re : grady re 11tet as ji

>
> Robert!
> If the 22 tone step is 54.54 cents. the maximum tolerance anyone could
> allow is 27 cents. At this point you would run into the next tone. Now if
> you allow 15 cents away as in the 25/24 you are more than half way to the
> 27 cent maximum. This means that any ratio would have more than a 50%
> chance of fitting. Don't you think this is a little all-inclusive?
>

Very good point. You are correct that the .01 term as an initial
error term is fairly tolerant for 22tet.

And now I see how to choose a 'correct' initial error term at
less than half the width of the tone step.

This will give a justifiable mathematic best fit, and (I think)
will result in all et notes getting unique ji notes assigned to them.

On the other hand, if one wanted the ji note that represented
"most likely to be heard as" then this results in things that
don't match what would be heard. Using an initial error term
of only slightly more correctness for 31tet (.006)
shows the note number 18 note being approximated first by

18 : =118/79

rather than a 3/2, so there should be another function in there,
constraining things to human pitch discrimination (30c?), looking
at limits, or how much of a jump there is to the next
approximation, etc.

And as I pondered in the rest of the mail, having unique ratios
for each et note doesn't completely match the usage, since punning
means exploiting two ratios represented by the same et step.

thanks,

Bob Valentine

[Robert C Valentine:]
>Using an initial error term of only slightly more correctness for
31tet (.006) shows the note number 18 note being approximated first by

18 : =118/79

rather than a 3/2, so there should be another function in there,
constraining things to human pitch discrimination (30c?), looking at
limits, or how much of a jump there is to the next approximation,
etc.

Using the (simple but arbitrary) .0 process I posted earlier this week
(TD 284.4), 31e would look like this:

1/1 0� 0.0

44/43 +1� 0.03
45/44 0� 0.01
46/45 -1� 0.02
47/46 -1� 0.04

22/21 +3� 0.04
23/22 0� 0.01

15/14 +3� 0.0
16/15 -4� 0.0

12/11 -4� 0.0

19/17 -1� 0.0

8/7 -1� 0.0

7/6 -4� 0.0

6/5 6� 0.0

11/9 -1� 0.0

5/4 -1� 0.0

9/7 +9� 0.0

17/13 0� 0.0

4/3 -5� 0.0

15/11 -5� 0.0

7/5 +2� 0.0

Dan