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
... -> 0
|-
| '''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 192: 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]]

Latest revision as of 14:07, 20 January 2025

Not to be confused with Metallic harmony.

The sequence of metallic means can be used as a variation on the harmonic series.

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

... → 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

! 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

Retrieved from "https://en.xen.wiki/index.php?title=Metallic_harmonic_series&oldid=177280"