SandBox: Difference between revisions

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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, ~~respectively~~ 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 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.

==Untempered version (arranged in quasi-Pythagorean fashion)==

{| class="wikitable"

|+

Period = <math>δ_s</math>

! colspan="3" |Intervals with hyper-generator <math>φ</math>

! colspan="3" |Intervals with hypo-generator <math>φ^{-1}</math>

Line 18:

Line 19:

|0

|<math>δ_s</math>

|<math>~~\frac{~~1 + sqrt~~\{2}~~{2}}</math>

|<math>1 + \sqrt{2}</math>

|1525.854

|-

|

|<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>

|

|692.764

|-

|

|<math>{δ_s^{-1}} {φ^2}</math>

|

|<math>\frac{(3 + sqrt{5})(sqrt{2} - 1)}{2}</math>

|

|140.326

|

|<math>{δ_s^{2}} {φ^{-2}}</math>

|

|<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>

|

|1385.528

|}

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 {{SandBox please edit after this line}}
-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, respectively 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 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.
 ==Untempered version (arranged in quasi-Pythagorean fashion)==
 {| class="wikitable"
 |+
+Period = <math>δ_s</math>
 ! colspan="3" |Intervals with hyper-generator <math>φ</math>
 ! colspan="3" |Intervals with hypo-generator <math>φ^{-1}</math>
@@ Line 18: / Line 19: @@
 |0
 |<math>δ_s</math>
-|<math>\frac{1 + sqrt\{2}{2}}</math>
+|<math>1 + \sqrt{2}</math>
 |1525.854
 |-
-|
+|<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>
-|
+|692.764
 |-
-|
+|<math>{δ_s^{-1}} {φ^2}</math>
-|
+|<math>\frac{(3 + sqrt{5})(sqrt{2} - 1)}{2}</math>
-|
+|140.326
-|
+|<math>{δ_s^{2}} {φ^{-2}}</math>
-|
+|<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>
-|
+|1385.528
 |}