Over here! (long)

[Clark]

Hi, Ed,

> In fact, the meantone tunings are simply subsets of the higher ET's. 
> i.e.  a 54 TET has all the notes found in 12 TET as well as those we 
> form by moving from ET to well-temperaments.

Umm, 54 is divisible by 6 9 times so an (n+9)/n interval is 200 cents,
so it contains half of 12tET:

n	cents
1	0
2	22.2
3	44.4
4	66.666
5	88.8
6	111.1
7	133.3
8	155.6
9	177.8
10	200.0
...

Only those et's divisible by 12 contain all the familiar 12tET pitches,
i.e. 24, 36, 48, 60, 72... which are the temperaments Carillo explored.
Ezra Sims and Joe Maneri use 72tET (along with like, half of Boston) but
whose consistent proximity to just intervals (17-limit...) makes it a
versatile system.

Since Huygens, it has been possible to associate meantone tunings with
equal temperaments; in 1693, he wrote an elegant essay showing the
proximity of 31tET to ¼-comma mt, to exclusion of the anecdotal "proof"
of its musical uselessness furnished by Mersenne and Zarlino (seemingly
a well-used device in this subject), and which he does with logs. The
same case can be made for 19tET and 1/3 cmt, 12tET and 1/11cmt, &c -
their distinctions arguably are theoretical and imperceptible (though
Huygens doesn't refrain from pointing them out).

Vicentino's pseudo-adaptive system was not without its controversy. Here
is a reconstruction of his arcicembalo:
<http://www.infosys.it/pamparato/ima/ma/ma81/tastiere.html>


Clark