Metallic harmonic series: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{Distinguish|Metallic | {{Distinguish|Metallic harmony}} | ||
The sequence of [[wikipedia:Metallic_means|metallic means]] can be used as a variation on the [[harmonic series]]. | The sequence of [[wikipedia:Metallic_means|metallic means]] can be used as a variation on the [[harmonic series]]. | ||
| Line 22: | Line 22: | ||
! Change (¢) | ! Change (¢) | ||
|- | |- | ||
|'''0''' | | '''0''' | ||
|<math>\frac{0 + \sqrt{0^2 + 4}}{2}</math> | | <math>\frac{0 + \sqrt{0^2 + 4}}{2}</math> | ||
|1 | | 1 | ||
|0 | | 0 | ||
| | | — | ||
| | | — | ||
| | | — | ||
| | | — | ||
| | | — | ||
|- | |- | ||
|'''1''' | | '''1''' | ||
|<math>\frac{1 + \sqrt{1^2 + 4}}{2}</math> | | <math>\frac{1 + \sqrt{1^2 + 4}}{2}</math> | ||
|1.618033989 | | 1.618033989 | ||
|833.090296 | | 833.090296 | ||
|833.0902964 | | 833.0902964 | ||
|1.000000 | | 1.000000 | ||
|0 | | 0 | ||
| | | — | ||
|833.0902964 | | 833.0902964 | ||
|- | |- | ||
|'''2''' | | '''2''' | ||
|<math>\frac{2 + \sqrt{2^2 + 4}}{2}</math> | | <math>\frac{2 + \sqrt{2^2 + 4}}{2}</math> | ||
|2.414213562 | | 2.414213562 | ||
|1525.863964 | | 1525.863964 | ||
|692.7736674 | | 692.7736674 | ||
|2.000000 | | 2.000000 | ||
|1200 | | 1200 | ||
|1200 | | 1200 | ||
|325.8639638 | | 325.8639638 | ||
|- | |- | ||
|'''3''' | | '''3''' | ||
|<math>\frac{3 + \sqrt{3^2 + 4}}{2}</math> | | <math>\frac{3 + \sqrt{3^2 + 4}}{2}</math> | ||
|3.302775638 | | 3.302775638 | ||
|2068.414762 | | 2068.414762 | ||
|542.5507985 | | 542.5507985 | ||
|3.000000 | | 3.000000 | ||
|1901.955001 | | 1901.955001 | ||
|701.9550009 | | 701.9550009 | ||
|166.4597615 | | 166.4597615 | ||
|- | |- | ||
|'''4''' | | '''4''' | ||
|<math>\frac{4 + \sqrt{4^2 + 4}}{2}</math> | | <math>\frac{4 + \sqrt{4^2 + 4}}{2}</math> | ||
|4.236067977 | | 4.236067977 | ||
|2499.270889 | | 2499.270889 | ||
|430.8561267 | | 430.8561267 | ||
|4.000000 | | 4.000000 | ||
|2400 | | 2400 | ||
|498.0449991 | | 498.0449991 | ||
|99.27088907 | | 99.27088907 | ||
|- | |- | ||
|'''5''' | | '''5''' | ||
|<math>\frac{5 + \sqrt{5^2 + 4}}{2}</math> | | <math>\frac{5 + \sqrt{5^2 + 4}}{2}</math> | ||
|5.192582404 | | 5.192582404 | ||
|2851.742647 | | 2851.742647 | ||
|352.4717582 | | 352.4717582 | ||
|5.000000 | | 5.000000 | ||
|2786.313714 | | 2786.313714 | ||
|386.3137139 | | 386.3137139 | ||
|65.42893342 | | 65.42893342 | ||
|- | |- | ||
|'''6''' | | '''6''' | ||
|<math>\frac{6 + \sqrt{6^2 + 4}}{2}</math> | | <math>\frac{6 + \sqrt{6^2 + 4}}{2}</math> | ||
|6.16227766 | | 6.16227766 | ||
|3148.156427 | | 3148.156427 | ||
|296.4137793 | | 296.4137793 | ||
|6.000000 | | 6.000000 | ||
|3101.955001 | | 3101.955001 | ||
|315.6412870 | | 315.6412870 | ||
|46.20142576 | | 46.20142576 | ||
|- | |- | ||
|'''7''' | | '''7''' | ||
|<math>\frac{7 + \sqrt{7^2 + 4}}{2}</math> | | <math>\frac{7 + \sqrt{7^2 + 4}}{2}</math> | ||
|7.140054945 | | 7.140054945 | ||
|3403.122211 | | 3403.122211 | ||
|254.9657848 | | 254.9657848 | ||
|7.000000 | | 7.000000 | ||
|3368.825906 | | 3368.825906 | ||
|266.8709056 | | 266.8709056 | ||
|34.296305 | | 34.296305 | ||
|- | |- | ||
|'''8''' | | '''8''' | ||
|<math>\frac{8 + \sqrt{8^2 + 4}}{2}</math> | | <math>\frac{8 + \sqrt{8^2 + 4}}{2}</math> | ||
|8.123105626 | | 8.123105626 | ||
|3626.437685 | | 3626.437685 | ||
|223.3154734 | | 223.3154734 | ||
|8.000000 | | 8.000000 | ||
|3600 | | 3600 | ||
|231.1740935 | | 231.1740935 | ||
|26.43768483 | | 26.43768483 | ||
|- | |- | ||
|'''9''' | | '''9''' | ||
|<math>\frac{9 + \sqrt{9^2 + 4}}{2}</math> | | <math>\frac{9 + \sqrt{9^2 + 4}}{2}</math> | ||
|9.109772229 | | 9.109772229 | ||
|3824.897979 | | 3824.897979 | ||
|198.4602946 | | 198.4602946 | ||
|9.000000 | | 9.000000 | ||
|3803.910002 | | 3803.910002 | ||
|203.9100017 | | 203.9100017 | ||
|20.98797765 | | 20.98797765 | ||
|- | |- | ||
|'''10''' | | '''10''' | ||
|<math>\frac{10 + \sqrt{10^2 + 4}}{2}</math> | | <math>\frac{10 + \sqrt{10^2 + 4}}{2}</math> | ||
|10.09901951 | | 10.09901951 | ||
|4003.371993 | | 4003.371993 | ||
|178.4740134 | | 178.4740134 | ||
|10.000000 | | 10.000000 | ||
|3986.313714 | | 3986.313714 | ||
|182.4037121 | | 182.4037121 | ||
|17.05827891 | | 17.05827891 | ||
|- | |- | ||
|'''11''' | | '''11''' | ||
|<math>\frac{11 + \sqrt{11^2 + 4}}{2}</math> | | <math>\frac{11 + \sqrt{11^2 + 4}}{2}</math> | ||
|11.09016994 | | 11.09016994 | ||
|4165.451482 | | 4165.451482 | ||
|162.079489 | | 162.079489 | ||
|11.000000 | | 11.000000 | ||
|4151.317942 | | 4151.317942 | ||
|165.0042285 | | 165.0042285 | ||
|14.13353942 | | 14.13353942 | ||
|- | |- | ||
|'''12''' | | '''12''' | ||
|<math>\frac{12 + \sqrt{12^2 + 4}}{2}</math> | | <math>\frac{12 + \sqrt{12^2 + 4}}{2}</math> | ||
|12.08276253 | | 12.08276253 | ||
|4313.854124 | | 4313.854124 | ||
|148.4026419 | | 148.4026419 | ||
|12.000000 | | 12.000000 | ||
|4301.955001 | | 4301.955001 | ||
|150.6370585 | | 150.6370585 | ||
|11.89912281 | | 11.89912281 | ||
|- | |- | ||
|'''13''' | | '''13''' | ||
|<math>\frac{13 + \sqrt{13^2 + 4}}{2}</math> | | <math>\frac{13 + \sqrt{13^2 + 4}}{2}</math> | ||
|13.07647322 | | 13.07647322 | ||
|4450.681905 | | 4450.681905 | ||
|136.8277809 | | 136.8277809 | ||
|13.000000 | | 13.000000 | ||
|4440.527662 | | 4440.527662 | ||
|138.5726609 | | 138.5726609 | ||
|10.15424279 | | 10.15424279 | ||
|- | |- | ||
|'''14''' | | '''14''' | ||
|<math>\frac{14 + \sqrt{14^2 + 4}}{2}</math> | | <math>\frac{14 + \sqrt{14^2 + 4}}{2}</math> | ||
|14.07106781 | | 14.07106781 | ||
|4577.591891 | | 4577.591891 | ||
|126.9099868 | | 126.9099868 | ||
|14.000000 | | 14.000000 | ||
|4568.825906 | | 4568.825906 | ||
|128.2982447 | | 128.2982447 | ||
|8.76598492 | | 8.76598492 | ||
|- | |- | ||
|'''15''' | | '''15''' | ||
|<math>\frac{15 + \sqrt{15^2 + 4}}{2}</math> | | <math>\frac{15 + \sqrt{15^2 + 4}}{2}</math> | ||
|15.06637298 | | 15.06637298 | ||
|4695.912293 | | 4695.912293 | ||
|118.3204019 | | 118.3204019 | ||
|15.000000 | | 15.000000 | ||
|4688.268715 | | 4688.268715 | ||
|119.4428083 | | 119.4428083 | ||
|7.643578517 | | 7.643578517 | ||
|- | |- | ||
|'''16''' | | '''16''' | ||
|<math>\frac{16 + \sqrt{16^2 + 4}}{2}</math> | | <math>\frac{16 + \sqrt{16^2 + 4}}{2}</math> | ||
|16.06225775 | | 16.06225775 | ||
|4806.723349 | | 4806.723349 | ||
|110.8110554 | | 110.8110554 | ||
|16.000000 | | 16.000000 | ||
|4800 | | 4800 | ||
|111.7312853 | | 111.7312853 | ||
|6.7233486789 | | 6.7233486789 | ||
... | ... → 0 | ||
|} | |} | ||
| Line 198: | Line 198: | ||
Some interesting combination tones may result from this series. | Some interesting combination tones may result from this series. | ||
== Some Scala | == 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 == | == Listening == | ||