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]

Revision as of 05:51, 5 February 2020

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 (¢) frequency multiplier (decimal) pitch (¢)
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 1.000000 0 833.0902964
2 [math]\displaystyle{ \frac{2 + \sqrt{2^2 + 4}}{2} }[/math] 2.414213562 1525.863964 2.000000 1200 325.8639638
3 [math]\displaystyle{ \frac{3 + \sqrt{3^2 + 4}}{2} }[/math] 3.302775638 2068.414762 3.000000 1901.955001 166.4597615
4 [math]\displaystyle{ \frac{4 + \sqrt{4^2 + 4}}{2} }[/math] 4.236067977 2499.270889 4.000000 2400 99.27088907
5 [math]\displaystyle{ \frac{5 + \sqrt{5^2 + 4}}{2} }[/math] 5.192582404 2851.742647 5.000000 2786.313714 65.42893342
6 [math]\displaystyle{ \frac{6 + \sqrt{6^2 + 4}}{2} }[/math] 6.16227766 3148.156427 6.000000 3101.955001 46.20142576
7 [math]\displaystyle{ \frac{7 + \sqrt{7^2 + 4}}{2} }[/math] 7.140054945 3403.122211 7.000000 3368.825906 34.296305
8 [math]\displaystyle{ \frac{8 + \sqrt{8^2 + 4}}{2} }[/math] 8.123105626 3626.437685 8.000000 3600 26.43768483
9 [math]\displaystyle{ \frac{9 + \sqrt{9^2 + 4}}{2} }[/math] 9.109772229 3824.897979 9.000000 3803.910002 20.98797765
10 [math]\displaystyle{ \frac{10 + \sqrt{10^2 + 4}}{2} }[/math] 10.09901951 4003.371993 10.000000 3986.313714 17.05827891
11 [math]\displaystyle{ \frac{11 + \sqrt{11^2 + 4}}{2} }[/math] 11.09016994 4165.451482 11.000000 4151.317942 14.13353942
12 [math]\displaystyle{ \frac{12 + \sqrt{12^2 + 4}}{2} }[/math] 12.08276253 4313.854124 12.000000 4301.955001 11.89912281
13 [math]\displaystyle{ \frac{13 + \sqrt{13^2 + 4}}{2} }[/math] 13.07647322 4450.681905 13.000000 4440.527662 10.15424279
14 [math]\displaystyle{ \frac{14 + \sqrt{14^2 + 4}}{2} }[/math] 14.07106781 4577.591891 14.000000 4568.825906 8.76598492
15 [math]\displaystyle{ \frac{15 + \sqrt{15^2 + 4}}{2} }[/math] 15.06637298 4695.912293 15.000000 4688.268715 7.643578517
16 [math]\displaystyle{ \frac{16 + \sqrt{16^2 + 4}}{2} }[/math] 16.06225775 4806.723349 16.000000 4800 6.7233486789

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

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