# flu

[Gene Ward Smith, Yahoo tuning group message 57606 (Fri Mar 18, 2005 12:55 pm)]

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.

. . . . . . . . .
[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

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:

• exactly 125/5754 (= ~0.021724018 = ~1/46 ) of a millioctave
• exactly 64/2877 (= ~0.022245395 = ~1/45 ) of a yamaha-unit
• exactly 25/959 (= ~0.026068822 = ~1/38 ) of a cent
• exactly 1325/5754 (= ~0.230274592 = ~1/4 ) of a türk-sent
• exactly 30103/46032 (= ~0.653958116 = ~2/3 ) of a jot
• approximately 4/5 (= ~0.800065785) of a temperament-unit
• exactly 1 65/959 (= ~1.067778936 = ~1 1/15 ) 12mus (dodekamus)
• exactly 260/959 (= ~4.271115746 = ~4 1/4 ) 14mus (tetradekamus)

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
```
. . . . . . . . .

### flus calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = flus

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

 support level donor: \$5 USD friend: \$25 USD patron: \$50 USD savior: \$100 USD angel of tuning: \$500 USD microtonal god: \$1000 USD