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The Electrum temperaments are a type 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 scale (arranged in quasi-Pythagorean fashion)
| Hyper scale, generator = [math]\displaystyle{ φ }[/math] | Hypo scale, 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 |
| [math]\displaystyle{ 1 }[/math] | 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 |
| [math]\displaystyle{ {δ_s^{-2}} {φ^5} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^5 (3 - 2\sqrt{2})}{32} }[/math] | 1113.724 | [math]\displaystyle{ {δ_s^{3}} {φ^{-5}} }[/math] | [math]\displaystyle{ \frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4} }[/math] | 412.140 |
| [math]\displaystyle{ {δ_s^{-3}} {φ^6} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^6 (5\sqrt{2} - 7)}{64} }[/math] | 420.950 | [math]\displaystyle{ {δ_s^{4}} {φ^{-6}} }[/math] | [math]\displaystyle{ \frac{64(1 + \sqrt{2})^4}{(1 + \sqrt{5})^6} }[/math] | 1104.914 |
| [math]\displaystyle{ {δ_s^{-3}} {φ^7} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^7 (5 - 7\sqrt{2})}{128} }[/math] | 1254.040 | [math]\displaystyle{ {δ_s^{4}} {φ^{-7}} }[/math] | [math]\displaystyle{ \frac{128(1 + \sqrt{2})^4}{(1 + \sqrt{5})^7} }[/math] | 271.824 |
| [math]\displaystyle{ {δ_s^{-4}} {φ^8} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^8}{256(1 + sqrt{2})^4} }[/math] | 561.267 | [math]\displaystyle{ {δ_s^{5}} {φ^{-8}} }[/math] | [math]\displaystyle{ \frac{256(1 + \sqrt{2})^5}{(1 + \sqrt{5})^8} }[/math] | 964.597 |
| [math]\displaystyle{ {δ_s^{-4}} {φ^9} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^9}{512(1 + sqrt{2})^4} }[/math] | 1394.357 | [math]\displaystyle{ {δ_s^{5}} {φ^{-9}} }[/math] | [math]\displaystyle{ \frac{512(1 + \sqrt{2})^5}{(1 + \sqrt{5})^9} }[/math] | 131.507 |
| [math]\displaystyle{ {δ_s^{-5}} {φ^{10}} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^10}{1024(1 + sqrt{2})^5} }[/math] | 701.583 | [math]\displaystyle{ {δ_s^{6}} {φ^{-10}} }[/math] | [math]\displaystyle{ \frac{1024(1 + \sqrt{2})^6}{(1 + \sqrt{5})^10} }[/math] | 824.281 |
| [math]\displaystyle{ {δ_s^{-6}} {φ^{11}} }[/math] | [math]\displaystyle{ \frac{(1 + \sqrt{5})^11}{2048(1 + sqrt{2})^6} }[/math] | 8.809 | [math]\displaystyle{ {δ_s^{6}} {φ^{-11}} }[/math] | [math]\displaystyle{ \frac{1024(1 + \sqrt{2})^5}{(1 + \sqrt{5})^10} }[/math] | -8.809 |
Tempering out the comma
As shown above, the largest comma between the hyper-scale and the hypo-scale, produced by the two generators [math]\displaystyle{ φ }[/math] and [math]\displaystyle{ δ_s }[/math], is a mere [math]\displaystyle{ {δ_s^{-6}} {φ^{11}} }[/math] ≈ 8.809¢, much smaller and more imperceptible than both the Pythagorean comma (23.460¢) and the Syntonic comma (81/80, 21.506¢). Hence, it is practically safe to temper it out:
[math]\displaystyle{ {δ_s^{-6}} {φ^{11}} → 1 }[/math]
[math]\displaystyle{ φ^{11} → δ_s^{6} }[/math]
[math]\displaystyle{ φ → {\sqrt[11]{δ_s}}^{6} }[/math] OR [math]\displaystyle{ δ_s → {\sqrt[6]{φ}}^{11} }[/math]
The solution on the left provides for an equal division of [math]\displaystyle{ δ_s }[/math] into 11 notes to approximate [math]\displaystyle{ φ }[/math] as step 6\11. Reversely, the alternative solution provides for an equal division of [math]\displaystyle{ φ }[/math] into 6 notes to approximate the period [math]\displaystyle{ δ_s }[/math] with 5 extra steps above [math]\displaystyle{ φ }[/math]. The former equal temperament puts more weight on the silver ratio while the latter preserves the golden ratio.
Considering the archaeological analogy that electrum found in modern Anatolia contains more gold (70–90%) than electrum coins made in ancient Lydia (45–55%), 6ed-[math]\displaystyle{ φ }[/math] and 11ed-[math]\displaystyle{ δ_s }[/math] may be nicknamed "Anatolian Electrum" and "Lydian Electrum" respectively. Their intervals and differences with the untempered Electrum scales are listed below.
| Step | In terms of [math]\displaystyle{ φ }[/math] | Absolute Cents | Closest hyper-interval (¢) | Closest hypo-interval (¢) | Difference (¢) |
|---|---|---|---|---|---|
| 1 | [math]\displaystyle{ \sqrt[6]{φ} }[/math] | 138.848 | 140.317 | 131.507 | -1.47, +7.34 |
| 2 | [math]\displaystyle{ \sqrt[3]{φ} }[/math] | 277.700 | 280.633 | 271.824 | -2.93, +5.88 |
| 3 | [math]\displaystyle{ \sqrt{φ} }[/math] | 416.545 | 420.950 | 412.140 | -4.41, +4.41 |
| 4 | [math]\displaystyle{ {\sqrt[3]{φ}}^2 }[/math] | 555.394 | 561.267 | 552.457 | -5.87, +2.94 |
| 5 | [math]\displaystyle{ {\sqrt[6]{φ}}^5 }[/math] | 694.242 | 701.583 | 692.774 | -7.34, +1.47 |
| 6 | [math]\displaystyle{ φ }[/math] | 833.090 | 824.281 | 0, +8.81 | |
| ... | |||||
| 11 | [math]\displaystyle{ {\sqrt[6]{φ}}^{11} }[/math] | 1527.332 | - | 1525.864 | +1.47 |
| Step | In terms of [math]\displaystyle{ δ_s }[/math] | Absolute Cents | Closest hyper-interval (¢) | Closest hypo-interval (¢) | Difference (¢) |
|---|---|---|---|---|---|
| 1 | [math]\displaystyle{ \sqrt[11]{δ_s} }[/math] | 138.715 | 140.317 | 131.507 | -1.60, +7.21 |
| 2 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^2 }[/math] | 277.425 | 280.633 | 271.824 | -3.21, +5.60 |
| 3 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^3 }[/math] | 416.145 | 420.950 | 412.140 | -4.81, +4.01 |
| 4 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^4 }[/math] | 554.860 | 561.267 | 552.457 | -6.41, +2.40 |
| 5 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^5 }[/math] | 693.575 | 701.583 | 692.774 | -8.01, +0.80 |
| 6 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^6 }[/math] | 832.289 | 833.090 | 824.281 | -0.80, +8.01 |
| 7 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^7 }[/math] | 971.004 | 973.407 | 964.597 | -2.40, +6.41 |
| 8 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^8 }[/math] | 1109.719 | 1113.724 | 1104.914 | -4.01, +4.81 |
| 9 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^9 }[/math] | 1248.434 | 1254.040 | 1245.231 | -5.60, +3.21 |
| 10 | [math]\displaystyle{ {\sqrt[11]{δ_s}}^{10} }[/math] | 1387.149 | 1394.357 | 1385.547 | -7.21, +1.60 |
| 11 | [math]\displaystyle{ δ_s }[/math] | 1525.864 | - | 1525.864 | 0 |