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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, 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.
+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>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 scale (arranged in quasi-Pythagorean fashion)==
 {| class="wikitable"
 |+
 Period = <math>δ_s</math>
-! colspan="3" |Intervals with hyper-generator <math>φ</math>
+! colspan="3" |Hyper scale, generator = <math>φ</math>
-! colspan="3" |Intervals with hypo-generator <math>φ^{-1}</math>
+! colspan="3" |Hypo scale, generator = <math>φ^{-1}</math>
 |-
 !In terms of metallic ratios
@@ Line 16: / Line 16: @@
 !Absolute cents
 |-
-| colspan="2" |1
+| colspan="2" |<math>1</math>
 |0
 |<math>δ_s</math>
@@ Line 44: / Line 44: @@
 |-
 |<math>{δ_s^{-2}} {φ^4}</math>
-|<math>\frac{(1 + \sqrt{5})^4 (3 -2\sqrt{2})}{2}</math>
+|<math>\frac{(1 + \sqrt{5})^4 (3 - 2\sqrt{2})}{2}</math>
 |280.633
 |<math>{δ_s^{3}} {φ^{-4}}</math>
 |<math>\frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4}</math>
 |1245.231
+|-
+|<math>{δ_s^{-2}} {φ^5}</math>
+|<math>\frac{(1 + \sqrt{5})^5 (3 - 2\sqrt{2})}{32}</math>
+|'''1113.724'''
+|<math>{δ_s^{3}} {φ^{-5}}</math>
+|<math>\frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4}</math>
+|'''412.140'''
+|-
+|<math>{δ_s^{-3}} {φ^6}</math>
+|<math>\frac{(1 + \sqrt{5})^6 (5\sqrt{2} - 7)}{64}</math>
+|'''420.950'''
+|<math>{δ_s^{4}} {φ^{-6}}</math>
+|<math>\frac{64(1 + \sqrt{2})^4}{(1 + \sqrt{5})^6}</math>
+|'''1104.914'''
+|-
+|<math>{δ_s^{-3}} {φ^7}</math>
+|<math>\frac{(1 + \sqrt{5})^7 (5 - 7\sqrt{2})}{128}</math>
+|1254.040
+|<math>{δ_s^{4}} {φ^{-7}}</math>
+|<math>\frac{128(1 + \sqrt{2})^4}{(1 + \sqrt{5})^7}</math>
+|271.824
+|-
+|<math>{δ_s^{-4}} {φ^8}</math>
+|<math>\frac{(1 + \sqrt{5})^8}{256(1 + sqrt{2})^4}</math>
+|561.267
+|<math>{δ_s^{5}} {φ^{-8}}</math>
+|<math>\frac{256(1 + \sqrt{2})^5}{(1 + \sqrt{5})^8}</math>
+|964.597
+|-
+|<math>{δ_s^{-4}} {φ^9}</math>
+|<math>\frac{(1 + \sqrt{5})^9}{512(1 + sqrt{2})^4}</math>
+|1394.357
+|<math>{δ_s^{5}} {φ^{-9}}</math>
+|<math>\frac{512(1 + \sqrt{2})^5}{(1 + \sqrt{5})^9}</math>
+|131.507
+|-
+|<math>{δ_s^{-5}} {φ^10}</math>
+|<math>\frac{(1 + \sqrt{5})^10}{1024(1 + sqrt{2})^5}</math>
+|701.583
+|<math>{δ_s^{6}} {φ^{-10}}</math>
+|<math>\frac{1024(1 + \sqrt{2})^6}{(1 + \sqrt{5})^10}</math>
+|824.281
+|-
+|<math>{δ_s^{-6}} {φ^11}</math>
+|<math>\frac{(1 + \sqrt{5})^11}{2048(1 + sqrt{2})^6}</math>
+|''8.809''
+|<math>{δ_s^{6}} {φ^{-11}}</math>
+|<math>\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>φ</math> and <math>δ_s</math>, is a mere <math>{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>{δ_s^{-6}} {φ^11} = 1</math>
+<math>φ^11 = δ_s^{6}</math>
+<math>φ  = (\sqrt[11]{δ_s})^{6}</math> OR <math>δ_s  = (\sqrt[6]{φ})^{11}</math>
+The solution on the left provides for an equal division of <math>δ_s</math> into 11 notes to approximate <math>φ</math> as step 6\11. Reversely, the alternative solution provides for an equal division of <math>φ</math> into 6 notes to approximate the period <math>δ_s</math> with 5 extra steps above <math>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>φ</math> and 11ed-<math>δ_s</math> may be nicknamed "Anatolian Electrum" and "Lydian Electrum" respectively. Their intervals and differences with the untempered Electrum scales are listed below.
+{| class="wikitable"
+|+Anatolian Electrum (6ed-<math>φ</math>)
+!Step
+!In terms of <math>φ</math>
+!Absolute Cents
+!Closest hyper-interval (¢)
+!Closest hypo-interval (¢)
+!Difference (¢)
+|-
+|1
+|<math>\sqrt[6]{φ}</math>
+|'''138.848'''
+|140.317
+|131.507
+| -1.47, +7.34
+|-
+|2
+|<math>\sqrt[3]{φ}</math>
+|'''277.700'''
+|280.633
+|271.824
+| -2.93, +5.88
+|-
+|3
+|<math>\sqrt{φ}</math>
+|'''416.545'''
+|420.950
+|412.140
+| -4.41, +4.41
+|-
+|4
+|<math>(\sqrt[3]{φ})^2</math>
+|'''555.394'''
+|561.267
+|552.457
+| -5.87, +2.94
+|-
+|5
+|<math>(\sqrt[6]{φ})^5</math>
+|'''694.242'''
+|701.583
+|692.774
+| -7.34, +1.47
+|-
+|6
+|<math>φ</math>
+| colspan="2" |'''833.090'''
+|824.281
+|0, +8.81
+|-
+| colspan="6" |...
+|-
+|''11''
+|<math>(\sqrt[6]{φ})^11</math>
+|''1527.332''
+| -
+|1525.864
+| +1.47
+|}
+{| class="wikitable"
+|+Lydian Electrum (11ed-<math>δ_s</math>)
+!Step
+!In terms of <math>δ_s</math>
+!Absolute Cents
+!Closest hyper-interval (¢)
+!Closest hypo-interval (¢)
+!Difference (¢)
+|-
+|1
+|<math>\sqrt[11]{δ_s}</math>
+|'''138.715'''
+|140.317
+|131.507
+| -1.60, +7.21
+|-
+|2
+|<math>(\sqrt[11]{δ_s})^2</math>
+|'''277.425'''
+|280.633
+|271.824
+| -3.21, +5.60
+|-
+|3
+|<math>(\sqrt[11]{δ_s})^3</math>
+|'''416.145'''
+|420.950
+|412.140
+| -4.81, +4.01
+|-
+|4
+|<math>(\sqrt[11]{δ_s})^4</math>
+|'''554.860'''
+|561.267
+|552.457
+| -6.41, +2.40
+|-
+|5
+|<math>(\sqrt[11]{δ_s})^5</math>
+|'''693.575'''
+|701.583
+|692.774
+| -8.01, +0.80
+|-
+|6
+|<math>(\sqrt[11]{δ_s})^6</math>
+|'''832.289'''
+|833.090
+|824.281
+| -0.80, +8.01
+|-
+|7
+|<math>(\sqrt[11]{δ_s})^7</math>
+|'''971.004'''
+|973.407
+|964.597
+| -2.40, +6.41
+|-
+|8
+|<math>(\sqrt[11]{δ_s})^8</math>
+|'''1109.719'''
+|1113.724
+|1104.914
+| -4.01, +4.81
+|-
+|9
+|<math>(\sqrt[11]{δ_s})^9</math>
+|'''1248.434'''
+|1254.040
+|1245.231
+| -5.60, +3.21
+|-
+|10
+|<math>(\sqrt[11]{δ_s})^10</math>
+|'''1387.149'''
+|1394.357
+|1385.547
+| -7.21, +1.60
+|-
+|11
+|<math>δ_s</math>
+|'''1525.864'''
+| -
+|1525.864
+|0
 |}

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