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| {{SandBox please edit after this line}} | | [[Category:Sandboxes]] {{SandBox please edit after this line}} |
| 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.
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| ==Untempered scale (arranged in quasi-Pythagorean fashion)==
| | ~~<noinclude />~~ |
| {| class="wikitable"
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| |+
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| Period = <math>δ_s</math>
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| ! colspan="3" |Hyper scale, generator = <math>φ</math>
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| ! colspan="3" |Hypo scale, generator = <math>φ^{-1}</math>
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| |-
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| !In terms of metallic ratios
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| !In surd form
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| !Absolute cents
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| !In terms of metallic ratios
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| !In surd form
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| !Absolute cents
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| |-
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| | colspan="2" |<math>1</math>
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| |0
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| |<math>δ_s</math>
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| |<math>1 + \sqrt{2}</math>
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| |1525.864
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| |-
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| |<math>φ</math>
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| |<math>\frac{1 + \sqrt{5}}{2}</math>
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| |833.090
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| |<math>{δ_s} {φ^{-1}}</math>
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| |<math>\frac{(1 + \sqrt{2})(\sqrt{5} - 1)}{2}</math>
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| |692.774
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| |-
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| |<math>{δ_s^{-1}} {φ^2}</math>
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| |<math>\frac{(3 + \sqrt{5})(\sqrt{2} - 1)}{2}</math>
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| |140.317
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| |<math>{δ_s^{2}} {φ^{-2}}</math>
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| |<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>
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| |1385.547
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| |-
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| |<math>{δ_s^{-1}} {φ^3}</math>
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| |<math>(2 + \sqrt{5})(\sqrt{2} - 1)</math>
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| |973.407
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| |<math>{δ_s^{2}} {φ^{-3}}</math>
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| |<math>\frac{(3 + 2 \sqrt{2})(3 - \sqrt{5})}{2}</math>
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| |552.457
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| |-
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| |<math>{δ_s^{-2}} {φ^4}</math>
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| |<math>\frac{(1 + \sqrt{5})^4 (3 - 2\sqrt{2})}{2}</math>
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| |280.633
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| |<math>{δ_s^{3}} {φ^{-4}}</math>
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| |<math>\frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4}</math>
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| |1245.231
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| |-
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| |<math>{δ_s^{-2}} {φ^5}</math>
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| |<math>\frac{(1 + \sqrt{5})^5 (3 - 2\sqrt{2})}{32}</math>
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| |'''1113.724'''
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| |<math>{δ_s^{3}} {φ^{-5}}</math>
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| |<math>\frac{16(7 + 5 \sqrt{2})}{(1 + \sqrt{5})^4}</math>
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| |'''412.140'''
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| |-
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| |<math>{δ_s^{-3}} {φ^6}</math>
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| |<math>\frac{(1 + \sqrt{5})^6 (5\sqrt{2} - 7)}{64}</math>
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| |'''420.950'''
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| |<math>{δ_s^{4}} {φ^{-6}}</math>
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| |<math>\frac{64(1 + \sqrt{2})^4}{(1 + \sqrt{5})^6}</math>
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| |'''1104.914'''
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| |-
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| |<math>{δ_s^{-3}} {φ^7}</math>
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| |<math>\frac{(1 + \sqrt{5})^7 (5 - 7\sqrt{2})}{128}</math>
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| |1254.040
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| |<math>{δ_s^{4}} {φ^{-7}}</math>
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| |<math>\frac{128(1 + \sqrt{2})^4}{(1 + \sqrt{5})^7}</math>
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| |271.824
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| |-
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| |<math>{δ_s^{-4}} {φ^8}</math>
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| |<math>\frac{(1 + \sqrt{5})^8}{256(1 + sqrt{2})^4}</math>
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| |561.267
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| |<math>{δ_s^{5}} {φ^{-8}}</math>
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| |<math>\frac{256(1 + \sqrt{2})^5}{(1 + \sqrt{5})^8}</math>
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| |964.597
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| |-
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| |<math>{δ_s^{-4}} {φ^9}</math>
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| |<math>\frac{(1 + \sqrt{5})^9}{512(1 + sqrt{2})^4}</math>
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| |1394.357
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| |<math>{δ_s^{5}} {φ^{-9}}</math>
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| |<math>\frac{512(1 + \sqrt{2})^5}{(1 + \sqrt{5})^9}</math>
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| |131.507
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| |-
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| |<math>{δ_s^{-5}} {φ^{10}}</math>
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| |<math>\frac{(1 + \sqrt{5})^10}{1024(1 + sqrt{2})^5}</math>
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| |701.583
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| |<math>{δ_s^{6}} {φ^{-10}}</math>
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| |<math>\frac{1024(1 + \sqrt{2})^6}{(1 + \sqrt{5})^10}</math>
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| |824.281
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| |-
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| |<math>{δ_s^{-6}} {φ^{11}}</math>
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| |<math>\frac{(1 + \sqrt{5})^11}{2048(1 + sqrt{2})^6}</math>
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| |''8.809''
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| |<math>{δ_s^{6}} {φ^{-11}}</math>
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| |<math>\frac{1024(1 + \sqrt{2})^5}{(1 + \sqrt{5})^10}</math>
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| |''-8.809''
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| |}
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|
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| == Tempering out the comma ==
| | 833.090{{c}} |
| 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:
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| <math>{δ_s^{-6}} {φ^{11}} = 1</math>
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| <math>φ^11 = δ_s^{6}</math>
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| <math>φ = (\sqrt[11]{δ_s})^{6}</math> OR <math>δ_s = (\sqrt[6]{φ})^{11}</math>
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| 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.
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| 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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| {| class="wikitable"
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| |+Anatolian Electrum (6ed-<math>φ</math>)
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| !Step
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| !In terms of <math>φ</math>
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| !Absolute Cents
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| !Closest hyper-interval (¢)
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| !Closest hypo-interval (¢)
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| !Difference (¢)
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| |-
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| |1
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| |<math>\sqrt[6]{φ}</math>
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| |'''138.848'''
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| |140.317
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| |131.507
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| | -1.47, +7.34
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| |-
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| |2
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| |<math>\sqrt[3]{φ}</math>
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| |'''277.700'''
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| |280.633
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| |271.824
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| | -2.93, +5.88
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| |-
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| |3
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| |<math>\sqrt{φ}</math>
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| |'''416.545'''
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| |420.950
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| |412.140
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| | -4.41, +4.41
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| |-
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| |4
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| |<math>{\sqrt[3]{φ}}^2</math>
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| |'''555.394'''
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| |561.267
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| |552.457
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| | -5.87, +2.94
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| |-
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| |5
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| |<math>{\sqrt[6]{φ}}^5</math>
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| |'''694.242'''
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| |701.583
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| |692.774
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| | -7.34, +1.47
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| |-
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| |6
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| |<math>φ</math>
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| | colspan="2" |'''833.090'''
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| |824.281
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| |0, +8.81
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| |-
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| | colspan="6" |...
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| |-
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| |''11''
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| |<math>{\sqrt[6]{φ}}^11</math>
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| |''1527.332''
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| | -
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| |1525.864
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| | +1.47
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| |}
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| {| class="wikitable"
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| |+Lydian Electrum (11ed-<math>δ_s</math>)
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| !Step
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| !In terms of <math>δ_s</math>
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| !Absolute Cents
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| !Closest hyper-interval (¢)
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| !Closest hypo-interval (¢)
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| !Difference (¢)
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| |-
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| |1
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| |<math>\sqrt[11]{δ_s}</math>
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| |'''138.715'''
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| |140.317
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| |131.507
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| | -1.60, +7.21
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| |-
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| |2
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| |<math>{\sqrt[11]{δ_s}}^2</math>
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| |'''277.425'''
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| |280.633
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| |271.824
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| | -3.21, +5.60
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| |-
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| |3
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| |<math>{\sqrt[11]{δ_s}}^3</math>
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| |'''416.145'''
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| |420.950
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| |412.140
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| | -4.81, +4.01
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| |-
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| |4
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| |<math>{\sqrt[11]{δ_s}}^4</math>
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| |'''554.860'''
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| |561.267
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| |552.457
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| | -6.41, +2.40
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| |-
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| |5
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| |<math>{\sqrt[11]{δ_s}}^5</math>
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| |'''693.575'''
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| |701.583
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| |692.774
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| | -8.01, +0.80
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| |-
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| |6
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| |<math>{\sqrt[11]{δ_s}}^6</math>
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| |'''832.289'''
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| |833.090
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| |824.281
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| | -0.80, +8.01
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| |-
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| |7
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| |<math>{\sqrt[11]{δ_s}}^7</math>
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| |'''971.004'''
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| |973.407
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| |964.597
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| | -2.40, +6.41
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| |-
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| |8
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| |<math>{\sqrt[11]{δ_s}}^8</math>
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| |'''1109.719'''
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| |1113.724
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| |1104.914
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| | -4.01, +4.81
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| |-
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| |9
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| |<math>{\sqrt[11]{δ_s}}^9</math>
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| |'''1248.434'''
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| |1254.040
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| |1245.231
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| | -5.60, +3.21
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| |-
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| |10
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| |<math>{\sqrt[11]{δ_s}}^{10}</math>
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| |'''1387.149'''
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| |1394.357
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| |1385.547
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| | -7.21, +1.60
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| |-
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| |11
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| |<math>δ_s</math>
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| |'''1525.864'''
| |
| | -
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| |1525.864
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| |0
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| |}
| |