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. | The sequence of metallic means can be used as a variation on the harmonic series. | ||
{| class="wikitable" | |||
|+ | |||
| rowspan="2" |'''step''' | |||
| colspan="3" |'''metallic harmonic series''' | |||
| colspan="2" |'''harmonic series''' | |||
| rowspan="2" |'''difference (¢)''' | |||
|- | |||
|'''frequency multiplier (definition)''' | |||
|'''frequency multiplier (decimal)''' | |||
!'''pitch (¢)''' | |||
|'''frequency multiplier (decimal)''' | |||
|'''pitch (¢)''' | |||
|- | |||
|'''0''' | |||
|<nowiki><math>\frac{0+sqrt{0^2 + 4}}{2}</math></nowiki> | |||
|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, <span><math>μ_0</math></span>, is equal to 1, and for each subsequent metallic mean: | The zeroth metallic mean, <span><math>μ_0</math></span>, is equal to 1, and for each subsequent metallic mean: | ||