Diophantine clarity

I mentioned before that you can get better tuning accuracy and still
have nice divisors for the Pythagorean and Didymus commas by dividing
the octave by 29460, with 576 parts to the Pythagorean comma and 528
parts to the Didymus comma; this gets you much closer to the poptimal
range of 5-limit atomic and away from lame expedients like 60.004 TU
for the schisma. However, we can do even better; the first division I
find actually *in* the poptimal range is 46032, and by good luck it
happens to be excellent so far as divisors for the commas go. If we
give these units a name (say "flu" for five-limit units) we get a
system suited both to expressing 5-limit pure intervals with great
exactitude, supporting atomic temperament, and with nicely divisible
commas.

We have:

Pythagorean comma: 900 flus 900=2^2 3^2 5^2
Didymus comma: 825 flus 825 = 3 5^2 11
Schisma: 75 flus
Major third: 14819 flus
Fifth: 26927 flus
Octave: 46032 flus
1/12 octave: 3836 flus
Cent: 38.36 flus exactly

If "flu" makes you ill, there is always "dc" for Diophantine clarity
units. 825 is an odd number, and we need 206.25 for how much the fifth
is flattended fdrom 26927 to get a 1/4 comma fifth, but we get some
fractions of a Didymus comma to come out integrally, and do even
better with the Pythagorean comma.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
However, we can do even better; the first division I
> find actually *in* the poptimal range is 46032, and by good luck it
> happens to be excellent so far as divisors for the commas go. If we
> give these units a name (say "flu" for five-limit units) we get a
> system suited both to expressing 5-limit pure intervals with great
> exactitude, supporting atomic temperament, and with nicely divisible
> commas.

We also can convert flus to TUs simply by multiplying by 4/5, and this
gives the following for TUs:

Pythagorean comma: 720 TU
Didymus comma: 660 TU
Schisma: 60 TU
Major third: 11855.2 TU
Fifth: 21541.6 TU
Octave: 36825.6 TU

If you set TUs to these precise values, TUs and flus differ only
proportionally. We now would have a system where any 5-limit interval
would work out to a precisely defined number of TUs, which seems to me
to be much better than simply leaving the matter undefined, or
arbitarily deciding the Pythagorean comma must be tuned pure.