SandBox

This is the sandbox. To experiment with editing, click on the [Edit] tab.

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)

Period = [math]\displaystyle{ δ_s }[/math]

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.

Anatolian Electrum (6ed-[math]\displaystyle{ φ }[/math])

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

Lydian Electrum (11ed-[math]\displaystyle{ δ_s }[/math])

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