Defining Meantone and Pythagorean

To me, "meantone" means being able to assume, at the very least, that the
best approximation to a 4:5 major third is obtained by a chain of 4 fifths
(octave reduced), when chains of no more than 11 fifths (or fourths) are
considered (i.e. at most a 12 note scale). This makes 12-tET the borderline
of meantone in one direction, since beyond that -8 fifths is a better
approximation. And a fifth size of around 691.51c is the border in the
other direction (26-tET is in, but 33-tET is out), since 11 fifths becomes
a better approximation. This requirement also ensures a similar property
for approximations to the 5:6 minor third.

Of course, the term "meantone" was coined to mean that the whole-tones of
the tuning were close to the mean of the major and minor whole-tones, 9/8
and 10/9, and the popular ones ranged from about 1/6 to 1/3 comma, centered
on 1/4 comma (the exact mean).

I see people (myself included) casting about for a term that covers all
tunings based on a single chain of approximate fifths. Thus we get
travesties like so-called "negative meantones", and the term "Pythagorean"
being used for this purpose in Scala.

What's wrong with "chain-of-fifths". Anyone have another suggestion?

We have people saying that Pythagorean is contained in meantone and others
saying that meantone is contained in Pythagorean. Can't we agree that
Pythagorean and meantone are disjoint or at least share only 12-tET as the
borderline case of both. We can go further.

If I were asked to determine the other limit of the Pythagorean region I'd
put it at 17-tET, 705.8c (also 34-tET). This is where the optimum 4:5 flips
from -8 to +9 fifths. I'm not sure what to call the next region, but I'd
say it goes from 17-tET to 59-tET, 711.8c (22-tET and 27-tET are both in
this region. Could be called the 22-tET region or the decatonic region or
the Erlich region. Who was the earliest known proponent of 22-tET or
tunings in general where 9 fifths is a major third?

So the known-world of chain-of-fifths tunings can be usefully divided into
3 kingdoms:
Chain-of-fifths
/ | \
| Meantone | Pythagorean | Erlichean? |
691.5c 700.0c 705.8c 711.8c Borderline fifth size
59-tET* 12-tET 17-tET 59-tET Borderline ETs
11 4 -8 9 14 Fifths in a maj third

* It's kind of annoying that my proposed lower border of meantone doesn't
correspond to (the best fifth in) any ET, but it turns out to be extremely
close to the second-best fifth in 59-tET and may be conveniently considered
as such, so we've got 59-tET at both ends!

And beyond there be dragons (well wolves anyway).

Regards,
-- Dave Keenan
http://dkeenan.com

At 01:36 PM 4/23/99 +1000, Dave Keenan wrote:

>I see people (myself included) casting about for a term that covers all
>tunings based on a single chain of approximate fifths. Thus we get
>travesties like so-called "negative meantones", and the term "Pythagorean"
>being used for this purpose in Scala.

>What's wrong with "chain-of-fifths".

It assumes that the repeated interval falls on the fifth note of the scale.
In China, it is the fourth note. In Joseph Yasser's 19-TET, it is the
eighth. Frequently but not necessarily, the repeated interval is the dominant.

>Anyone have another suggestion?

"_Regular Temperament_ -- A temperament in which all the fifths save one are
of the same size, such as the Pythagorean tuning or the meantone
temperament. (Equal temperament, with all fifths equal, is also a regular
temperament, and so are the closed systems of multiple division.)"--James
Murray Barbour, _Tuning and Temperament_, 1951, glossary.

Although one could infer from Barbour's initial choice of words that there
must be a wolf, his specific inclusion of ET makes that odd interval
optional. Barbour applies the terms "just" and "meantone" strictly,
requiring both to have a pure 5:4 major third.

Tunings based on repetition of rational fifths such as 112/75, 121/81,
148/99, and 160/107 are certainly regular. They fall under some definitions
of just and meantone, but not Barbour's. What is the name of their type?

--Charlie Jordan <http://www.rev.net/~aloe/music/tuning.html>

Greetings,
Charlie writes:
>Tunings based on repetition of rational fifths such as 112/75, 121/81,
>148/99, and 160/107 are certainly regular. They fall under some definitions
>of just and meantone, but not Barbour's. What is the name of their type?

At the moment, these tunings are referred to as "microtonal" tunings by
most of the uneducated, ( or partially exposed........). This is the overall
view that the public seems to to hold of anything but the status quo. As far
as a technical description, I also favor regular or irregular as
descriptive. The numbers that can be attached
I am selling Mr. Kirnberger and Young's ideas out there, but must cloak
them in the mossy, classical drapes of "culture" to get past the fences.
This is changing as more and more professional tuners become
multi-temperament capable. There has been a dearth of temperament change
between 1900 and oh, say 1990. A few, far-reaching tangents, (Partch, Ives
etal) didn't really hit the mainstream, but there are growing numbers of new
music lovers that are exposed to alternatives early. To wit:
My 16 year old son has been listening to some of the most incredible
garbage for several years. Thrash music from Maryilyn manson, etc. Is it
not wonderful that he be so taken with Neil H'stick's "Acoustic Stick?"
and Bill Sethares' "Xentonality"?
I would never have guessed it, but he is moving out of the common teen
mud music and into some of the most avante-garde stuff I have heard recorded.
Though he is not a musician, there *will* be future composers that arise out
of a harmonic environment that we were, for the most part, denied. This is
progress!
Regards,
Ed Foote
Precision Piano Works
Nashville, Tn.

Charlie Jordan wrote:

>At 01:36 PM 4/23/99 +1000, Dave Keenan wrote:
>
>>I see people (myself included) casting about for a term that covers all
>>tunings based on a single chain of approximate fifths. Thus we get
>>travesties like so-called "negative meantones", and the term "Pythagorean"
>>being used for this purpose in Scala.
>
>>What's wrong with "chain-of-fifths".
>
>It assumes that the repeated interval falls on the fifth note of the scale.
>In China, it is the fourth note. In Joseph Yasser's 19-TET, it is the
>eighth. Frequently but not necessarily, the repeated interval is the
>dominant.

I think that in western writing on alternative tunings, much as we may not
like it, the unqualified term "fifth" has come to mean an approximate 2:3
more strongly than it means the fifth degree of a scale.

>>Anyone have another suggestion?
>
>"_Regular Temperament_ -- A temperament in which all the fifths save one are
>of the same size,

In which all the _whats_ ... are the same size? :-)

>such as the Pythagorean tuning or the meantone
>temperament. (Equal temperament, with all fifths equal, is also a regular
>temperament, and so are the closed systems of multiple division.)"--James
>Murray Barbour, _Tuning and Temperament_, 1951, glossary.
>
>Although one could infer from Barbour's initial choice of words that there
>must be a wolf, his specific inclusion of ET makes that odd interval
>optional.

Thanks Charlie. The definition may be sloppy but I like the term. However,
I worry that it might be taken to apply to tunings generated by repeating
intervals other than an approximate 2:3 (e.g. an approximate 4:5).

>Barbour applies the terms "just" and "meantone" strictly,
>requiring both to have a pure 5:4 major third.

I don't think such a strict definition of meantone is historically tenable.

>Tunings based on repetition of rational fifths such as 112/75, 121/81,
>148/99, and 160/107 are certainly regular. They fall under some definitions
>of just and meantone, but not Barbour's. What is the name of their type?

I don't think the fact that they are rational has any particular
significance. I would classify them according to their size, as meantone,
quasi-Pythagorean, Erlichean, or other.

Regards,
-- Dave Keenan
http://dkeenan.com

At 06:22 AM 4/23/99 EDT, Ed Foote <A440A@aol.com> wrote:

>Charlie writes:
>>Tunings based on repetition of rational fifths such as 112/75, 121/81,
>>148/99, and 160/107 are certainly regular. They fall under some definitions
>>of just and meantone, but not Barbour's. What is the name of their type?
>
>
> At the moment, these tunings are referred to as "microtonal" tunings by
>most of the uneducated, ( or partially exposed........). This is the overall
>view that the public seems to to hold of anything but the status quo.

Your public must be better-read than mine, to even utter such words.

Are we discussing the same tunings? Each of the fractions I mentioned
produces a 12-tone variety comparable to conventional meantones. Who would
consider these microtonal?

112/75 {129 188 318 377 506 565 694 823 883 1012 1071}
121/81 {126 190 316 379 505 569 695 821 884 1010 1074}
148/99 {119 192 312 384 504 577 696 816 888 1008 1081}
160/107 {117 193 310 386 503 579 697 814 890 1007 1083}

--Charlie Jordan <http://www.rev.net/~aloe/music>