What kind of near just intervals...?

[David Beardsley:]
> Sounds cool. What kind of near just intervals 11tet close to?

Recently John A. deLaubenfels posted a tuning math tutorial which
contained the following:

"To convert cents interval to frequency ratio:

Freq_ratio = 2 ^ (delta_cents / 1200)

That's "two to the power". Again, do an example: a 12-tET minor
third, 3 semitones or 300 cents, divided by 1200 gives 0.25 (octaves).
Two (2.0) raised to that power gives 1.1892; compare to 1.2000 for a
perfect 6/5 minor third ratio."

This shows one way to quickly answer this sort of a 'what kind of near
just intervals is such and such equal division of the octave close to'
question... By taking sequential multiples of the frequency ratio and
rounding
them to the nearest integer (e.g., using John's 0.25 quarter of an
octave example, 1:1.1892 becomes 1/1, 2:2.3784 becomes 2/2, 3:3.5676
becomes 4/3, etc.), and then choosing some simple criteria for
matching the fraction of an octave to one of these rounded ratios - I
think that the first difference between the numerators of the
sequential multiples and the rounded integers to round to 0.0 is a
simple enough criteria.

As the intervals of equal divisions of the octave all (save the
half-octave) have an exact mirror inversion (or complement), if you
carry this process past the half-octave,* 0.0 no longer has the same
meaning - so this process need not be carried out past the
half-octave... In 11e that would mean 1/1, 16/15, 17/15, 6/5, 9/7,
11/8, and their inversions 16/11, 14/9, 5/3, 30/17, 15/8, 2/1 would
all fulfill the (simple but arbitrary) 0.0 criteria.

While this type of a fraction of an octave into interval ratios rarely
if ever sounds any better (or to some, any different) than the
original equal divisions of the octaves, it can offer some simple ways
to start _seeing_ and manipulating certain relationships between the
two.

Dan
_________
*For what needs doing here you could use n/[(log24-log17)*(12/log2)],
where "n" is the equal division of the octave.