Intuitive Well-Tempering

Hey all,

An open question I'm exploring is: How mathematically precise a historical
temperament needs to be in practice to be usable and have the general qualities
ascribe to and need for 18th and 19th century music in well-tempered tunings?
My answer: not very.

As an example, I came up with a quite usable 'intuitive tuning' last night, and test
all the keys and played some of the WTC, and it was quite pleasant and usable.
I simply did a quick and dirty 1/4 comma-meantone where the fifths from C to pure
'5/4 E' beat equally (if you do the math, it turns out to be about 3.0779 hz), then , I
halved this beat rate, reasoning that it would 'tame the wolf' and give me more
usable keys. So I did an equal-beating 8 up, 3 down; and then a 7 down 4 up
version using this beat rate. Then, I kind of brushed up the A-flat and E major triad
relationship, to make them both usable....it was quite easy actually. It sounded
functionally like an equal-beating 1/6 or 1/7 comma-meantone, and/or like a kind-of
intuited Young well-temperament, and was warm and interesting. It had the same
general qualities of which keys are more active, etc., that Werkmeister does. I
think that my hypothesis would be that these kind of ad-hoc 'utility tunings'
happened more often than not during the 19th century, and, they have the added
bonus of being easy to tune. In other words, it was close enough that most ears
wouldn't be able to tell the difference in performance (would any ears?) between it,
and a mathematically fussy and theoretically correct well-temperament.

<blasphemy> In fact, there is nothing whatsoever special about that particular beat
rate---one could choose any slow enough beat rate and have an arbitrary, usable
well-temperament by following the general principle of '7 down, 4 up' or
what-have-you. </blasphemy>

I hear the rebuttal: how do instrumentalists find the notes when playing with the
keyboard? I reply: how did they find it when there was NO standard western
tuning?

An interesting aside: when you tune a theoretically correct 1/4 comma meantone,
and half THAT beat rate, and do an equally-beating well-temperament, you get a
remarkably close 12-tet approximation. Try it in Scala sometime.....

Best,
Aaron.

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> Hey all,
>
> An open question I'm exploring is: How mathematically precise a
>historical
> temperament needs to be in practice to be usable and have the
>general qualities
> ascribe to and need for 18th and 19th century music in well-
>tempered tunings?
> My answer: not very.

agreed.

> I
> think that my hypothesis would be that these kind of ad-
>hoc 'utility tunings'
> happened more often than not during the 19th century,

agreed again. jorgensen's _tuning_ tome details plenty of these kinds
of historically documented ad-hoc strategies, and often supplies
precise numerical tables for each *just by way of example*. some
critics of jorgensen that have been quoted here have completely
ignored the text and gone on to rail against the precision of the
numerical tables. they should read the text.

a nice paper you might want to read, which intelligently argues
against the over-application of certain anachronistically
precise, "stock" temperaments commonly found on keyboards and tuners,
while making other points which will surely interest you, is:

http://ourworld.compuserve.com/homepages/paulpoletti/T4D.PDF

> An interesting aside: when you tune a theoretically correct 1/4
>comma meantone,
> and half THAT beat rate, and do an equally-beating well-
>temperament, you get a
> remarkably close 12-tet approximation.

wouldn't that depend on where the original fifth lies within your
bearing compass?

In a message dated 9/19/03 5:24:29 PM Eastern Daylight Time,
perlich@aya.yale.edu writes:

> > Hey all,
> >
> > An open question I'm exploring is: How mathematically precise a
> >historical
> > temperament needs to be in practice to be usable and have the
> >general qualities
> > ascribe to and need for 18th and 19th century music in well-
> >tempered tunings?
> > My answer: not very.
>
> agreed.
>
>

Question, How mathematically precise does equal temperament need to be in
practice to be usable?

also not very, and it will have the natural quality generally ascribed to
equal temperament.

Johnny

On Friday 19 September 2003 05:31 pm, Afmmjr@aol.com wrote:
> In a message dated 9/19/03 5:24:29 PM Eastern Daylight Time,
>
> perlich@aya.yale.edu writes:
> > > Hey all,
> > >
> > > An open question I'm exploring is: How mathematically precise a
> > >historical
> > > temperament needs to be in practice to be usable and have the
> > >general qualities
> > > ascribe to and need for 18th and 19th century music in well-
> > >tempered tunings?
> > > My answer: not very.
> >
> > agreed.
>
> Question, How mathematically precise does equal temperament need to be in
> practice to be usable?
>
> also not very, and it will have the natural quality generally ascribed to
> equal temperament.
>
> Johnny

Indeed!

-Aaron.

On Friday 19 September 2003 04:22 pm, Paul Erlich wrote:

> a nice paper you might want to read, which intelligently argues
> against the over-application of certain anachronistically
> precise, "stock" temperaments commonly found on keyboards and tuners,
> while making other points which will surely interest you, is:
>
> http://ourworld.compuserve.com/homepages/paulpoletti/T4D.PDF

thanks for the linnk...it is good. I skipped over the basic info, but there
were some valuable, and not widely taked about, points in there....

> > An interesting aside: when you tune a theoretically correct 1/4
> >comma meantone,
> > and half THAT beat rate, and do an equally-beating well-
> >temperament, you get a
> > remarkably close 12-tet approximation.
>
> wouldn't that depend on where the original fifth lies within your
> bearing compass?

No. The importatn thing is the ratio between them, not their absolute
position.

If you started, say, an octave lower for your bearing, the same ratios apply.

Best,
Aaron.

On Friday 19 September 2003 05:31 pm, Afmmjr@aol.com wrote:
> In a message dated 9/19/03 5:24:29 PM Eastern Daylight Time,
>
> perlich@aya.yale.edu writes:
> > > Hey all,
> > >
> > > An open question I'm exploring is: How mathematically precise a
> > >historical
> > > temperament needs to be in practice to be usable and have the
> > >general qualities
> > > ascribe to and need for 18th and 19th century music in well-
> > >tempered tunings?
> > > My answer: not very.
> >
> > agreed.
>
> Question, How mathematically precise does equal temperament need to be in
> practice to be usable?
>
> also not very, and it will have the natural quality generally ascribed to
> equal temperament.
>
> Johnny

Another thing I might hazard to add (although I haven't really thought it out,
I'll say it as an instinct) - Just Intonation is probably the only tuning
system whose identity is dependant on accuracy (at least this is true for
simpler ratios, but that is like saying that one could tell the difference
between an irrational ratio and an arbitrarily complex rational one--I remain
skeptical)

Reprhased, most tuning systems have a built in, tolerant, margin of error, and
the mathematical descriptions of them are maps which get you the park....put
the map in the glove compartment when you arrive, and lay out your blanket,
and enjoy the picnic!!!! (The tuning-math folks perhaps don't agree with this
sentiment....) The ear, in the end, is what should value the system or not,
not the blackboard.....

-Aaron.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akj@r...> wrote:

> > > An interesting aside: when you tune a theoretically correct 1/4
> > >comma meantone,
> > > and half THAT beat rate, and do an equally-beating well-
> > >temperament, you get a
> > > remarkably close 12-tet approximation.
> >
> > wouldn't that depend on where the original fifth lies within your
> > bearing compass?
>
> No. The importatn thing is the ratio between them, not their
>absolute
> position.

right, but do you go up-down-down or down-up-down, etc.? these things
will affect the ratios, and thus the beat ratios for a particular
tuning.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akj@r...> wrote:

> Reprhased, most tuning systems have a built in, tolerant, margin of
>error, and
> the mathematical descriptions of them are maps which get you the
>park....put
> the map in the glove compartment when you arrive, and lay out your
>blanket,
>> and enjoy the picnic!!!! (The tuning-math folks perhaps don't
>agree with this
> sentiment....)

i think the vast majority of us *would* agree with this sentiment. i
most certainly do, which is why i tend to focus more on the
*structure* of tempered tuning systems (say, meantone in general)
rather than *particular instances* of that structure (say, 4/25-comma
meantone, various irregular meantones, etc). probably most of what's
on tuning-math concerns these structures.

regular temperament is really preferable to irregular temperament
only in cases where the error being distributed is quite large (per
consonant interval) to begin with and anything but an equal
distribution will yield one or more intolerable intervals. in some
cases, regular temperament is equal temperament because you're
distributing enough independent commas (equal in number to the
dimensionality of your consonant-interval basis).

ok, math-torture over!