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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.
{{Distinguish|Metallic harmony}}


The zeroth metallic mean, <span><math>μ_0</math></span>, is equal to 1, and for each subsequent metallic mean:
The sequence of [[wikipedia:Metallic_means|metallic means]] can be used as a variation on the [[harmonic series]]. 


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


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.
With each successive metallic mean we converge closer to the harmonic series.


Combination tones from this scale.
Some interesting combination tones may result from this series.
 
== 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>
 
<pre>
! 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
</pre>
 
== 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]
 
[[Category:Harmonic series]]
[[Category:Xenharmonic series]]
[[Category:Math]]
[[Category:Tables]]
Retrieved from "https://en.xen.wiki/w/Metallic_harmonic_series"