Iwuqety: /* Tempering out the comma */

2021-06-12T12:08:03Z

Tempering out the comma

← Older revision		Revision as of 12:08, 12 June 2021
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	<math>φ → {\sqrt[11]{δ_s}}^{6}</math> OR <math>δ_s → {\sqrt[6]{φ}}^{11}</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.		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. The reverse 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.		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.

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2021-06-12T10:29:43Z

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The '''Electrum temperaments''' are a proposed type of rank-2 temperaments, 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)==
{| 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==
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
|}

User:Iwuqety/Electrym - Revision history

Iwuqety: /* Tempering out the comma */

Iwuqety: Migrated from sandbox