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


Line 191: 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
! 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>


! Metallic Harmonic Series - First Four Octaves.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
!
  16
Metallic Harmonic Series - First Four Octaves - Octave Reduced
!
16
  833.0902964
!
  1525.863964
26.43768483
  2068.414762
99.27088907
  2499.270889
224.8979794
  2851.742647
325.8639638
  3148.156427
403.3719928
  3403.122211
451.7426473
  3626.437685
565.4514818
  3824.897979
713.8541237
  4003.371993
748.1564266
  4165.451482
833.0902964
  4313.854124
850.6819046
  4450.681905
868.4147623
  4577.591891
977.5918914
  4695.912293
1003.122211
  4800.000000
1095.912293
1200.000000
! Metallic Harmonic Series - Octave Reduced.scl
</pre>
! 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 ==
== 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"