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Metallic harmonic series: Difference between revisions

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The sequence of [[wikipedia:Metallic_means|metallic means]] can be used as a variation on the [[wikipedia:Harmonic_series_(music)|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"
{| class="wikitable"
|+
| rowspan="2" |'''step'''
| colspan="3" |'''metallic harmonic series'''
| colspan="2" |'''harmonic series'''
| rowspan="2" |'''difference (¢)'''
|-
|-
|'''frequency multiplier (definition)'''
! rowspan="3" | Step
|'''frequency multiplier (decimal)'''
! colspan="4" | Metallic harmonic series
!'''pitch (¢)'''
! colspan="3" | Harmonic series
|'''frequency multiplier (decimal)'''
! rowspan="3" | Difference (¢)
|'''pitch (¢)'''
|-
! colspan="2" | Frequency multiplier
! colspan="2" | Pitch
! rowspan="2" | Frequency multiplier<br />(decimal)
! colspan="2" | Pitch
|-
! Definition
! Decimaal
! Cents
! Change (¢)
! Cents
! Change (¢)
|-
|-
|'''0'''
| '''0'''
|<math>\frac{0 + \sqrt{0^2 + 4}}{2}</math>
| <math>\frac{0 + \sqrt{0^2 + 4}}{2}</math>
|1
| 1
|0
| 0
| -
| —
| -
| —
| -
| —
| —
| —
|-
|-
|'''1'''
| '''1'''
|<math>\frac{1 + \sqrt{1^2 + 4}}{2}</math>
| <math>\frac{1 + \sqrt{1^2 + 4}}{2}</math>
|1.618033989
| 1.618033989
|833.090296
| 833.090296
|1.000000
| 833.0902964
|0
| 1.000000
|833.0902964
| 0
| —
| 833.0902964
|-
|-
|'''2'''
| '''2'''
|<math>\frac{2 + \sqrt{2^2 + 4}}{2}</math>
| <math>\frac{2 + \sqrt{2^2 + 4}}{2}</math>
|2.414213562
| 2.414213562
|1525.863964
| 1525.863964
|2.000000
| 692.7736674
|1200
| 2.000000
|325.8639638
| 1200
| 1200
| 325.8639638
|-
|-
|'''3'''
| '''3'''
|<math>\frac{3 + \sqrt{3^2 + 4}}{2}</math>
| <math>\frac{3 + \sqrt{3^2 + 4}}{2}</math>
|3.302775638
| 3.302775638
|2068.414762
| 2068.414762
|3.000000
| 542.5507985
|1901.955001
| 3.000000
|166.4597615
| 1901.955001
| 701.9550009
| 166.4597615
|-
|-
|'''4'''
| '''4'''
|<math>\frac{4 + \sqrt{4^2 + 4}}{2}</math>
| <math>\frac{4 + \sqrt{4^2 + 4}}{2}</math>
|4.236067977
| 4.236067977
|2499.270889
| 2499.270889
|4.000000
| 430.8561267
|2400
| 4.000000
|99.27088907
| 2400
| 498.0449991
| 99.27088907
|-
|-
|'''5'''
| '''5'''
|<math>\frac{5 + \sqrt{5^2 + 4}}{2}</math>
| <math>\frac{5 + \sqrt{5^2 + 4}}{2}</math>
|5.192582404
| 5.192582404
|2851.742647
| 2851.742647
|5.000000
| 352.4717582
|2786.313714
| 5.000000
|65.42893342
| 2786.313714
| 386.3137139
| 65.42893342
|-
|-
|'''6'''
| '''6'''
|<math>\frac{6 + \sqrt{6^2 + 4}}{2}</math>
| <math>\frac{6 + \sqrt{6^2 + 4}}{2}</math>
|6.16227766
| 6.16227766
|3148.156427
| 3148.156427
|6.000000
| 296.4137793
|3101.955001
| 6.000000
|46.20142576
| 3101.955001
| 315.6412870
| 46.20142576
|-
|-
|'''7'''
| '''7'''
|<math>\frac{7 + \sqrt{7^2 + 4}}{2}</math>
| <math>\frac{7 + \sqrt{7^2 + 4}}{2}</math>
|7.140054945
| 7.140054945
|3403.122211
| 3403.122211
|7.000000
| 254.9657848
|3368.825906
| 7.000000
|34.296305
| 3368.825906
| 266.8709056
| 34.296305
|-
|-
|'''8'''
| '''8'''
|<math>\frac{8 + \sqrt{8^2 + 4}}{2}</math>
| <math>\frac{8 + \sqrt{8^2 + 4}}{2}</math>
|8.123105626
| 8.123105626
|3626.437685
| 3626.437685
|8.000000
| 223.3154734
|3600
| 8.000000
|26.43768483
| 3600
| 231.1740935
| 26.43768483
|-
|-
|'''9'''
| '''9'''
|<math>\frac{9 + \sqrt{9^2 + 4}}{2}</math>
| <math>\frac{9 + \sqrt{9^2 + 4}}{2}</math>
|9.109772229
| 9.109772229
|3824.897979
| 3824.897979
|9.000000
| 198.4602946
|3803.910002
| 9.000000
|20.98797765
| 3803.910002
| 203.9100017
| 20.98797765
|-
|-
|'''10'''
| '''10'''
|<math>\frac{10 + \sqrt{10^2 + 4}}{2}</math>
| <math>\frac{10 + \sqrt{10^2 + 4}}{2}</math>
|10.09901951
| 10.09901951
|4003.371993
| 4003.371993
|10.000000
| 178.4740134
|3986.313714
| 10.000000
|17.05827891
| 3986.313714
| 182.4037121
| 17.05827891
|-
|-
|'''11'''
| '''11'''
|<math>\frac{11 + \sqrt{11^2 + 4}}{2}</math>
| <math>\frac{11 + \sqrt{11^2 + 4}}{2}</math>
|11.09016994
| 11.09016994
|4165.451482
| 4165.451482
|11.000000
| 162.079489
|4151.317942
| 11.000000
|14.13353942
| 4151.317942
| 165.0042285
| 14.13353942
|-
|-
|'''12'''
| '''12'''
|<math>\frac{12 + \sqrt{12^2 + 4}}{2}</math>
| <math>\frac{12 + \sqrt{12^2 + 4}}{2}</math>
|12.08276253
| 12.08276253
|4313.854124
| 4313.854124
|12.000000
| 148.4026419
|4301.955001
| 12.000000
|11.89912281
| 4301.955001
| 150.6370585
| 11.89912281
|-
|-
|'''13'''
| '''13'''
|<math>\frac{13 + \sqrt{13^2 + 4}}{2}</math>
| <math>\frac{13 + \sqrt{13^2 + 4}}{2}</math>
|13.07647322
| 13.07647322
|4450.681905
| 4450.681905
|13.000000
| 136.8277809
|4440.527662
| 13.000000
|10.15424279
| 4440.527662
| 138.5726609
| 10.15424279
|-
|-
|'''14'''
| '''14'''
|<math>\frac{14 + \sqrt{14^2 + 4}}{2}</math>
| <math>\frac{14 + \sqrt{14^2 + 4}}{2}</math>
|14.07106781
| 14.07106781
|4577.591891
| 4577.591891
|14.000000
| 126.9099868
|4568.825906
| 14.000000
|8.76598492
| 4568.825906
| 128.2982447
| 8.76598492
|-
|-
|'''15'''
| '''15'''
|<math>\frac{15 + \sqrt{15^2 + 4}}{2}</math>
| <math>\frac{15 + \sqrt{15^2 + 4}}{2}</math>
|15.06637298
| 15.06637298
|4695.912293
| 4695.912293
|15.000000
| 118.3204019
|4688.268715
| 15.000000
|7.643578517
| 4688.268715
| 119.4428083
| 7.643578517
|-
|-
|'''16'''
| '''16'''
|<math>\frac{16 + \sqrt{16^2 + 4}}{2}</math>
| <math>\frac{16 + \sqrt{16^2 + 4}}{2}</math>
|16.06225775
| 16.06225775
|4806.723349
| 4806.723349
|16.000000
| 110.8110554
|4800
| 16.000000
|6.7233486789
| 4800
| 111.7312853
| 6.7233486789
... &rarr; 0
|}
|}


Line 155: Line 198:
Some interesting combination tones may result from this series.
Some interesting combination tones may result from this series.


== Some Scala Files ==
== Some Scala files ==
 
<pre>
! Metallic Harmonic Series - First Four Octaves.scl
! Metallic Harmonic Series - First Four Octaves.scl
! Created using Scale Workshop 1.0.2
! Created using Scale Workshop 1.0.2
!
!
Metallic Harmonic Series - First Four Octaves
Metallic Harmonic Series - First Four Octaves
  16
16
!
!
  833.0902964
833.0902964
  1525.863964
1525.863964
  2068.414762
2068.414762
  2499.270889
2499.270889
  2851.742647
2851.742647
  3148.156427
3148.156427
  3403.122211
3403.122211
  3626.437685
3626.437685
  3824.897979
3824.897979
  4003.371993
4003.371993
  4165.451482
4165.451482
  4313.854124
4313.854124
  4450.681905
4450.681905
  4577.591891
4577.591891
  4695.912293
4695.912293
  4800.000000
4800.000000
</pre>


! Metallic Harmonic Series - Octave Reduced.scl
<pre>
! Created using Scale Workshop 1.0.2
! Metallic Harmonic Series - Octave Reduced.scl
!
! Created using Scale Workshop 1.0.2
Metallic Harmonic Series - First Four Octaves - Octave Reduced
!
  16
Metallic Harmonic Series - First Four Octaves - Octave Reduced
!
16
  26.43768483
!
  99.27088907
26.43768483
  224.8979794
99.27088907
  325.8639638
224.8979794
  403.3719928
325.8639638
  451.7426473
403.3719928
  565.4514818
451.7426473
  713.8541237
565.4514818
  748.1564266
713.8541237
  833.0902964
748.1564266
  850.6819046
833.0902964
  868.4147623
850.6819046
  977.5918914
868.4147623
  1003.122211
977.5918914
  1095.912293
1003.122211
  1200.000000
1095.912293
1200.000000
</pre>


== Listening ==
== 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 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]
[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]]
Retrieved from "https://en.xen.wiki/w/Metallic_harmonic_series"