 + 4}}{2}</math>
|7.140054945
|7.140054945
|3403.122211
|3403.122211
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|-
|-
|'''8'''
|'''8'''
|
|<math>\frac{8 + \sqrt{8^2 + 4}}{2}</math>
|8.123105626
|8.123105626
|3626.437685
|3626.437685
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|-
|-
|'''9'''
|'''9'''
|
|<math>\frac{9 + \sqrt{9^2 + 4}}{2}</math>
|9.109772229
|9.109772229
|3824.897979
|3824.897979
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|-
|-
|'''10'''
|'''10'''
|
|<math>\frac{10 + \sqrt{10^2 + 4}}{2}</math>
|10.09901951
|10.09901951
|4003.371993
|4003.371993
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|-
|-
|'''11'''
|'''11'''
|
|<math>\frac{11 + \sqrt{11^2 + 4}}{2}</math>
|11.09016994
|11.09016994
|4165.451482
|4165.451482
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|-
|-
|'''12'''
|'''12'''
|
|<math>\frac{12 + \sqrt{12^2 + 4}}{2}</math>
|12.08276253
|12.08276253
|4313.854124
|4313.854124
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|-
|-
|'''13'''
|'''13'''
|
|<math>\frac{13 + \sqrt{13^2 + 4}}{2}</math>
|13.07647322
|13.07647322
|4450.681905
|4450.681905
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|-
|-
|'''14'''
|'''14'''
|
|<math>\frac{14 + \sqrt{14^2 + 4}}{2}</math>
|14.07106781
|14.07106781
|4577.591891
|4577.591891
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|-
|-
|'''15'''
|'''15'''
|
|<math>\frac{15 + \sqrt{15^2 + 4}}{2}</math>
|15.06637298
|15.06637298
|4695.912293
|4695.912293
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|-
|-
|'''16'''
|'''16'''
|
|<math>\frac{16 + \sqrt{16^2 + 4}}{2}</math>
|16.06225775
|16.06225775
|4806.723349
|4806.723349
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|6.7233486789
|6.7233486789
|}
|}
The zeroth metallic mean, <span><math>μ_0</math></span>, is equal to 1, and for each subsequent metallic mean:


<math>
With each successive metallic mean we converge closer to the harmonic series.
n < μ_n < n + 1
</math>


Each successive metallic mean is closer to the lower of its two bounding integers than the previous metallic mean was, so eventually you'll converge onto the traditional harmonic series. But at the beginning this sequence is quite a bit different; the first metallic mean, the golden mean, is ≈1.618, is actually closer to 2 than it is to 1. And the second metallic mean, the silver mean, is ≈2.414, also still a ways off from 2. The sequence continues ≈3.303, ≈4.236, ≈5.193, ≈6.162, ≈7.140, ≈8.123, ≈9.110, ≈10.099, ≈11.090, ≈12.083, ≈13.076, ≈14.071, ≈15.066, etc. So in other words it starts out sounding quite like its own thing, but eventually starts to sound like the traditional harmonic series.
Some interesting combination tones may result from this series.


Combination tones from this scale.
== 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 ==
 
[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]
Retrieved from "https://en.xen.wiki/w/Metallic_harmonic_series"