6edϕ: Difference between revisions
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'''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]]. | '''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]]. | ||
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 | 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 second steps if acoustic phi is taken as an equivalence. | ||
== Interval table == | == Interval table == | ||
{{Interval table}} | {{Interval table}} | ||
{{Stub}} | {{Stub}} | ||
Revision as of 07:48, 4 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.
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 second steps if acoustic phi is taken as an equivalence.
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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