Second-best fifths

I suggested a while back that the cutoff for an interval counting as
a more or less decent approximation to a JI interval was around nine
cents. If so, 71 looks kind of nice in a way, as it is the smallest
edo with two "more or less decent fifths" by this definition.

Here is a list of the top 20 among the first 100 in terms of the size
of the error in cents of the second-best fifth:

100 6.044999
88 7.135908
95 7.218159
93 7.722418
83 7.979097
98 8.249081
76 8.571315
90 8.621668
71 8.997254
81 9.156110
97 9.171496
78 9.647309
86 9.672906
91 10.132911
85 10.190295
59 10.429577
96 10.544999
96 10.544999
92 10.650653
99 11.045910

The smallest edo on this list is 59; the second-smallest 71.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> 96 10.544999
> 96 10.544999
> 92 10.650653
> 99 11.045910
>
> The smallest edo on this list is 59; the second-smallest 71.

I have to amend this a little. This is because 96 appears twice; one of
those times it should really have been 64, which has exactly the same
size of second-best fifth as 96.

64edo is an excellent (9-limit poptimal) flattone tuning using its best
fifth of 693.75 cents, and a sharp superpyth tuning using its second-
best fifth of 712.5 cents.

96edo, of the bells, has a best fifth of 700 cents. Its second-best
fifth is the same 712.5 cent sharp superpyth fifth as 64; both derive
from 32.

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> > wrote:
> >> 96 10.544999
>> 96 10.544999
>> 92 10.650653
>> 99 11.045910
>>
>> The smallest edo on this list is 59; the second-smallest 71.
> > I have to amend this a little. This is because 96 appears twice; one of > those times it should really have been 64, which has exactly the same > size of second-best fifth as 96.
> > 64edo is an excellent (9-limit poptimal) flattone tuning using its best > fifth of 693.75 cents, and a sharp superpyth tuning using its second-
> best fifth of 712.5 cents. > > 96edo, of the bells, has a best fifth of 700 cents. Its second-best > fifth is the same 712.5 cent sharp superpyth fifth as 64; both derive > from 32.

I _was_ wondering why 64 didn't show up on that list...

http://www.io.com/~hmiller/midi/pavane-64.mid