12

Is it accurate to say that 12-tet lies sort of in between meantone (better
3rds) and Pythagorean (better 5ths), and that's another reason for its
popularity.

Or am I getting it wrong, and meantones themselves are in between
better 3rds, and better 5ths?

i have seen it this way also

Christopher Bailey wrote:

>Is it accurate to say that 12-tet lies sort of in between meantone (better >3rds) and Pythagorean (better 5ths), and that's another reason for its >popularity.
>
>Or am I getting it wrong, and meantones themselves are in between >better 3rds, and better 5ths?
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> >Yahoo! Groups Links
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--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In MakeMicroMusic@yahoogroups.com, Christopher Bailey
<chris@m...> wrote:
>
>
> Is it accurate to say that 12-tet lies sort of in between
> meantone (better 3rds) and Pythagorean (better 5ths), and
> that's another reason for its popularity.
>
> Or am I getting it wrong, and meantones themselves are
> in between better 3rds, and better 5ths?

12-ET is a member of both the pythagorean and meantone
temperament families.

You're certainly on the right track: the fact that 12-edo
belongs to both of those historically very important
tuning families is most definitely one of the main
reasons for its popularity, at least in Western music.

Another main reason for its popularity is that it is
by far the smallest-cardinality tuning which gives
a decent approximation of the entire interval set of
most of the West's historically important JI tunings,
including pythagorean and a lot of 5-limit schemes.

Low cardinality equates to easier playability and
presumably to easier notation.

I also speculate that 12-edo has been around for a
very long time, possibly originating with the Sumerians
c.3000 BC.

Various webpages of mine which touch on these topics:

http://tonalsoft.com/enc/number/12edo.aspx

http://tonalsoft.com/enc/f/family.aspx
http://tonalsoft.com/enc/e/equal-temperament.aspx
http://tonalsoft.com/enc/b/bingo.aspx

http://tonalsoft.com/monzo/sumerian/sumerian-tuning.htm
http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

Paul Erlich's "zoom" graphics on the "equal-temperament"
page show vividly to which families 12-edo belongs in
the 5-limit.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Chris,

> --- In MakeMicroMusic@yahoogroups.com, Christopher Bailey
> <chris@m...> wrote:

> Or am I getting it wrong, and meantones themselves are
> in between better 3rds, and better 5ths?

Sorry, i also meant to point you to some of my webpages
which address this.

http://tonalsoft.com/enc/m/meantone.aspx
http://tonalsoft.com/enc/m/major-3rd.aspx
http://tonalsoft.com/enc/p/perfect-5th.aspx

(i see that i never got around to making the meantone
table and graphic for the minor-3rd ... oh well, someday ...)

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hello Chris,

This question is really better suited for the tuning list. I'll reply
here, but anything further should go elsewhere, as we are getting a
bit removed from the making of microtonal music (which I love) with
these questions (which I love too).

--- In MakeMicroMusic@yahoogroups.com, "monz" <monz@t...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, Christopher Bailey
> <chris@m...> wrote:
> >
> >
> > Is it accurate to say that 12-tet lies sort of in between
> > meantone (better 3rds) and Pythagorean (better 5ths), and
> > that's another reason for its popularity.
> >
> > Or am I getting it wrong, and meantones themselves are
> > in between better 3rds,

I'll take this to mean both major and minor thirds (and thus their
inversions, minor and major sixths).

> and better 5ths?

That's more like it. Most real-life meantones are optimal for some
degree of weighting the 5ths *more* than the 3rds; most others
correspond with weighting both about equally. But the earliest form
of meantone ever precisely specified, 2/7-comma meantone (Zarlino,
1577 or so), optimizes the thirds only, letting the fifths fall where
they may (which turns out to be ~695.8 cents) . . . perhaps as a
reaction to the medieval preponderance of fifths and fourths and a
desire for the *new* consonances, thirds and sixths, to be as pure as
possible given that 81:80 is to vanish.

> 12-ET is a member of both the pythagorean and meantone
> temperament families.

There's no such thing as the pythagorean temperament family.
Pythagorean is a form of JI, not temperament.

> Paul Erlich's "zoom" graphics on the "equal-temperament"
> page show vividly to which families 12-edo belongs in
> the 5-limit.

And Pythagorean isn't one of them, since Pythagorean is 3-limit JI,
not a 5-limit temperament. "Aristoxenean" on your chart,
and "Compton" in my paper, both refer to the 5-limit system where the
Pythagorean comma is tempered out; normally this consists of multiple
12-equal chains, but 12-equal itself is a sort of degenerate example
of this family.

I think it's fair to assume Christopher meant Pythagorean tuning per
se, i.e. 3-limit JI.

And again, I also think anything further on this should be posted to
another list . . .

Hi Paul,

--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In MakeMicroMusic@yahoogroups.com, "monz" <monz@t...> wrote:

> > 12-ET is a member of both the pythagorean and meantone
> > temperament families.
>
> There's no such thing as the pythagorean temperament family.
> Pythagorean is a form of JI, not temperament.

Right ... i meant "Aristoxenean". sorry.

I missed the switch to "Compton" ... you can tell me about
that on the tuning list, or privately when we work out the
Encyclopedia error corrections.

-monz