Re: [RE: Scales found within 19 eq and 31 eq]

Joe Monzo <monz@juno.com> wrote:
> From: Joe Monzo <monz@juno.com>
>
> > [Paul Erlich, TD 294.19]
> >
> > 19 and 31 are meantone tunings; hence any conventionally
> > notated scales that do not rely on enharmonic equivalence
> > will work in 19 and 31.
>
>
> While the rest of Paul's post accurately describes the
> implications or approximations of 19- and 31-eq to meantone
> tunings, it is important to note that, while these ETs
> are close enough that for all intents and purposes they can
> be assumed to be identical to their respective meantones,
> strictly speaking, by mathematical definition, 19- and 31-eq
> are *not* meantone tunings.
>
> Nitpicking, perhaps, but one should be careful about making
> statements that 'x and y ARE z', when in fact an approximation
> is involved (albeit, in this case, an extremely close one).
>
> -monz

The term "meantone" has become a bit murky as of late (during the last half
century). In its strictest meaning, it refers to a regular temperament (all
like intervals _exactly_ the same size) where the whole step (tone)is
_exactly_ the geometric mean between 9:8 and 10:9. This is the 1/4-(syntonic)
comma meantone temperament. At some point, other regular temperaments
reducing the fifths by other fractions of a comma (anywhere from 1/8 to 1/3)
began to be called "meantone" temperaments, presumably because their major
seconds were *somewhere* between 9:8 and 10:9. Certainly 12-et and 19-et fall
into this category, and if all major seconds in 31-et are the same size, 31-et
also. Because of this fuzziness, I prefer to reserve the term "meantone" for
only 1/4- comma, and call the others M/N-comma regular temperaments. BTW, even
though 12-eq is _extremely_ close to 1/11-comma, as 19-et is _extremely_ close
to 1/3-comma, they are _all_ regular temperaments.

Fred Reinagel

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> From: Fred Reinagel <freinagel@netscape.net>

> Joe Monzo <monz@juno.com> wrote:

> > > [Paul Erlich, TD 294.19]
> > > 19 and 31 are meantone tunings; hence any conventionally

> > strictly speaking, by mathematical definition, 19- and 31-eq
> > are *not* meantone tunings.

> The term "meantone" has become a bit murky as of late (during the last half

> seconds were *somewhere* between 9:8 and 10:9.

Thanks for bringing this up, Fred, I personally am a bit of an expert on
meantone tuning, renaissance and baroque style, and I don't have a clue
what you guys are talking about.

The essence of meantone as I use it isn't that it's quarter comma; it can
be third-comma or fifth comma or sixth comma (which gets a little silly
and academic). It's that you come out with a good average approximation
for the intervals that really matter, and let the wolves howl away on the
other intervals. Somewhere in the Baroque, people started using wolves for
fun (Bach, Farewell suite to his brother going on a trip, very youthful
work), but that was a gimmick, pretty much a sign that meantone had worn
out its welcome.

What do you mean by calling these mathematically derived equal emperaments
'mean tone tunings'? Is it really just the size of the seconds?

Judy

On Fri, 27 Aug 1999, Judith Conrad wrote:
> I personally am a bit of an expert on
> meantone tuning, renaissance and baroque style, and I don't have a clue
> what you guys are talking about.

The term "meantone" as I use it refers to any tuning in which eleven
fifths are tempered narrow by the same amount, typically between 1/3 and
1/6 syntonic comma, and the last fifth (diminished sixth, most commonly
G#-Eb but movable as necessary) takes up the slack and becomes a wolf.
(Or extensions in which extra notes from the spiral of fifths are added
on at either end, but still keeping the size of the fifths consistent,
resulting in split accidentals, as was done on many historical
instruments.) I think this is pretty consistent with the use I see of
the term on this list and elsewhere.

[snip]

> What do you mean by calling these mathematically derived equal emperaments
> 'mean tone tunings'? Is it really just the size of the seconds?

More the size of the fifths. Paul and Joe are referring to the fact,
well-known among the sufficiently temperament-geeky, that the
appropriate 12-note subsets of 19TET and 31TET are aurally
indistinguishable from 1/3- and 1/4-comma meantone. (And 43TET from
1/5-comma MT, and 55TET from 1/6-comma, etc. These and more are all
listed in Barbour, but 19 and 31 are the best- and longest-known.
Huygens? Salinas? Vicentino?)

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
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