an introduction to meta meantone

A basic criteria of meantone is that 4 fifths will be equal to a third.
In order for this relationship to be a proportional triad, the
difference tones between the upper and lower tones in relation to the
middle tone must be the same. If we look at 4-5-6 we notice that the
difference tones are one. Now if we continue to feed a recurrent
sequence with the formula of meta-meantone (expressed in simplest terms
(A+B)x2=E we converge upon a certain number. This formula says that A
which we could call the root plus B otherwise known as the "fifth" is
equal to E the third. Why E? Because in the chain, E is four fifths
above. Now it is possible to seed this formula how one wishes. In Erv's
example on page 4 he starts 1 2.5 3 5 7 11 but it might be
clearer if we start with 16 24 36 54 80 as A, B, C, D, E. we see that
(16+ 24)x2=80 so next we treat 24 as A and then 36 as B which generates
120 and so on. Going back to Page 1 we see the series carried out beyond
that on page 4 . Already with the last few numbers we can see that it
closing in on converging on the stated ratio. Which comes out BTW to be
695.6304373 cents and the triads will be proportional and equal beating.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> A basic criteria of meantone is that 4 fifths will be equal to a
third.
> In order for this relationship to be a proportional triad, the
> difference tones between the upper and lower tones in relation to
the
> middle tone must be the same. If we look at 4-5-6 we notice that the
> difference tones are one. Now if we continue to feed a recurrent
> sequence with the formula of meta-meantone (expressed in simplest
terms
> (A+B)x2=E we converge upon a certain number. This formula says that
A
> which we could call the root plus B otherwise known as the "fifth"
is
> equal to E the third. Why E? Because in the chain, E is four fifths
> above. Now it is possible to seed this formula how one wishes. In
Erv's
> example on page 4 he starts 1 2.5 3 5 7 11 but it might
be
> clearer if we start with 16 24 36 54 80 as A, B, C, D, E. we see
that
> (16+ 24)x2=80 so next we treat 24 as A and then 36 as B which
generates
> 120 and so on. Going back to Page 1 we see the series carried out
beyond
> that on page 4 . Already with the last few numbers we can see that
it
> closing in on converging on the stated ratio. Which comes out BTW
to be
> 695.6304373 cents and the triads will be proportional and equal
beating.

thanks for the analysis, kraig! i'm sure this will very valuable. i
wonder how would gene and robert react to this tuning?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> > A basic criteria of meantone is that 4 fifths will be equal to a
> third.
> > In order for this relationship to be a proportional triad, the
> > difference tones between the upper and lower tones in relation to
> the
> > middle tone must be the same. If we look at 4-5-6 we notice that
the
> > difference tones are one. Now if we continue to feed a recurrent
> > sequence with the formula of meta-meantone (expressed in simplest
> terms
> > (A+B)x2=E we converge upon a certain number. This formula says
that
> A
> > which we could call the root plus B otherwise known as
the "fifth"
> is
> > equal to E the third. Why E? Because in the chain, E is four
fifths
> > above. Now it is possible to seed this formula how one wishes. In
> Erv's
> > example on page 4 he starts 1 2.5 3 5 7 11 but it
might
> be
> > clearer if we start with 16 24 36 54 80 as A, B, C, D, E. we see
> that
> > (16+ 24)x2=80 so next we treat 24 as A and then 36 as B which
> generates
> > 120 and so on. Going back to Page 1 we see the series carried out
> beyond
> > that on page 4 . Already with the last few numbers we can see
that
> it
> > closing in on converging on the stated ratio. Which comes out BTW
> to be
> > 695.6304373 cents and the triads will be proportional and equal
> beating.
>
> thanks for the analysis, kraig! i'm sure this will very valuable. i
> wonder how would gene and robert react to this tuning?

Bob:
I don't get this at all. Must be missing something. I just see powers
of 1.5 multiplied by 16 in the above series and it doesn't converge
toward anything. ???

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> thanks for the analysis, kraig! i'm sure this will very valuable. i
> wonder how would gene and robert react to this tuning?

It looks quite interesting, but I'm still trying to sort out my
miscalculations wrt stuff like Wendell Well. :(

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:

[ . . . the generator of meta-meantone is]
> > > 695.6304373 cents and the triads will be proportional and equal
> > > beating.
> >
> > thanks for the analysis, kraig! i'm sure this will very valuable.
i
> > wonder how would gene and robert react to this tuning?
>
> Bob:
> I don't get this at all. Must be missing something. I just see
powers
> of 1.5 multiplied by 16 in the above series and it doesn't converge
> toward anything. ???

well, bob, leaving aside the derivation for the moment, i actually
did a calculation and found that the beat rate ratio you have been
focusing on, that of the minor third and the major third in the root-
position close-voiced major triad, is exactly 1/1 in this "meta-
meantone" tuning that kraig describes above.

meanwhile, in 2/7-comma meantone, it's about 3/2, or more precisely,
it's 1.49734.

hmm . . .

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > thanks for the analysis, kraig! i'm sure this will very valuable.
i
> > wonder how would gene and robert react to this tuning?
>
> It looks quite interesting, but I'm still trying to sort out my
> miscalculations wrt stuff like Wendell Well. :(

well, i wish you good luck, and godspeed, with the sorting-out!

but it appears that wilson's meta-meantone, at least as described by
kraig here, is the same thing as what you've been referring to,
approximately (though still quite accurately), as 5/17-comma meantone
on the tuning-math list.

as i recall, you were hoping to take credit as the inventor of this
impressively beating-synchronized meantone tuning, but wilson seems
to have beat you to it.

what year did wilson come up with it? i think it should be added to
monz's list of historical meantones, given its importance with
respect to the criteria bob and yourself have been considering . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> well, i wish you good luck, and godspeed, with the sorting-out!

I'm going to start over from the ground up, and hopefully get it right
this time.

> but it appears that wilson's meta-meantone, at least as described by
> kraig here, is the same thing as what you've been referring to,
> approximately (though still quite accurately), as 5/17-comma meantone
> on the tuning-math list.
>
> as i recall, you were hoping to take credit as the inventor of this
> impressively beating-synchronized meantone tuning, but wilson seems
> to have beat you to it.

Actually, all I said about it was that it bears investigation, and Erv
certainly seemes to have done that. You are probably thinking of
5/19 comma meantone, by which I really meant the whole 81-et,
brat = -6, minor brat = -4 complex.

For any meantone with fifth f, the brat will be

b = f(3f^3 - 10) / 2(f^4 - 5)

so that f satisfies the 4th degree polynomial

f(3f^3 - 4) - 2b(f^4 - 5) = 0

If b = -1, then this gives us

f(3f^4-4)+2(f^4-5) = 5(f^4 - 2f - 2) = 0

from whence the characteristic polynomial of Erv's recurrence stems.
We get a nearby fifth by taking the minor brat c = b/f to be exactly
-2/3, with polynomial 4f^4 + 9f^3 -50, for which, however, the ratios
do not coverge.

By way of comparison, brat=-6 meantone, or about 5/19-comma, has a
recurrence relationship given by x^4 - 2/3x - 4, brat=-3/2 meantone,
or about 2/7-comma, has x^4 - 5/3x - 5/2. The rations for these *do*
converge, but the recurrence relationships we get from them do not
have integer terms; the polynomials are not "monic".

The four brats which do give monic characteristic polynomials are
b=-1, 1, 2, and 4; of these only Erv's b=-1 has ratios which converge.
It's beginning to look rather special.

>> well, i wish you good luck, and godspeed, with the sorting-out!
>
>I'm going to start over from the ground up, and hopefully get it
>right this time.

Where did you go wrong? Did I miss a post?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> well, i wish you good luck, and godspeed, with the sorting-out!
> >
> >I'm going to start over from the ground up, and hopefully get it
> >right this time.
>
> Where did you go wrong? Did I miss a post?

I thought I'd set things up so that the product of the fifths came out
to 128, giving octave equivalence, but somehow this equation got lost
in the shuffle. Incidentally, something I haven't mentioned is that we
can find lots of systems where the octave is not exactly 2, but this
of course screws up the equal-beating aspect for other versions of the
triad.

>I thought I'd set things up so that the product of the fifths came out
>to 128, giving octave equivalence, but somehow this equation got lost
>in the shuffle. Incidentally, something I haven't mentioned is that we
>can find lots of systems where the octave is not exactly 2, but this
>of course screws up the equal-beating aspect for other versions of the
>triad.

Thanks.

-Carl

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
wrote:
> > I don't get this at all. Must be missing something. I just see
> powers
> > of 1.5 multiplied by 16 in the above series and it doesn't
converge
> > toward anything. ???
>
Paul:
> well, bob, leaving aside the derivation for the moment, i actually
> did a calculation and found that the beat rate ratio you have been
> focusing on, that of the minor third and the major third in the
root-
> position close-voiced major triad, is exactly 1/1 in this "meta-
> meantone" tuning that kraig describes above.
>
> meanwhile, in 2/7-comma meantone, it's about 3/2, or more
precisely,
> it's 1.49734.
>
> hmm . . .

Bob:
So does this have any implications for a 12-tone well temperament
that is mild enough to work with any music?

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
> wrote:
> > > I don't get this at all. Must be missing something. I just see
> > powers
> > > of 1.5 multiplied by 16 in the above series and it doesn't
> converge
> > > toward anything. ???
> >
> Paul:
> > well, bob, leaving aside the derivation for the moment, i
actually
> > did a calculation and found that the beat rate ratio you have
been
> > focusing on, that of the minor third and the major third in the
> root-
> > position close-voiced major triad, is exactly 1/1 in this "meta-
> > meantone" tuning that kraig describes above.
> >
> > meanwhile, in 2/7-comma meantone, it's about 3/2, or more
> precisely,
> > it's 1.49734.
> >
> > hmm . . .
>
> Bob:
> So does this have any implications for a 12-tone well temperament

no.

> that is mild enough to work with any music?

any music? are you suggesting that 12-tone well temperaments work for
*any* music??

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> any music? are you suggesting that 12-tone well temperaments work
for
> *any* music??

Bob:
Well, if by "any music" you don't mean 19-limit just intonation or
some other kind of microtonal composition, YES! At least a mild well
temperament, one like my Very Mild Synchronous, or the Moore from the
19th century, will work for anything as well or better than ET will.
I don't mean to imply that a performance of Bach is
100% "historically informed" if it is in some temperament other than
what he used (apparently Werkmiester III), but it will be more
historically informed by far in my synchonous wells than in equal
temperament.

After all, my wells parallel quite plainly the general pattern of key
color in Werkmeister III, as do a huge number of historical
temperaments after Werkmeister. Also, the advantage of synchronicity
is two-fold in this context:

1) It sounds better in tune (less harsh) in the remote keys than it
would without beat synchrony

2) At the same time, the synchrony ironically makes the key color
more obvious even in very mild temperaments, since the "singing"
quality the vibrato-like synchrony lends to the voicing also
manifests as an increasing harmonic/melodic intensity as the beat
rate increases and leading tones narrow in more remote keys.

Also, there is no twentieth-century music, jazz, broadway, Bartok or
anything else, that doesn't sound just fine, even enhanced, in this
temperament. I tuned a friend's piano with my mild synchronous well
and he felt Chopin sounded "way better" in this temperament than in
ET. This is for the thirds on everything but C falling in a range
from +9.7 to +17.2 cents (-4.0 to +3.5 cents deviation from ET), and
the C is at +7.9. This is quite a mild temperament, and sounds even
smoother and milder owing to the pleasant beat synchrony.

Please bear in mind that the average piano tuner tunes "reverse
wells". This means they tend to tune the fifths near C too cleanly
and then iterate backwards to patch up the temperament setting and
absorb the cumulative error. This places the MOST TEMPERED FIFTHS IN
THE MOST REMOTE KEYS AND THE LEAST TEMPERED FIFTHS IN THE KEYS NEAR
C. This is what Bill Bremmer calls a "reverse well".

Even if the errors are random, the typical tuning errors made by
average, mediocre tuning professionals lie OUTSIDE THE RANGE of my
mildest temperament mentioned above. So no one has to even
tell a client he's tuning my temperament instead of ET if he wishes
to avoid a clash. There is a lot of conceptual resistance, as you
must know, to anything other than ET, even if they can't hear diddly
and would never know the difference if left to judge by ear.

So one strategy some leading-edge tuners are now implementing is to
tune something mild without informing the client and then check the
reaction. This was the impetus underlying the development of my
mildest well. One top tuner kept pushing me for a highly synchronous
well he could use this way. I'm getting excellent feedback from
several top professionals that clients love the tuning. Once you've
won them over, you can then take your time and gradually inform them
about the alternatives, and if they resist, go back to ET. They will
generally object, and say it sounds dead and flat. Thumbs UP! You've
won the day.

This is real feedback from people who are doing it. I am keenly
interested in microtonal music, but I'm also interesting in
influencing the mainstream, common-practice musicians to think more
about new tonal possibilities. This is part of my peaceful sneak
attack. :)

Cheers,

Bob