Dear Claudio and all


> Ibo wrote:
>
> ET in Vienna  But I tend keep all my sent mails and I do not find any
>> such info that I had sent ,to you or anybody else.
>> Also, I do not find any such  information about ET in Froberger's time in
>> Vienna, at all, nor am I aware of such information.
>
>
2017-12-04 16:42 GMT+01:00 J. Claudio Di Veroli <[log in to unmask]>:

> Indeed, I cannot find any concrete information either!
>
> I just searched and found only a post to this list where you mention the
> matter, on 7/12/2012. Let me quote:
>
[Ibo:]

"Froberger was part of the enharmonic stronghold that the Habsburg courts
>> were ... Whether Froberger's alleged piece went through the whole cycle
>> is therefore not known (and even less, whether he could have used a 12-ET,
>> 19- or even 31-ET — all of which would have been possible because the
>> imperial court provided instruments with the suitable number of keys) and a
>> lovely subject of speculation without evidence. ;)"
>
>  Claudio:

> In this post, needless to say, you are NOT stating that ET was in use in
> Vienna, just that it was a possibility!
>
>
My point with this was, that *all *these figures for keys per octave (*12,
 19,  31*) are also divisors that result in (usable) Equal temperaments.*

12-ET based on the 5th
19-ET based on the (practically just) minor 3rd
31-ET based on the (practically just) major 3rd

and therefore there exists a *theoretical possibility* that even 12-ET was
explored.
If we, however, look at the great weight that *Just Intonation* had and
continued to have in musical practice before and until  12-ET  succeeded**,
it appears to me to have very little likeliness  that this new
temperament came into practical use. Don't forget that  12-ET was called
"new" during the whole *later* 18th century!

The difficulties of tuning 12-tone ET exactly enough should not be
underestimated, and the advantages of 19 and 31 need to be considered.

A working 19-ET can indeed easily be achieved by tuning a series of pure
minor 3rds (better tune major sixths!): The difference of "off-ness" is
0.15 cent per minor 3rd. The downside of 19-ET is the rather bad fifth
(-7.2 cents or practically minus a third of a Syntonic comma), and the
major 3rd is smaller than just by ca one third of a comma. It will work
only on very fundamental sounding instruments.

31-ET is tunable by the major 3rd - even if the latter needs to be in
average only 0.8 cent larger. In other words: If one tunes thirty-one
consecutive pure major 3rds, one will be short by approximatelyone third of
a (syntonic) comma after tuning the full circle.  Or: each of your standard
1/4-comma meantone 5ths needs to be slightly better by only 0.24 cent – ikn
tuning practice that is so small as not to matter. Of course 31-ET is not
very practical to handle - 31-note keyboards were more likely used with
just intonation schemes.

12-ET in comparison is more prone to deviations, the 5th is 2 cent off -
and - of course both its 3rds are far from being just.


* Don't want to elaborate here more, but isn't it striking:

  keys per octave 12  +  19   = 31

  intervals             5th - min3 = maj3

**ET succeeded only in a very slow process from the time around 1700 until
the 19th century



If my wording in my quoted post was confusing, I am the one to blame. But
there was certainly no implication intended whatsoever  (since any evidence
is missing!) that 12-tone ET was in use in Vienna and related areas, i.e.
areas under Habsburg cultural influence.


Best

Ibo

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