6edϕ: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each | '''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each. It is well-approximated by every third step of [[26edo]], and somewhat less accurately by [[5edf]]. | ||
It serves as a basic tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are | It serves as a basic tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (acoustic phi or the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are approximated by the 11th and 15th steps respectively, which reduce to the fifth and third steps if acoustic phi is taken as an equivalence. The fourth metallic harmonic is also represented exactly because it is the cube of acoustic phi. However, the fifth metallic harmonic is poorly approximated, falling about halfway between two and three steps when reduced. | ||
== Interval table == | == Interval table == | ||
{{Interval table}} | {{Interval table}} | ||
{{Stub}} | |||
Latest revision as of 08:39, 18 April 2024
| ← 5edϕ | 6edϕ | 7edϕ → |
6 equal divisions of acoustic phi (6edϕ) is the nonoctave tuning system derived by dividing acoustic phi into 6 equal steps of 138.8 cents each. It is well-approximated by every third step of 26edo, and somewhat less accurately by 5edf.
It serves as a basic tempered approximation for the metallic harmonic series, and the associated metallic intonation. The first metallic harmonic (acoustic phi or the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are approximated by the 11th and 15th steps respectively, which reduce to the fifth and third steps if acoustic phi is taken as an equivalence. The fourth metallic harmonic is also represented exactly because it is the cube of acoustic phi. However, the fifth metallic harmonic is poorly approximated, falling about halfway between two and three steps when reduced.
Interval table
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 138.8 | 10/9, 11/10, 14/13, 15/14, 20/19, 21/19 |
| 2 | 277.7 | 13/11, 15/13, 17/14 |
| 3 | 416.5 | 13/10, 14/11, 17/13, 19/15, 21/17 |
| 4 | 555.4 | 4/3, 7/5, 15/11, 19/14 |
| 5 | 694.2 | 3/2, 17/11, 19/13 |
| 6 | 833.1 | 5/3, 11/7, 21/13 |
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