Metallic harmonic series
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>\frac{0+sqrt{0^2 + 4}}{2}</math> | 1 | 0 | - | - | - |
| 1 | 1.618033989 | 833.090296 | 1.000000 | 0 | 833.0902964 | |
| 2 | 2.414213562 | 1525.863964 | 2.000000 | 1200 | 325.8639638 | |
| 3 | 3.302775638 | 2068.414762 | 3.000000 | 1901.955001 | 166.4597615 | |
| 4 | 4.236067977 | 2499.270889 | 4.000000 | 2400 | 99.27088907 | |
| 5 | 5.192582404 | 2851.742647 | 5.000000 | 2786.313714 | 65.42893342 | |
| 6 | 6.16227766 | 3148.156427 | 6.000000 | 3101.955001 | 46.20142576 | |
| 7 | 7.140054945 | 3403.122211 | 7.000000 | 3368.825906 | 34.296305 | |
| 8 | 8.123105626 | 3626.437685 | 8.000000 | 3600 | 26.43768483 | |
| 9 | 9.109772229 | 3824.897979 | 9.000000 | 3803.910002 | 20.98797765 | |
| 10 | 10.09901951 | 4003.371993 | 10.000000 | 3986.313714 | 17.05827891 | |
| 11 | 11.09016994 | 4165.451482 | 11.000000 | 4151.317942 | 14.13353942 | |
| 12 | 12.08276253 | 4313.854124 | 12.000000 | 4301.955001 | 11.89912281 | |
| 13 | 13.07647322 | 4450.681905 | 13.000000 | 4440.527662 | 10.15424279 | |
| 14 | 14.07106781 | 4577.591891 | 14.000000 | 4568.825906 | 8.76598492 | |
| 15 | 15.06637298 | 4695.912293 | 15.000000 | 4688.268715 | 7.643578517 | |
| 16 | 16.06225775 | 4806.723349 | 16.000000 | 4800 | 6.7233486789 | |
The zeroth metallic mean, [math]\displaystyle{ μ_0 }[/math], is equal to 1, and for each subsequent metallic mean:
[math]\displaystyle{ n \lt μ_n \lt 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.
Combination tones from this scale.