Phith root of phi: Difference between revisions

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== Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==

The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>, or approximately 514.878 cents) is ~~another~~ useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.

The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>, or approximately 514.878 cents) is a useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.

<math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:

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[[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (acoustic phi) *and* because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢ (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.

[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.{{Interval Edo Approximation|interval=134636/100000|interval_name=<math>\varphi^{1/\varphi}</math>}}

[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.{{Interval Edo Approximation|interval=134636/100000|interval_name=<math>\sqrt[\varphi]{\varphi}</math>}}

== Compositions based on the golden ratio ==

* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)

* ''[https://www.youtube.com/watch?v=gXMqkyVbFmI Durationplex]'' - [[Sean Archibald]] (2025)

== External links ==

* [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]]

[[Category:Irrational intervals]]

[[Category:Golden ratio]]

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 == Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==
-The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>,  or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.
+The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>,  or approximately 514.878 cents) is a useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.
 <math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:
@@ Line 10: / Line 10: @@
 [[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (acoustic phi) *and* because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢  (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.
-[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.{{Interval Edo Approximation|interval=134636/100000|interval_name=<math>\varphi^{1/\varphi}</math>}}
+[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.{{Interval Edo Approximation|interval=134636/100000|interval_name=<math>\sqrt[\varphi]{\varphi}</math>}}
 == Compositions based on the golden ratio ==
 * ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)
+* ''[https://www.youtube.com/watch?v=gXMqkyVbFmI Durationplex]'' - [[Sean Archibald]] (2025)
 == External links ==
 * [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]]
+[[Category:Irrational intervals]]
+[[Category:Golden ratio]]