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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>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 scale (arranged in quasi-Pythagorean fashion)==
{{nowrap|''Iθ''{{"}} + ''bθ''{{``}} + ''mgL''&thinsp;sin(''θ'') {{=}} 0}}
{| class="wikitable"
|+
Period = <math>δ_s</math>
! colspan="3" |Hyper scale, generator = <math>φ</math>
! colspan="3" |Hypo scale, generator = <math>φ^{-1}</math>
|-
!In terms of metallic ratios
!In surd form
!Absolute cents
!In terms of metallic ratios
!In surd form
!Absolute cents
|-
| colspan="2" |<math>1</math>
|0
|<math>δ_s</math>
|<math>1 + \sqrt{2}</math>
|1525.864
|-
|<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>
|692.774
|-
|<math>{δ_s^{-1}} {φ^2}</math>
|<math>\frac{(3 + \sqrt{5})(\sqrt{2} - 1)}{2}</math>
|140.317
|<math>{δ_s^{2}} {φ^{-2}}</math>
|<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>
|1385.547
|-
|<math>{δ_s^{-1}} {φ^3}</math>
|<math>(2 + \sqrt{5})(\sqrt{2} - 1)</math>
|973.407
|<math>{δ_s^{2}} {φ^{-3}}</math>
|<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>
|552.457
|-
|<math>{δ_s^{-2}} {φ^4}</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 ==
702
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>
fafshdtharsgdasjdkhajsgdh


<math>φ^11 = δ_s^{6}</math>
Test


<math>φ  = (\sqrt[11]{δ_s})^{6}</math> OR <math>δ_s  = (\sqrt[6]{φ})^{11}</math>
hi


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.
== 29L 12s, ig. ==


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.
{{Infobox MOS|Scale Signature=29L 12s|debug=1}}
{| class="wikitable"
 
|+Anatolian Electrum (6ed-<math>φ</math>)
{{MOS intro|Scale Signature = 29L 12s}}
!Step
 
!In terms of <math>φ</math>
=== Scale properties ===
!Absolute Cents
{{TAMNAMS use|Scale Signature=29L 12s}}
!Closest hyper-interval (¢)
 
!Closest hypo-interval (¢)
==== Intervals ====
!Difference (¢)
{{MOS intervals|Scale Signature=29L 12s}}
|-
 
|1
==== Generator chain ====
|<math>\sqrt[6]{φ}</math>
{{MOS genchain|Scale Signature=29L 12s}}
|'''138.848'''
 
|140.317
==== Modes ====
|131.507
{{MOS mode degrees|Scale Signature=29L 12s}}
| -1.47, +7.34
 
|-
=== Scale tree ===
|2
{{MOS tuning spectrum|Scale Signature=29L 12s}}
|<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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