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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] | 1525.864 | |
| [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] | 692.774 |
| [math]\displaystyle{ {δ_s^{-1}} {φ^2} }[/math] | [math]\displaystyle{ \frac{(3 + \sqrt{5})(\sqrt{2} - 1)}{2} }[/math] | 140.317 | [math]\displaystyle{ {δ_s^{2}} {φ^{-2}} }[/math] | [math]\displaystyle{ \frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2} }[/math] | 1385.547 |
| [math]\displaystyle{ {δ_s^{-1}} {φ^3} }[/math] | [math]\displaystyle{ (2 + \sqrt{5})(\sqrt{2} - 1) }[/math] | 973.407 | [math]\displaystyle{ {δ_s^{2}} {φ^{-3}} }[/math] | [math]\displaystyle{ \frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2} }[/math] | 552.457 |
| [math]\displaystyle{ {δ_s^{-2}} {φ^4} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^4 (3 -2\sqrt{2})}{2} }[/math] | 280.633 | [math]\displaystyle{ {δ_s^{3}} {φ^{-4}} }[/math] | [math]\displaystyle{ \frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4} }[/math] | 1245.231 |