Why the Golden Ratio is golden

Why the Golden Ratio is golden

Dan
Dozens Disciple
Dan
Dozens Disciple
Joined: Aug 8 2005, 02:45 PM

May 9 2018, 03:54 AM #1

I saw this Numberphile video and thought it was interesting.

https://www.youtube.com/watch?v=sj8Sg8qnjOg

Of course, I already knew that φ equals the continued fraction [1; 1, 1, 1, 1, ...], but didn't realize what this had to do with "optimizing" the seed arrangement of a sunflower.

If φ is "the most irrational number", it raises the question of how to quantify what makes one real number "more irrational" than another.  Any ideas for how to define "relative irrationality"?

One practical use of "almost rational" irrational numbers is musical tuning.  For example, the equal-tempered major third has a frequency ratio of the cube root of two (1.259921 decimal, 1.315188 dozenal), approximating the just intonation ratio 5/4.  So, would a tuning based on setting the minor sixth to φ (or equivalently, setting the major fifth to (8/φ)^(1/4), or 691.7274 cents) sound dissonant?
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Kodegadulo
Obsessive poster
Kodegadulo
Obsessive poster
Joined: Sep 10 2011, 11:27 PM

May 9 2018, 04:25 AM #2

Dan wrote: I saw this Numberphile video and thought it was interesting.

https://www.youtube.com/watch?v=sj8Sg8qnjOg

Of course, I already knew that φ equals the continued fraction [1; 1, 1, 1, 1, ...], but didn't realize what this had to do with "optimizing" the seed arrangement of a sunflower.

If φ is "the most irrational number", it raises the question of how to quantify what makes one real number "more irrational" than another.  Any ideas for how to define "relative irrationality"?
This was explained in the video. φ is maximally irrational precisely because its continued fraction coefficients are all the lowest possible value: 1.  Any natural number denominator greater than one will tend towards dominating the result, such that you could take just the terms up to that point, and use them to construct a rational number that is a "ridiculously good" approximation of the irrational number, so that it would take many additional terms to calculate a significantly better approximation.

So I suppose we could define a relative scale of "rationality" by weighting the coefficients by their size and perhaps their position in the series (giving greater weight to earlier coefficients).
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Paul Rapoport
Dozens Disciple
Paul Rapoport
Dozens Disciple
Joined: Dec 26 2012, 01:59 AM

May 10 2018, 03:28 PM #3

For example, the equal-tempered major third has a frequency ratio of the cube root of two (1.259921 decimal, 1.315188 dozenal), approximating the just intonation ratio 5/4.  So, would a tuning based on setting the minor sixth to φ (or equivalently, setting the major fifth to (8/φ)^(1/4), or 691.7274 cents) sound dissonant?
It turns out that the ET major third is a problematic approximation of the JI ratio 5/4 (being nearly 14 cents too big). The piano masks its out-of-tune thirds and sixths by its inharmonic overtones, which are not integer multiples of fundamentals. That works fairly well much of the time.

A perfect fifth of 691.7274 cents is about 10 cents too narrow but within the range of recognizability for that interval. Given the right circumstances, it's usable, without too much beating, and therefore not discordant (or dissonant) in the usual sense, although in various contexts it may sound out of tune. Easley Blackwood has demonstrated in his 12 Microtonal Etudes and elsewhere that you may get away with (have a range of recognizability of) perfect fifths between 685.714 and 720 cents.

All of which may or may not answer your question. (I'm not sure yet of the assumptions in arriving at your interval for the perfect fifth, including where other notes are.)
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