User:Iwuqety/Electrym: Difference between revisions

Line 106:

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.

@@ Line 106: / Line 106: @@
 <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.