Phith root of phi - Revision history

Eliora: since phith rooth of phi ist alked about, the converse, phi to to the phi, should be talked about.

2026-06-17T20:44:42Z

since phith rooth of phi ist alked about, the converse, phi to to the phi, should be talked about.

← Older revision		Revision as of 20:44, 17 June 2026
Line 17:		Line 17:

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

			== See also ==
			* [[User:Eliora/Phi to the phi\|Phi to the phi]] - interval ''to'' which acoustic phi instead is dividing ''it'' by logarithmic phi

	== External links ==		== External links ==

Hotcrystal0: add missing category

2026-04-28T15:14:43Z

add missing category

← Older revision		Revision as of 15:14, 28 April 2026
Line 23:		Line 23:

	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]
			[[Category:Golden ratio]]

BudjarnLambeth: Uncategorised page

2025-12-28T00:12:08Z

Uncategorised page

← Older revision		Revision as of 00:12, 28 December 2025
Line 21:		Line 21:

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

			[[Category:Irrational intervals]]

Pailiaq: ypo/* Compositions based on the golden ratio */ typo

2025-11-26T23:57:57Z

ypoCompositions based on the golden ratio: typo

← Older revision		Revision as of 23:57, 26 November 2025
Line 16:		Line 16:
	* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)		* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)

	* ''[https://www.youtube.com/watch?v=gXMqkyVbFmI Durationplex]'' - [[Sean Archibald]] (~~2021~~)		* ''[https://www.youtube.com/watch?v=gXMqkyVbFmI Durationplex]'' - [[Sean Archibald]] (2025)

	== External links ==		== External links ==

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

Pailiaq: /* Compositions based on the golden ratio */

2025-11-26T21:54:50Z

Compositions based on the golden ratio

← Older revision		Revision as of 21:54, 26 November 2025
Line 15:		Line 15:

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

			* ''[https://www.youtube.com/watch?v=gXMqkyVbFmI Durationplex]'' - [[Sean Archibald]] (2021)

	== External links ==		== External links ==

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

Pailiaq at 09:12, 26 November 2025

2025-11-26T09:12:15Z

← Older revision		Revision as of 09:12, 26 November 2025
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.		[[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 ==		== Compositions based on the golden ratio ==

Pailiaq: /* Phith root of Phi ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]) */

2025-11-26T09:05:33Z

Phith root of Phi ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math])

← Older revision		Revision as of 09:05, 26 November 2025
Line 1:		Line 1:
	== Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==		== 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:		<math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:

Pailiaq: created a phith root of phi page, moving it from the golden ratio page

2025-11-26T09:03:55Z

created a phith root of phi page, moving it from the golden ratio page

New page

== 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.

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

: 833.09¢ ÷ φ = 514.878¢

When using acoustic phi (~833.09¢) as the [[equave]] instead of the octave, <math>\varphi^{1/\varphi}</math> generates MOS scales at Fibonacci cardinalities (3, 5, 8, 13...). This creates fractal, self-similar scales where every interval relationship exhibits the golden ratio, which is useful for compositions that maximize golden properties while avoiding octave equivalence entirely.

[[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>}}

== Compositions based on the golden ratio ==

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

== External links ==

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