Music/sound's relationship to the golden ratio without alluding to any conspiracy https://www.goldennumber.net/music/ <--- linked story includes easily to digest illustrations and tables.
The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as:
There are 13 notes in the span of any note through its octave.
A scale is composed of 8 notes, of which the
5th and 3rd notes create the basic foundation of all chords, and
are based on a tone which are combination of 2 steps and 1 step from the root tone, that is the 1st note of the scale.
Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2.While some might “note” that there are only 12 “notes” in the scale, if you don’t have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes. The word “octave” comes from the Latin word for 8, referring to the eight tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.
In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave. This provides an added instance of Fibonacci numbers in key musical relationships. Interestingly, 8/13 is .61538, which approximates phi. What’s more, the typical three chord song in the key of A is made up of A, its Fibonacci & phi partner E, and D, to which A bears the same relationship as E does to A. This is analogous to the “A is to B as B is to C” basis for the golden section, or in this case “D is to A as A is to E.”
Here’s another view of the Fibonacci relationship presented by Gerben Schwab in his YouTube video.
https://www.youtube.com/watch?v=2pbEarwdusc&feature=youtu.beFirst, number the 8 notes of the octave scale. Next, number the 13 notes of the chromatic scale. The Fibonacci numbers, in red on both scales, fall on the same keys in both methods (C, D, E, G and C). This creates the Fibonacci ratios of 1:1, 2:3, 3:5, 5:8 and 8:13:
Mathematics is a game of rules. Follow the rules and you can't lose.