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The '''Electrum temperaments''' are a group of rank-2 temperaments proposed by Iwuqety, inspired by the idea of using the [[acoustic phi]] (golden ratio <math>φ</math>) and acoustic silver ratio <math>δ_s</math> as generators, | The '''Electrum temperaments''' are a group of rank-2 temperaments proposed by Iwuqety, inspired by the idea of using the [[acoustic phi]] (golden ratio <math>φ</math>) and acoustic silver ratio <math>δ_s</math> as generators, replacing the [[3/2]] perfect fifth and the [[2/1]] octave used in common practice music. [[wikipedia:Electrum|Electrum]] refers to naturally-occurring alloy which is mainly made up of gold and silver. | ||
==Untempered version (arranged in quasi-Pythagorean fashion)== | ==Untempered version (arranged in quasi-Pythagorean fashion)== | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
Period = <math>δ_s</math> | |||
! colspan="3" |Intervals with hyper-generator <math>φ</math> | ! colspan="3" |Intervals with hyper-generator <math>φ</math> | ||
! colspan="3" |Intervals with hypo-generator <math>φ^{-1}</math> | ! colspan="3" |Intervals with hypo-generator <math>φ^{-1}</math> | ||
| Line 18: | Line 19: | ||
|0 | |0 | ||
|<math>δ_s</math> | |<math>δ_s</math> | ||
|<math> | |<math>1 + \sqrt{2}</math> | ||
|1525.854 | |1525.854 | ||
|- | |- | ||
| | |<math>φ</math> | ||
| | |<math>\frac{1 + \sqrt{5}}{2}</math> | ||
| | |833.090 | ||
| | |<math>{δ_s} {φ^{-1}}</math> | ||
| | |<math>\frac{(1+ \sqrt{2})(\sqrt{5} - 1)}{2}</math> | ||
| | |692.764 | ||
|- | |- | ||
| | |<math>{δ_s^{-1}} {φ^2}</math> | ||
| | |<math>\frac{(3 + sqrt{5})(sqrt{2} - 1)}{2}</math> | ||
| | |140.326 | ||
| | |<math>{δ_s^{2}} {φ^{-2}}</math> | ||
| | |<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math> | ||
| | |1385.528 | ||
|} | |} | ||
Revision as of 08:10, 12 June 2021
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The Electrum temperaments are a group of rank-2 temperaments proposed by Iwuqety, inspired by the idea of using the acoustic phi (golden ratio [math]\displaystyle{ φ }[/math]) and acoustic silver ratio [math]\displaystyle{ δ_s }[/math] as generators, replacing the 3/2 perfect fifth and the 2/1 octave used in common practice music. Electrum refers to naturally-occurring alloy which is mainly made up of gold and silver.
Untempered version (arranged in quasi-Pythagorean fashion)
| Intervals with hyper-generator [math]\displaystyle{ φ }[/math] | Intervals with hypo-generator [math]\displaystyle{ φ^{-1} }[/math] | ||||
|---|---|---|---|---|---|
| In terms of metallic ratios | In surd form | Absolute cents | In terms of metallic ratios | In surd form | Absolute cents |
| 1 | 0 | [math]\displaystyle{ δ_s }[/math] | [math]\displaystyle{ 1 + \sqrt{2} }[/math] | 1525.854 | |
| [math]\displaystyle{ φ }[/math] | [math]\displaystyle{ \frac{1 + \sqrt{5}}{2} }[/math] | 833.090 | [math]\displaystyle{ {δ_s} {φ^{-1}} }[/math] | [math]\displaystyle{ \frac{(1+ \sqrt{2})(\sqrt{5} - 1)}{2} }[/math] | 692.764 |
| [math]\displaystyle{ {δ_s^{-1}} {φ^2} }[/math] | [math]\displaystyle{ \frac{(3 + sqrt{5})(sqrt{2} - 1)}{2} }[/math] | 140.326 | [math]\displaystyle{ {δ_s^{2}} {φ^{-2}} }[/math] | [math]\displaystyle{ \frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2} }[/math] | 1385.528 |