--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
> <akjmicro@c...> wrote:
>
> > > So where do "very close to JI" and "quasi-just" fall in terms of
> cents?
> >
> > Probably somewhere around 3.1415927 cents for me.....
>
> You might consider joining the many fans of 72-equal. It is the
first
> et with errors under pi cents up to the 7 limit, and only the 4
cents
> flat of the 9 is over up to the 11 limit. If the 5-limit suffices,
> then good old 53 will do it for you. 99 gets you up to the 9-limit,
> 118 up to the 11-limit, and 130 up to the 13-limit. All of these, of
> course, are important systems anyway.

For 29-HTT (the tuning in the subject line) the max error is ~1.6237c
at the 7 limit and ~3.2474 at the 15 limit, with the error of each
consonance being either zero or one of those two figures (or very
close).

A listening test will show that 72-ET can't touch it, either at the 7-
limit or the 11-limit (and I have midi files to prove it). 72-ET
clearly sounds like a low-error temperament, while 29-HTT sounds like
JI without phase-locking, very similar to 130-ET. Beating that
occurs with approximated 7-limit consonances is much more noticeable
than at higher limits, so you had better cut your allowable error in
half at the 7 limit.

Gene, for your enlightenment: 29-HTT consists (except for one filler
tone) of 3 chains of fifths of ~703.5787c, or exactly (504/13)^
(1/9). The 3 chains of fifths contain tones 1/1, 5/4, and 7/4,
respectively, and the tones in each chain are taken to as many places
as are required to result in otonal ogdoads on roots Bb, F, C, G, D,
and A. This also gives very-near-just diatonic (5-limit) scales in 5
different keys.

The greatest advantage: the total number of tones/octave is small
enough to make acoustic instruments practical (the smallest intervals
being ~24 cents).

--George