Phi as a generator: Difference between revisions
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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 [http://en.wikipedia.org/wiki/Lucas_number 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 [[Kleismic_family#Sqrtphi|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it. | 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 [http://en.wikipedia.org/wiki/Lucas_number 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 [[Kleismic_family#Sqrtphi|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 [[ | Let's use the archexample of [[46edo]]. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (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. | 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. | ||
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Period=[[ | Period=[[2edo]] | ||
ET: [[6edo|6]] (133:233) | ET: [[6edo|6]] (133:233) | ||
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Period=[[ | Period=[[3edo]] | ||
ET: [[36edo|36]] (33:36) | ET: [[36edo|36]] (33:36) | ||
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Period=[[ | Period=[[4edo]] | ||
ET: [[36edo|36]] (32:34) | ET: [[36edo|36]] (32:34) | ||
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Period=[[ | Period=[[5edo]] | ||
ET: [[85edo| | "Elderthing" | ||
ET: [[85edo|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. | |||
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Period=[[ | Period=[[7edo]] | ||
ET: [[49edo|49]] (24:27) | ET: [[49edo|49]] (24:27) | ||
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Period=[[ | Period=[[8edo]] | ||
ET: [[16edo|16]] (67:83) | ET: [[16edo|16]] (67:83) | ||
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Period=[[ | Period=[[11edo]] | ||
ET: [[33edo|33]] (30:40) | ET: [[33edo|33]] (30:40) | ||
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Period=[[ | Period=[[Edt|3/1]] | ||
ET: [[16edt|16edt]] (126:110) | ET: [[16edt|16edt]] (126:110) | ||
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Period=[[ | Period=[[2edt]] | ||
ET: [[16edt|16edt]] (118:126) | ET: [[16edt|16edt]] (118:126) | ||
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Period=[[ | Period=[[3edt]] | ||
ET: [[9edt|9edt]] (199:236) | ET: [[9edt|9edt]] (199:236) | ||
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Period=[[ | Period=[[5edt]] | ||
ET: [[25edt|25edt]] (72:91) | ET: [[25edt|25edt]] (72:91) | ||
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Period: [[ | Period: [[6edt]] | ||
ET: [[48edt|48edt]] (37:45) | ET: [[48edt|48edt]] (37:45) | ||
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Period: [[ | Period: [[9edt]] | ||
ET: 153edt (122:157) | ET: 153edt (122:157) | ||
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Period: [[ | Period: [[Ed5|5/1]] | ||
ET: [[10ed5|10ed5]] (259:287) | |||
[[ | ET: [[97ed5|97ed5]] (280:347) | ||
{{todo|cleanup}} | |||
[[Category:Golden ratio]] | |||
[[Category:Golden]] | |||