SandBox: Difference between revisions
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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: | 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}} | <math>{δ_s^{-6}} {φ^{11}} → 1</math> | ||
<math>φ^{11} | <math>φ^{11} → δ_s^{6}</math> | ||
<math>φ | <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. 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. | ||