Yahoo Tuning Groups Ultimate Backup tuning Some magic meantones

I'm calling a beat ratio "magic" if all of the associated
beat ratios for both major and minor triads are not too
complicated, in the sense of not being too great a multiple of
beat whose frequency divides them all. Below are
24 "magic" q-comma meantones for all the different values of q
which keep things less than or equal to 200 times the base
beat. I give the q for the q-comma, the beat ratio, and the
maximum ratio of the base beat to the others. This is probably not
the best way to evaluate what should be considered suitably magic,
but it seems no matter how you slice it, a lot of historical
temperaments are turning up. This means that while Paul has given
cogent historical evidence against Robert's thesis, there is also this
sort of evidence in its favor. Temperaments which have that
fascinatin' rhythm do keep turning up.

Note also that while the q-values are good only for meantone, the beat
ratios are the same magic ones which would turn up in any other regular temperament. I list first the q-comma value, then the brat,
then my major and minor goodness values (as they stand now.)

q b maj min

0 3/2 3 1

1/7 2 50 144

1/6 9/4 200 24

2/11 5/2 125 45

1/5 3 45 8

2/9 9/2 135 9

1/4 infinity 5 2

5/19 -6 36 160

3/11 -3 135 24

5/18 -9/4 27 45

2/7 -3/2 45 1

5/17 -1 1 75

8/27 -9/10 128 80

10/33 -9/14 56 175

4/13 -1/2 32 12

5/16 -3/8 48 50

1/3 0 5 2

5/14 1/4 8 75

4/11 3/10 125 25

2/5 1/2 25 9

3/7 3/5 125 200

1/2 3/4 200 8

2/3 9/10 125 75

1 1 25 72

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

🔗manuel.op.de.coul@eon-benelux.com

🔗monz <monz@attglobal.net>

hi Gene,

> From: <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 28, 2003 2:27 AM
> Subject: [tuning] Re: Some magic meantones
>
>
> From Monzo's site I find that the following
> magic values have been proposed: 1/6, 1/5, 2/9,
> 1/4, 5/18, 2/7, 1/3, 1/2-comma. Except for 5/18
> these are so simple you can hardly claim it shows
> much support for the idea that synchronized beating
> ever played a role in this business. I note that
> Smith proposed 5/18, and now Smith is proposing
> 5/19-comma. Way to go, Smith! It is a little depressing
> to think no one has ever even had the gumption to
> go for 3/11-meantone, so I hope this list is not
> very complete.

i don't know if 3/14-comma is "magic", but it and
7/26-comma are the only fraction-of-a-comma meantones
in my list that you left out of yours ... was that
simply an oversight, or are those two not "magic"?

in addition, note that "golden meantone", advocated
by Kornerup and Dudon, is nearly identical to 4/15-comma,
and is also audibly indistinguishable from the
Woolhouse/Erlich 7/26-comma.

http://sonic-arts.org/dict/golden.htm

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:
> > I'm calling a beat ratio "magic" if all of the associated
> > beat ratios for both major and minor triads are not too
> > complicated,
>
> are you restricting your attention to close-voiced, root-position
> major and minor triads? if so, why?

Bob:
We can't have everything. That's why tempering exists in the first
place. My approach is to take the simplest, most fundamental
parameters, the ones that imply the most about everything else, and
work with them.

I have trouble buying that close- or spread-voiced matters, since all
that octave displacement does to an interval is invert it and double
or halve the beat rate on that interval. For example, if you use the
first inversion of a close-voiced, root-position triad in which the
m3/M3 beat ratio is 2.0, the major sixth that encompasses this
inversion will beat at exactly the same rate as the major third on
top does. This is very audible in the octave below middle C (C3-C4)
on real, rather inharmonic strings.

I consequently do NOT find it easy to believe that musicians as
sensitive to subtle differences in tuning as those of Zarlino's day
manifestly were, would fail to notice beat synchrony. If you've
played with it for real and not just on paper, it's very much in your
face! And anyone sophisticated enough to describe a temperament
accurately in mathematical terms, is a musician, and has actually
GOTTEN HIS HANDS DIRTY TUNING REAL STRINGS, is not likely to miss
that it's there! It's slow enough and audible enough to use as a
guide in both setting and checking accuracy in meantone temperaments.

At this stage, I have tuned many times on real acoustic strings, with
inharmonicity and all, a lot of different meantones, well
temperaments, and some modified meantones. I have designed six
temperaments that have received such strongly favorable views from
top professional tuners that six of them are going into the next
release of Verituner.

Verituner finally got to where when they heard through the grape vine
that I'd come up with a new temperament, they contacted me asked if
they could include it. I have received emails from professionals, one
who tuned conservatory pianos, who went bananas over my
temperaments...people I've never heard of. I get emails from
professional tuners who say their clients raved about "whatever they
did", and "please do it again", when the client had no idea that my
temperaments were on their instrument. [Many professionals into
alternative temperaments do not mention to clients they're doing
anything different the first time they put a sufficiently mild well
temperament on their piano. :) ]

I assure you, all this hullabaloo is no accident. As far as I'm
concerned, it's the proof of the proverbial pudding from practical
professionals and their clients who have actually tasted it. It
absolutely confirms my own aural experience, and I don't think there
is anyone on this list who is very familiar with me who worries much
about the accuracy of my ear.

Major triads in close-voiced root position with 1.50, and especially
2.00 beat ratios between the thirds sound wonderful in spread
position. Such a 2.0 triad spread out as C3-G3-E4, for example, will
have, for every five beats on the tenth encompassing the chord, ten
(2/1) on the major sixth at the top and one beat on the fifth at the
bottom. This sounds gorgeous; much better than the close-voiced chord
because the beats are slower. (The common harmonics occur lower
in the series, since the spread position imitates their corresponding
order in the harmonic series.)

So why all this concern about potentially invalid projections from
close-voiced position? The mathematics is elementary arithmetic, and
the ear can hear this without any problem. People love the
smooth "tremolo" voice it gives to the voicing of the instrument, and
the harmonic/melodic clarity and color.

The beat synchrony makes the key color of well temperaments more
obvious and more charming. Tuners repeatedly tell me, "That piano
never sounded so good!" We're talking about several different
synchronous temperaments, including my Synchronous ET Equivalent,
where the only significant difference from ET is the synchrony all
around the circle.

This is not me talking theory, dang it! I'm telling you what it
SOUNDS like. Try it, tune it, if you can, and then come back and tell
me it's irrelevant, and that these ancient musicians didn't even
notice it! I don't need any historical documentation to tell that
they did. If they weren't *#*@*&^@!! deaf, they HEARD it! And I bet
they used it like any intelligent tuner trying set those meantones by
ear would eventually end up learning to do even if no one ever
mentioned it to them.

:) Cheers,

Bob

P.S. I've played "flat third" chords (e.g., C-G-E-Bb-EB), "stacked
fourths" (e.g., Bb-E-A-D-G-C, which is harmonically equivalent to an
extended Cm7 with the root at the top and the seventh on the bottom),
etc. in every key on a couple of my milder synchronous wells and they
sound wonderful! They have a harmonic sheen and clarity that is
missing in equal temperament. They glisten, and it is magical. That's
the word, Gene! You hit it. It's plain magical! And there is no
harmony that doesn't work in such a mild, synchronous well.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > i don't know if 3/14-comma is "magic", but it and
> > 7/26-comma are the only fraction-of-a-comma meantones
> > in my list that you left out of yours ... was that
> > simply an oversight, or are those two not "magic"?
>
> "Magic" is a relative term. We have
>
> 3/14-comma brat = 15/4 major [15/4, -10/9, -6/15]
> minor [5/2, 3/10, 4/3]
>
> 7/26-comma brat = -15/4 major [-15/4, 10/21, -14/15]
> minor [-5/2, 7/10, -4/7]
>
> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
> minor [-3, -2/3, -1/2]
>
> Note that if any one of the three, q-comma, major ratios,
> minor ratios, is exact, the other two are approximate.
>
> We see 4/15-comma is good for minor triads and marginal
> for major ones, 3/14-comma is pretty good for minor triads
> and fairly awful for major triads, and 7/26 is marginal for
> minor triads and really stinks for major triads.

bob wendell has been using ratios like 2.08 in his published tunings,
because they're "close enough" to 2. (correct me if i'm wrong bob.)

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> > <gwsmith@s...> wrote:
> > > I'm calling a beat ratio "magic" if all of the associated
> > > beat ratios for both major and minor triads are not too
> > > complicated,
> >
> > are you restricting your attention to close-voiced, root-position
> > major and minor triads? if so, why?
>
> Bob:
> We can't have everything. That's why tempering exists in the first
> place. My approach is to take the simplest, most fundamental
> parameters, the ones that imply the most about everything else, and
> work with them.
>
> I have trouble buying that close- or spread-voiced matters, since
all
> that octave displacement does to an interval is invert it and
double
> or halve the beat rate on that interval. For example, if you use
the
> first inversion of a close-voiced, root-position triad in which the
> m3/M3 beat ratio is 2.0, the major sixth that encompasses this
> inversion will beat at exactly the same rate as the major third on
> top does. This is very audible in the octave below middle C (C3-C4)
> on real, rather inharmonic strings.

right but why not try starting with sixths in the first place and see
what you get? that's all i was trying to say!

> I consequently do NOT find it easy to believe that musicians as
> sensitive to subtle differences in tuning as those of Zarlino's day
> manifestly were, would fail to notice beat synchrony.

manifestly? on what basis do you claim that? here we're talking about
*fractions* of a cent!

> If you've
> played with it for real and not just on paper, it's very much in
your
> face! And anyone sophisticated enough to describe a temperament
> accurately in mathematical terms, is a musician, and has actually
> GOTTEN HIS HANDS DIRTY TUNING REAL STRINGS,

ahem. i've done 1/4-comma meantone before, and if you tell me it can
be done by ear, i'm ready to try 2/7-comma.

> is not likely to miss
> that it's there!

what kind of strings did you use?

> It's slow enough and audible enough to use as a
> guide in both setting and checking accuracy in meantone
>temperaments.

i'd like to see *any* historical mention of this fact from the 16th
century. just one.

> Verituner finally got to where when they heard through the grape
vine
> that I'd come up with a new temperament, they contacted me asked if
> they could include it. I have received emails from professionals,
one
> who tuned conservatory pianos, who went bananas over my
> temperaments...people I've never heard of. I get emails from
> professional tuners who say their clients raved about "whatever
they
> did", and "please do it again", when the client had no idea that my
> temperaments were on their instrument. [Many professionals into
> alternative temperaments do not mention to clients they're doing
> anything different the first time they put a sufficiently mild well
> temperament on their piano. :) ]

congratulations, and i raise a glass in celebration of your great
discoveries and successes!

> and that these ancient musicians didn't even
> notice it!

can you tune a harpsichord to this tuning by ear? a clavichord? with
no electronic tuning devices?

> I don't need any historical documentation to tell that
> they did. If they weren't *#*@*&^@!! deaf, they HEARD it!

zarlino and his cohorts (such as . . .?) wrote *way too much* about
their tunings to ever let such an observation go unmentioned. your
personal successes with pianos are deserving of much celebration, but
you should make sure the historical claims you're making fit the
documented historical facts.

> And I bet
> they used it like any intelligent tuner trying set those meantones
by
> ear would eventually end up learning to do even if no one ever
> mentioned it to them.

how would they eventually end up learning to do so? convince me. i've
got my tuning wrench in hand. what do i do, what do i listen for --
don't tell me *too* much, since no one has ever mentioned it to me :)

> in every key on a couple of my milder synchronous wells and they
> sound wonderful! They have a harmonic sheen and clarity that is
> missing in equal temperament. They glisten, and it is magical.
That's
> the word, Gene! You hit it. It's plain magical! And there is no
> harmony that doesn't work in such a mild, synchronous well.

that's awesome, bob!

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

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

> > are you restricting your attention to close-voiced, root-position
> > major and minor triads? if so, why?
>
> Bob:
> We can't have everything. That's why tempering exists in the first
> place.

Extending this business to higher prime limits presents real
problems, but octave relationships are another matter. I think
magicality might be evaluated something like the way we do
for octave-class relationships, where over on tuning-math we
might speak of the Tenney height of the odd part of a ratio.
Calling zero and infinity height one, we could extend the idea
beyond the positive rational numbers.

> I consequently do NOT find it easy to believe that musicians as
> sensitive to subtle differences in tuning as those of Zarlino's day
> manifestly were, would fail to notice beat synchrony.

It seems to me for 2/7-comma meantone, this might have been
particularly noticable for minor triads. It would be interesting
to hear from anyone else who might know what they think of the idea
that Zarlino might have picked on 2/7-comma and 1/4-comma because
they sound good.

If you've
> played with it for real and not just on paper, it's very much in your
> face! And anyone sophisticated enough to describe a temperament
> accurately in mathematical terms, is a musician, and has actually
> GOTTEN HIS HANDS DIRTY TUNING REAL STRINGS, is not likely to miss
> that it's there! It's slow enough and audible enough to use as a
> guide in both setting and checking accuracy in meantone temperaments.

Have you done much work with 2/7-comma meantone?

> At this stage, I have tuned many times on real acoustic strings, with
> inharmonicity and all, a lot of different meantones, well
> temperaments, and some modified meantones. I have designed six
> temperaments that have received such strongly favorable views from
> top professional tuners that six of them are going into the next
> release of Verituner.
>
> Verituner finally got to where when they heard through the grape vine
> that I'd come up with a new temperament, they contacted me asked if
> they could include it. I have received emails from professionals, one
> who tuned conservatory pianos, who went bananas over my
> temperaments...people I've never heard of. I get emails from
> professional tuners who say their clients raved about "whatever they
> did", and "please do it again", when the client had no idea that my
> temperaments were on their instrument. [Many professionals into
> alternative temperaments do not mention to clients they're doing
> anything different the first time they put a sufficiently mild well
> temperament on their piano. :) ]
>
> I assure you, all this hullabaloo is no accident. As far as I'm
> concerned, it's the proof of the proverbial pudding from practical
> professionals and their clients who have actually tasted it. It
> absolutely confirms my own aural experience, and I don't think there
> is anyone on this list who is very familiar with me who worries much
> about the accuracy of my ear.
>
> Major triads in close-voiced root position with 1.50, and especially
> 2.00 beat ratios between the thirds sound wonderful in spread
> position. Such a 2.0 triad spread out as C3-G3-E4, for example, will
> have, for every five beats on the tenth encompassing the chord, ten
> (2/1) on the major sixth at the top and one beat on the fifth at the
> bottom. This sounds gorgeous; much better than the close-voiced chord
> because the beats are slower.

For really slow beats, note that this whole business is not
confined to meantone!

> That's
> the word, Gene! You hit it. It's plain magical!

Thanks. When I get my computer problems strightened out, I will
definately be working on the practice side of this, as well as the
theory. It's certainly one of the most interesting tuning ideas I've
heard, whether it checks with history or not.

🔗monz <monz@attglobal.net>

someone correct me if i'm wrong ...

wasn't this equal-beating business (along with
various other numerological stuff) the criteria
upon which Herbert Anton Kellner based his
reconstruction of Bach's supposed preferred tuning
for his _Well-tempered Klavier_?

-monz

----- Original Message -----
From: <gwsmith@svpal.org>
To: <tuning@yahoogroups.com>
Sent: Friday, February 28, 2003 11:31 PM
Subject: [tuning] Re: Some magic meantones

> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
>
> > > are you restricting your attention to close-voiced, root-position
> > > major and minor triads? if so, why?
> >
> > Bob:
> > We can't have everything. That's why tempering exists in the first
> > place.
>
> Extending this business to higher prime limits presents real
> problems, but octave relationships are another matter. I think
> magicality might be evaluated something like the way we do
> for octave-class relationships, where over on tuning-math we
> might speak of the Tenney height of the odd part of a ratio.
> Calling zero and infinity height one, we could extend the idea
> beyond the positive rational numbers.
>
> > I consequently do NOT find it easy to believe that musicians as
> > sensitive to subtle differences in tuning as those of Zarlino's day
> > manifestly were, would fail to notice beat synchrony.
>
> It seems to me for 2/7-comma meantone, this might have been
> particularly noticable for minor triads. It would be interesting
> to hear from anyone else who might know what they think of the idea
> that Zarlino might have picked on 2/7-comma and 1/4-comma because
> they sound good.
>
>
> If you've
> > played with it for real and not just on paper, it's very much in your
> > face! And anyone sophisticated enough to describe a temperament
> > accurately in mathematical terms, is a musician, and has actually
> > GOTTEN HIS HANDS DIRTY TUNING REAL STRINGS, is not likely to miss
> > that it's there! It's slow enough and audible enough to use as a
> > guide in both setting and checking accuracy in meantone temperaments.
>
> Have you done much work with 2/7-comma meantone?
>
> > At this stage, I have tuned many times on real acoustic strings, with
> > inharmonicity and all, a lot of different meantones, well
> > temperaments, and some modified meantones. I have designed six
> > temperaments that have received such strongly favorable views from
> > top professional tuners that six of them are going into the next
> > release of Verituner.
> >
> > Verituner finally got to where when they heard through the grape vine
> > that I'd come up with a new temperament, they contacted me asked if
> > they could include it. I have received emails from professionals, one
> > who tuned conservatory pianos, who went bananas over my
> > temperaments...people I've never heard of. I get emails from
> > professional tuners who say their clients raved about "whatever they
> > did", and "please do it again", when the client had no idea that my
> > temperaments were on their instrument. [Many professionals into
> > alternative temperaments do not mention to clients they're doing
> > anything different the first time they put a sufficiently mild well
> > temperament on their piano. :) ]
> >
> > I assure you, all this hullabaloo is no accident. As far as I'm
> > concerned, it's the proof of the proverbial pudding from practical
> > professionals and their clients who have actually tasted it. It
> > absolutely confirms my own aural experience, and I don't think there
> > is anyone on this list who is very familiar with me who worries much
> > about the accuracy of my ear.
> >
> > Major triads in close-voiced root position with 1.50, and especially
> > 2.00 beat ratios between the thirds sound wonderful in spread
> > position. Such a 2.0 triad spread out as C3-G3-E4, for example, will
> > have, for every five beats on the tenth encompassing the chord, ten
> > (2/1) on the major sixth at the top and one beat on the fifth at the
> > bottom. This sounds gorgeous; much better than the close-voiced chord
> > because the beats are slower.
>
> For really slow beats, note that this whole business is not
> confined to meantone!
>
> > That's
> > the word, Gene! You hit it. It's plain magical!
>
> Thanks. When I get my computer problems strightened out, I will
> definately be working on the practice side of this, as well as the
> theory. It's certainly one of the most interesting tuning ideas I've
> heard, whether it checks with history or not.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:
> >
> > "Magic" is a relative term.

--- In tuning@yahoogroups.com, "wallyesterpaulrus
> bob wendell has been using ratios like 2.08 in his published
> tunings, because they're "close enough" to 2. (correct me if i'm
> wrong bob.)

Yes, you're right, but my Natural Synchronous Well has no compromised
ratios at all, unless you want to count the ratio in the C major
triad going from 2.00000.... to 2.0016 because I put the spare change
there (0.0058 cents) left over for the Pythagorean comma.

However, it isn't possible, I have pretty much concluded, to have
any other tuning in a cycle of 12 fifths limited to perfect 1.50 and
2.00 beat ratios. That's why I call it the Natural, because it just
works out that way. I didn't make it happen that way (no choice
involved; just a discovery). It's simply a mathematically beautiful
thing that it does.

Anything else is going to require some compromised ratios. Since the
whole idea of synchronized beating is to compensate that there is
inevitable beating in the first place (i.e., tempering
or "necessarily compromised tuning"). The most heavily mistuned
triads are the ones most in need of precisely synchronized beats.
Where the beats are slow and therefore relatively unimportant to the
ear, we can compromise the beat ratios most and get away with it.

So yes, 2.00 +-0.08 is the high side of my tolerance (the Wendell
Very Mild Synchronous Well) so far except on C, where I have
compromised with 1.86 versus 2.0. However, the C major is the
cleanest, nearest just of all the triads. It sounds lovely at +7.9
cents with a 1.86 beat ratio (m3/M3). This is the most conservative
of all my temperament designs, deliberately aimed at tuners who wish
to try an alternative to ET without feeling any need to advise their
clients. I have received great feedback on this from two top members
of the Piano Technicians Guild, one of whom has several temperaments
with his name on them.

So this very mild well uses a deliberate "tempering" of the beat
ratios in the opposite direction from the tempering of the
intonation. That is, the perfect ratios are in the most remote keys
where the tuning is most compromised and the most heavily compromised
ratios are on the side of the circle nearest C where the intonation
is purest.

I have noticed empirically that 2.0 offers the most room for
variation before it ceases to sound synchronous, since it is the
simplest of all the ratios except 1/1. Further, the slower the beat
rate, the less obvious any variation from 2.0 becomes. If you think a
second, you will realize that 2.08 implies 20.8 beats for every ten
instead of 20. When the beat rate is slow, this is not very
noticeble.

I find it interesting to observe that this phenomenon is quite the
opposite of what we find in intonation. The conceptual equivalents of
1/1 and 2/1 beat ratios, unisons and octaves, are the least tolerant
of any mistuning. In the case of beat synchrony, however, we are not
dealing with interference among the strongest, lowest-order common
harmonics or, in the case of unisons, directly between common
fundamentals, as we are in intonation. We are simply dealing with
rhythmic perception, and at a fairly subtle perceptual level to boot.

Cheers,

Bob

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

> --- In tuning@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
>
> > bob isn't really using it for meantone -- his contributions to
> > Verituner are all well-temperaments, i believe . . .

All but one, which is a modified meantone with the widest major third
on Db/C# at +21.5 cents, so if you don't mind fully Pythagorean
thirds, it doubles as a well. That was not intention, however. I
wanted a relatively homogeneous tuning from Bb major to A major (a
range of keys that encompass the triads on Eb and E), but wanted to
add a triad to the traditional 8 meantones both ends for a total of 8
good major and 5 good minor keys, versus 6 and 3 respectively for
straight, totally homogeneous meantones.

So mine began with 1/7-comma meantone as a basis, since it has 2.0
beat ratios for the m3/M3 of all playable major triads. I bent it at
the ends so that at Bb and Eb we'e just nipping the underside of 12-
tET and then it swoops sharply upward to max width for the third on
C#. The result is much nicer than ET in the six traditional meantone
keys, but still quite playable in the extensions to Eb major and E
major at opposite ends of the traditional range.

The purpose was to provide a modern practical performance option that
would work for almost any music originally written for meantone
without any need to retune or transpose, and yet remain much more
historically informed than it would be in ET.

>
> For which part of the circle of fifths is 1/7-comma meantone.
>
Having 2.0 beat ratios all the way around has nothing to do with 1/7-
comma meantone. It has everything to do with the fifth being 1/3 as
flat as the major third on the same pitch is sharp. Only that and
always that will result in a 2.0 beat ratio on the resulting major
triad. 1/7-comma does that, but in well temperament you can do that
on all the tempered fifths if you choose them as I did for the
Natural Synchronous Well.

The triad on G is closest to 1/7-comma at -3.27 cents on the fifth
(versus -3.07 = 1/7th syntonic comma), but the fifths range from 1/4-
comma to almost 1/9-comma and none are excatly 1/7-comma. They all
have fifths 1/3 as flat as the thirds are sharp, though. That is the
only necessary and also sufficient condition.

> but it does seem
> > that even beat ratios like 5:1 could be important . . . and bob
does
> > seem willing to use "tempered" beat ratios in this context . . .
so
> > in further research on this, why don't we use a harmonic entropy
> > function designed to mirror tenney height up to n*d = 6 or so,
but
> > which is continuous everywhere?
>
Even a 3.0 beat ratio of the minor to major third in a major triad,
which I use in my Bold Synchronous Well, uses up so much of the
available real estate in the requisite total of one Pythagorean comma
that you can only us it once in the boldest well. The fifth has to be
flat by the same amount as the third is sharp to get 3.0. If you
don't want a horribly flat fifth, that means you have to use up most
of your total Pythagorean real estate in four consecutive fifths.