The "Diophantine clarity" division, useful for discussing 5-limit tempering. If one step is a "flu", then a Pythagorean comma is 900 flus and a Didymus comma 825 flus, and therefore a schisma is 75 flus. The flu system is plenty accurate enough while tempering the atom out of the discussion. I recommend it as a replacement for Temperament Units.
An interval measurement invented by Gene Ward Smith as a replacement for Temperament units. It is especially accurate for giving integer values for 5-limit just intonation and some meantone temperaments; it is not as good for higher-limit rational intonation.
A flu is the logarithmic division of the octave into 46032 equal parts. It is calculated as the 46032nd root of 2 (46032√2, or 2(1/46032) ), with a ratio of approximately 1:1.000015058. It is an irrational number, and is one degree of 46032-edo. The formula for calculating the flu-value of any ratio is: cents = log10r * [46032 / log102] or cents = log2r * 46032, where r is the ratio.
A flu is:
Here are flu values for some intervals of 5-limit just-intonation and associated temperament measurement units:
interval 2,3,5,7,11-monzo ~flus
3/2 ratio [ -1 1 , 0 0 0 > 26926.99383
5/4 ratio [ -2 0 , 1 0 0 > 14818.99406
7/4 ratio [ -2 0 , 0 1 0 > 37164.16177
11/8 ratio [ -3 0 , 0 0 1 > 21148.55627
pythagorean comma [-19 12 , 0 0 0 > 899.9259984
syntonic comma [ -4 4 , -1 0 0 > 824.9812689
skhisma [-15 8 , 1 0 0 > 74.94472943
612-edo schisma [ 1/612 0 , 0 0 0 > 75.21568627
grad [-19/ 12 12/ 12, 0 0 0 > 74.9938332
temperament unit [-19/720 12/720, 0 0 0 > 1.24989722
kirnberger-atom [161 -84 , -12 0 0 > 0.589245253
It so happens that the generator "5ths" of 1/3-comma meantone, 1/5-comma meantone, and 1/11-comma meantone all come extremely close to an integer flu value. Because 46032 is exactly divisible by 12, all intervals in 12-edo have exact integer flu values. Here are some flu values for "5ths" of some meantone temperaments:
19-edo 5th 26650.10526 1/3-comma 5th 26652.00008 50-edo 5th 26698.56 7/26-comma 5th 26704.88349 1/4-comma 5th 26720.74852 31-edo 5th 26728.25806 1/5-comma 5th 26761.99758 43-edo 5th 26762.7907 55-edo 5th 26782.25455 1/6-comma 5th 26789.49696 1/11-comma 5th 26851.99554 12-edo 5th 26852
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