Phi as a generator

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Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the Lucas numbers, closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the sqrtphi temperament, a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.

Let's use the archexample of 46edo. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.

Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.

While this is but a rank-2 temperament, suppose more generators could be added, apply phi temperament to arbitrarily higher ranks? For the price of a 3/1 you get a 7/1 and so on. That's an interesting idea on its own but it gets even much better, when one considers periods. The period, as I imagine it and maybe I'm way off the mark mathematically, can be seen as an abstract, degenerate rank. It might not be immediately so, 600 cents hardly fills in for 701.955, but eventually it gets there, at very least with phi tunings. For the price of complexity one gets a different kind of simplicity.

(Heck maybe this is mathematical gibberish for you REAL mathematicians but it works for and makes intuitive sense to me.)

For various reasons, efficiency, elegance, symmetry, curiosity, je ne sais quoi, I feel compelled to extend these temperaments to their ultimate abstraction, rank-1, equal temperament. The search is for those which structurally match this phi temperament best. This is done by a simple process, finding those moments of symmetry of the phi/period rank-2 scale, noting which ones have the closest large and small step sizes, and then equally tempering these. The results are curious.

A couple days ago I worked out this list.

Generator = 833.0903 cents



"Modi Sephirotorum"

ET: 3 (366:466)

ET: 4 (99:366)

ET: 7 (99:267)

ET: 10 (99:168)

ET: 13 (69:99)

ET: 23 (30:69)

ET: 36 (30:39) best value!

ET: 121 (8.749 : 12.675)



ET: 6 (133:233)

ET: 10 (99:133)

ET: 16 (34:99)

ET: 26 (34:64)

ET: 36 (30:34)



ET: 36 (33:36)



ET: 36 (32:34)




ET: 85 (138:163)

Generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)

This temperament is named "elderthing" according to the map of rank-2 temperaments. It was first added to that page by Kosmorsky, so he may be the one who named it.



ET: 49 (24:27)



ET: 16 (67:83)

ET: 72 (18.37:16.18)



ET: 33 (30:40)

ET: 121 (982:1018)



ET: 16edt (126:110)

ET: 121edt (154:157) woo!



ET: 16edt (118:126)



ET: 9edt (199:236)

ET: 54edt (52:36)

ET: 102edt (16:21)



ET: 25edt (72:91)

ET: 105edt (16:19)


Period: 6edt

ET: 48edt (37:45)


Period: 9edt

ET: 153edt (122:157)


Period: 5/1

ET: 10ed5 (259:287)

ET: 97ed5 (280:347)