Yahoo Tuning Groups Ultimate Backup tuning Monzo lattice comparing 1/6-c MT and 5-limit JI

Everyone just got an automatic announcement... here's the personalized
version.

This is possibly the greatest lattice diagram I've done so far:
a comparison of 1/6-comma meantone (= ~55-EDO) with the 5-limit
JI pitches that are best implied by it.

/tuning/files/monz/6th-cmt-lattice.gif

I've also just added this lattice to my webpage on 55-EDO:

http://www.ixpres.com/interval/monzo/55edo/55edo.htm

The commentary introducing it on the webpage:

>> Below is a lattice illustrating the relationship of
>> 1/6-comma meantone with the 5-limit JI pitch-classes
>> it implies. (This particular example illustrates a
>> symmetrical 27-tone chain of 1/6-comma meantone "5th"s;
>> it could be extended in either direction.)

I know the text written on the diagram is difficult to read
... sorry about that. Here's the explanatory comment at the
bottom of the diagram:

>> - Along the meantone chain, +/-x indicates a ratio of 3^(x/3) * 5^(x/6).
>>
>> - Dotted lines connect the meantone pitch-classes with their
>> closest JI implied ratios on the 5-limit lattice.
>>
>> - The meantone pitch-class's position on the dotted line shows the
>> fraction-of-a-comma deviation [from the implied JI ratio].

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗monz <joemonz@yahoo.com>

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 2:15 PM
> Subject: [tuning] Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> This is possibly the greatest lattice diagram I've done so far:
> a comparison of 1/6-comma meantone (= ~55-EDO) with the 5-limit
> JI pitches that are best implied by it.
>
> /tuning/files/monz/6th-cmt-lattice.gif

So, the point of these meantone lattices has finally
become clear to me: they show that each fraction-of-a-comma
meantone has a particular flavor associated with it, because
each one also has a unique set of 5-limit JI rational implications
associated with it, and these rational implications follow
a general linear trend across the 2-dimensional 5-limit
ratio-space plane of the lattice, with the meantone itself
at the center.

So obviously, my theory is that each type of meantone has
a certain sound/feeling, which can be modeled as the set of
implied 5-limit ratios and their general trend.

Feedback sought.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> So, the point of these meantone lattices has finally
> become clear to me: they show that each fraction-of-a-comma
> meantone has a particular flavor associated with it, because
> each one also has a unique set of 5-limit JI rational implications
> associated with it,

To me, they all have _exactly the same_ set of 5-limit JI rational
implications associated with them.

> and these rational implications follow
> a general linear trend across the 2-dimensional 5-limit
> ratio-space plane of the lattice, with the meantone itself
> at the center.

Try doing this for LucyTuning or Golden Meantone or 55-tET and you'll
soon see the weakness of this approach.
>
> So obviously, my theory is that each type of meantone has
> a certain sound/feeling, which can be modeled as the set of
> implied 5-limit ratios and their general trend.

To me, the implied 5-limit ratios are exactly the same from one
meantone to another, and they each have a different sound/feeling
because of how they distribute the syntonic comma error differently
among the consonances.

> Feedback sought.

You sought it, you got it!

🔗monz <joemonz@yahoo.com>

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 2:15 PM
> Subject: [tuning] Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
>
> Everyone just got an automatic announcement... here's the personalized
> version.
>
> This is possibly the greatest lattice diagram I've done so far:
> a comparison of 1/6-comma meantone (= ~55-EDO) with the 5-limit
> JI pitches that are best implied by it.
>
> /tuning/files/monz/6th-cmt-lattice.gif
>
> ...
>
> Here's the explanatory comment at the bottom of the diagram:
>
> >> - Along the meantone chain, +/-x indicates a ratio of 3^(x/3) *
5^(x/6).
> >>
> >> - Dotted lines connect the meantone pitch-classes with their
> >> closest JI implied ratios on the 5-limit lattice.
> >>
> >> - The meantone pitch-class's position on the dotted line shows the
> >> fraction-of-a-comma deviation [from the implied JI ratio].

I thought some readers would appreciate a little more explanation.

First, the JI lattice plots the meantone pitch-classes precisely
according to their fractional prime-factorization. Thus, with
C = n^0 at the center as a reference, it can be easily seen that

G = 3^(1/3) * 5^(1/6),
D = 3^(2/3) * 5^(1/3),
A = 3^1 * 5^(1/2), etc.

To check the accuracy of this, use vector subtraction to
deduct 1/6 of a syntonic comma (comma = 81:80 ratio =
3^4 * 5^-1 in "8ve"-equivalent prime-factor notation):

3^(6/6) * 5^(0/6) = 3:2 ratio = 3^1
- 3^(4/6) * 5^(-1/6) = 1/6 of a syntonic comma
--------------------
3^(2/6) * 5^(1/6) = 1/6-comma meantone "5th"

Compare this result with the value for "G" plotted on the lattice.

Secondly, the dotted lines connecting the meantone pitch-classes
with the JI implied ratios, follow the axis of the syntonic comma.
Thus, it can be easily seen that

meantone G is 1/6-comma higher than 3:2,
meantone D is 1/3-comma higher than 9:8,
meantone A is 1/2-comma higher than 27:16
and 1/2-comma lower than 5/3, etc.

I include both JI ratios when the meantone pitches is exactly
midway between two of them (as with "A"), otherwise only the
closest approximated ratio is shown.

So one might facetiously, but more accurately, call 1/6-comma meantone
tuning "mean-6th tuning".

(A similar diagram for 1/4-comma meantone, whose linear trend
runs identical to the 5-axis, shows the meantone "D" plotted
exactly midway between the 9:8 and 10:9 ratios along the
syntonic-comma-axis, thus, literally, "mean-tone".)

(I plan to work up a whole set of these diagrams in a Java applet,
similar to the one that's already on my meantone Dictionary entry.)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗monz <joemonz@yahoo.com>

Hi Paul,

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 2:31 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > So, the point of these meantone lattices has finally
> > become clear to me: they show that each fraction-of-a-comma
> > meantone has a particular flavor associated with it, because
> > each one also has a unique set of 5-limit JI rational implications
> > associated with it,
>
> To me, they all have _exactly the same_ set of 5-limit JI rational
> implications associated with them.

That's what I used to think. But it's apparent to me that in
a broad sense (over the course of an entire movement of a
Mozart symphony, for example) the specific meantone chosen
with have a specific set of 5-limit approximations that are
*acoustically* (i.e., in pitch-height) the closest ones implied
by the meantone.

Of course, the musical context can do much to alter or negate
the boundaries of this set. But still, there are distinct
variations in the sets of JI ratios implied for each meantone.
And I think there's something to that... at least, it seems
to be a good subject for further musico-historical research.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

>
> > and these rational implications follow
> > a general linear trend across the 2-dimensional 5-limit
> > ratio-space plane of the lattice, with the meantone itself
> > at the center.
>
> Try doing this for LucyTuning or Golden Meantone or 55-tET and you'll
> soon see the weakness of this approach.
> >
> > So obviously, my theory is that each type of meantone has
> > a certain sound/feeling, which can be modeled as the set of
> > implied 5-limit ratios and their general trend.
>
> To me, the implied 5-limit ratios are exactly the same from one
> meantone to another, and they each have a different sound/feeling
> because of how they distribute the syntonic comma error differently
> among the consonances.
>
> > Feedback sought.
>
> You sought it, you got it!
>
>
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🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗monz <joemonz@yahoo.com>

> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 3:45 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > I've also just added this lattice to my webpage on 55-EDO:
> >
> > http://www.ixpres.com/interval/monzo/55edo/55edo.htm
>
> I'd recommed making it clear which Mozart you are referring to, since
> unless you say Leopold people will assume you mean Wolfgang Amadeus.

I do mean both Mozarts, father and son. There is documentary
evidence that both of them taught that flats should be higher
than sharps.

(But -- caveat -- as Paul has mentioned, W.A. Mozart
occasionally made us of "enharmonic equivalences" that would be
impossible if a meantone tuning were strictly observed.)

The evidence for Leopold is in his violin instruction book,
published 1756, the year of W.A.'s birth.

The evidence for W.A. is in the study-notes written down by
his student Thomas Atwood. It's all in my webpage, and the
Chesnut article.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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🔗monz <joemonz@yahoo.com>

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 3:00 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > That's what I used to think. But it's apparent to me that in
> > a broad sense (over the course of an entire movement of a
> > Mozart symphony, for example) the specific meantone chosen
> > with have a specific set of 5-limit approximations that are
> > *acoustically* (i.e., in pitch-height) the closest ones implied
> > by the meantone.
>
> But that will make no perceptual difference.

How can you be completely sure about that? I think it's
entirely possible that composers like Mozart may have composed
the musical phrases/tunes/harmonies they did *because* of
the pitches that would be implied by a particular meantone
tuning.

> And it's nonsense (in my view) to insist that Mozart was in,
> say, 1/6-comma meantone, but not in 1/5-comma meantone, or
> 55-tET, or a chain of 698-cent fifths.

I certainly suggest strongly that Mozart would have preferred
his non-keyboard music to have been performed in 1/6-comma MT
or 55-EDO, but I don't actually insist on it. My own opinion
is that 55-EDO is the most likely candidate because of its
conceptual simplicity.

1/5-comma or 43-EDO is certainly another possibility, but from
what I've read (chiefly in Lindley/Turner-Smith), that tuning
was outmoded by Mozart's time and 1/6-comma or 55-EDO was more
widely used then. Also, it seems to me that 1/5-c./43-EDO
was more widely used in France and 1/6-c./55-EDO more common
in Germany and Austria during the 1700s and 1800s.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_30953.html#30964

>
> That's what I used to think. But it's apparent to me that in
> a broad sense (over the course of an entire movement of a
> Mozart symphony, for example) the specific meantone chosen
> with have a specific set of 5-limit approximations that are
> *acoustically* (i.e., in pitch-height) the closest ones implied
> by the meantone.
>

Well, this is kinda interesting. But I'm not getting it. 1/4 comma
meantone, for example, is virtually just for 5-limit sonorities and
1/6th comma is off some for the 5-limit sonorities, yes?

But basically both systems are going to "imply" the same 5-limit
intervals, yes? Only one is a bit more "distorted" than the other
and maybe approximates (say in 1/6th comma) *more* 5-limit intervals
in various keys, yes?

But they are all the same intervals that are being approximated, no?

Help me if you can, I'm feeling down...

Joseph Pehrson

🔗genewardsmith@juno.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> First, the JI lattice plots the meantone pitch-classes precisely
> according to their fractional prime-factorization. Thus, with
> C = n^0 at the center as a reference, it can be easily seen that
>
> G = 3^(1/3) * 5^(1/6),
> D = 3^(2/3) * 5^(1/3),
> A = 3^1 * 5^(1/2), etc.

Since in meantone, 81/80~1, 2^(-4) 3^4 5^(-1)~1, 2^4~3^4 5^(-1) and
2 ~ 3 5^(-1/4). This leads to

G = 5^(1/4)
D = 3^(-1) * 5^(3/4)
A = 5/3

by my calculations.

> To check the accuracy of this, use vector subtraction to
> deduct 1/6 of a syntonic comma (comma = 81:80 ratio =
> 3^4 * 5^-1 in "8ve"-equivalent prime-factor notation):
>
> 3^(6/6) * 5^(0/6) = 3:2 ratio = 3^1
> - 3^(4/6) * 5^(-1/6) = 1/6 of a syntonic comma
> --------------------
> 3^(2/6) * 5^(1/6) = 1/6-comma meantone "5th"

It seems to me all meantones should come out the same.

🔗monz <joemonz@yahoo.com>

> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 8:36 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > First, the JI lattice plots the meantone pitch-classes precisely
> > according to their fractional prime-factorization. Thus, with
> > C = n^0 at the center as a reference, it can be easily seen that
> >
> > G = 3^(1/3) * 5^(1/6),
> > D = 3^(2/3) * 5^(1/3),
> > A = 3^1 * 5^(1/2), etc.
>
> Since in meantone, 81/80~1, 2^(-4) 3^4 5^(-1)~1, 2^4~3^4 5^(-1) and
> 2 ~ 3 5^(-1/4). This leads to
>
> G = 5^(1/4)
> D = 3^(-1) * 5^(3/4)
> A = 5/3
>
> by my calculations.

I don't fully understand this.

Your "G" is from 1/4-comma meantone.

Your "D" would place "G" at 3^(-1/2) * 5^(3/8), which
is 3/8-comma quasi-meantone.

Your "A" is from 1/3-comma quasi-meantone.

In 1/4-comma meantone, each
"5th" is 5^(1/4), so

G = 5^(1/4)
D = 5^(1/2)
A = 5^(3/4)
E = 5^1

>
> > To check the accuracy of this, use vector subtraction to
> > deduct 1/6 of a syntonic comma (comma = 81:80 ratio =
> > 3^4 * 5^-1 in "8ve"-equivalent prime-factor notation):
> >
> > 3^(6/6) * 5^(0/6) = 3:2 ratio = 3^1
> > - 3^(4/6) * 5^(-1/6) = 1/6 of a syntonic comma
> > --------------------
> > 3^(2/6) * 5^(1/6) = 1/6-comma meantone "5th"
>
> It seems to me all meantones should come out the same.

OK, OK... then we should be restricting the word "meantone"
to only 1/4-comma meantone, the only one which contains
the actual mean-tone.

So we have to use "quasi-meantone" for all the others.
So my words should have been:

>> First, the JI lattice plots the quasi-meantone pitch-classes
>> precisely according to their fractional prime-factorization.
>> Thus, with C = n^0 at the center as a reference, it can be
>> easily seen that
>>
>> G = 3^(1/3) * 5^(1/6), etc.

The values I give are for 1/6-comma quasi-meantone, as displayed
on the lattice I made at
/tuning/files/monz/6th-cmt-lattice.gif

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🔗monz <joemonz@yahoo.com>

Hi Paul (and Gene),

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 2:31 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > and these rational implications follow
> > a general linear trend across the 2-dimensional 5-limit
> > ratio-space plane of the lattice, with the meantone itself
> > at the center.
>
> Try doing this for LucyTuning or Golden Meantone or 55-tET and you'll
> soon see the weakness of this approach.

I anticipated you saying something like that... that's why
I was careful to specify that this latticing method is for
the fraction-of-a-comma types of quasi-meantones. The other
types you mention simply aren't calculated as fractions of
a comma, and therefore don't have anything to do with the
5-limit lattice _per se_.

So if something describes oranges well but apples poorly,
what's wrong with using it to describe oranges?

> >
> > So obviously, my theory is that each type of meantone has
> > a certain sound/feeling, which can be modeled as the set of
> > implied 5-limit ratios and their general trend.
>
> To me, the implied 5-limit ratios are exactly the same from one
> meantone to another, and they each have a different sound/feeling
> because of how they distribute the syntonic comma error differently
> among the consonances.

But those error distributions are going to be reflected in the
harmonic relationships by the types of lattice-movement that
can be seen in the "general trend" to which I referred. In other
words, for example, as a piece in 1/6-comma quasi-meantone
modulates to the flat or sharp end of the circle-of-5ths, the
implied JI proportions will tend to be those latticed along the
vicinity of the linear axis graphed by the 1/6-comma pitches.
Likewise, in 1/4-comma meantone, the relationships will tend
to follow the 5-axis, etc.

Paul, please elaborate on why you disagree with me on this.
It seems to me that the elegant mathematics portrayed on my
lattice would convince you of the validy of my line of reasoning.
Since Gene is also tending to disagree with me, I feel like
I must be missing something significant.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗genewardsmith@juno.com

🔗monz <joemonz@yahoo.com>

Hi Gene,

Re:
/tuning/files/monz/6th-cmt-lattice.gif

> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 10:21 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I don't fully understand this.
> >
> > Your "G" is from 1/4-comma meantone.
> >
> > Your "D" would place "G" at 3^(-1/2) * 5^(3/8), which
> > is 3/8-comma quasi-meantone.
> >
> > Your "A" is from 1/3-comma quasi-meantone.
>
> I'm writing everything (including the octave) as the product of two
> generators, one a meantone version of "3" and the other a meantone
> version of "5"; *which* meantone is not specified. I thought it would
> be interesting, because it does produce a planar lattice, which
> seemed to be what you wanted, and does allow such a calculation like
> yours to be made in a way which makes sense to me, at least.
>

Thanks for your comments. Now I understand your notation.

> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 10:15 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Since Gene is also tending to disagree with me, I feel like
> > I must be missing something significant.
>
> My calculations were along very different lines, so maybe we can
> figure something out along yours. You factor the generator into two
> factors you call "3^(1/3)" and "5^(1/6)", and I can't figure out
> where you go with this, since you always use them as a product
> afterwards.

It simply allows me to plot the quasi-meantones as linear axes
on the 3x5 planar lattice. This also provides a nice graphical
means of comparing the different fraction-of-a-comma quasi-meantones
and the JI systems they represent, allowing my contention that
the quasi-meantones (and regular meantone as well) imply a certain
particular set of JI pitch-classes.

Note that all of these fraction-of-a-comma quasi-meantones contain
certain pitch-classes where periodic coincidences between the
quasi-meantone and JI tunings intersect, for example, F# and Gb on
the 1/6-comma lattice. I know that you, Gene, would see this
right away as an inherent property of numerical relationships,
but thought it worth pointing out anyway.

I guess the main point I'm making with these lattices is that
there is a particular set of JI pitches associated with each
meantone chain, which I would think could be modeled as a
periodicity-block.

For instance, the 1/6-comma variety might have a 55-tone PB,
since that is the point in the 1/6-comma chain where a small
interval appears which would be a good candidate for a
unison-vector. It's important for me to point out that I
did not use this boundary on my lattice... it's a smaller
system that has an arbitrary boundary of -13...+13. By
comparison, the typical 12-tone version Eb...G# would be
-3...+8, which is also mathematically arbitrary (but good
from a practical point of view regarding instruments).

I also thought I might point out a parallel with a point
that I made last Spring in regard to Ben Johnston.
/tuning/topicId_21464.html#21464

That comment made note of how Johnston's modulatory patterns
tended to follow movement by "mediant" (5:3) relationships,
which I think might be partly a result of the way his notation
works. Joe Pehrson thought this was an interesting idea.
/tuning/topicId_21464.html#21485

So what I'm thinking is that a composer with sensitive ears
would know intuitively that meantone or a particular
quasi-meantone would imply a distinctive set of JI pitches,
and that set of pitches (periodicity-block?) might influence
his melodic and harmonic practice.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: Paul Erlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, December 03, 2001 3:00 PM
> > Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-
limit JI
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > That's what I used to think. But it's apparent to me that in
> > > a broad sense (over the course of an entire movement of a
> > > Mozart symphony, for example) the specific meantone chosen
> > > with have a specific set of 5-limit approximations that are
> > > *acoustically* (i.e., in pitch-height) the closest ones implied
> > > by the meantone.
> >
> > But that will make no perceptual difference.
>
>
> How can you be completely sure about that? I think it's
> entirely possible that composers like Mozart may have composed
> the musical phrases/tunes/harmonies they did *because* of
> the pitches that would be implied by a particular meantone
> tuning.

The _pitches_ that would be _implied_? Sorry, I only believe in the
pitches that are heard/played.

> > And it's nonsense (in my view) to insist that Mozart was in,
> > say, 1/6-comma meantone, but not in 1/5-comma meantone, or
> > 55-tET, or a chain of 698-cent fifths.
>
>
> I certainly suggest strongly that Mozart would have preferred
> his non-keyboard music to have been performed in 1/6-comma MT
> or 55-EDO, but I don't actually insist on it. My own opinion
> is that 55-EDO is the most likely candidate because of its
> conceptual simplicity.

Then what happens to your implied complex 5-limit ratios?

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_30953.html#30964
>
> >
> > That's what I used to think. But it's apparent to me that in
> > a broad sense (over the course of an entire movement of a
> > Mozart symphony, for example) the specific meantone chosen
> > with have a specific set of 5-limit approximations that are
> > *acoustically* (i.e., in pitch-height) the closest ones implied
> > by the meantone.
> >
>
> Well, this is kinda interesting. But I'm not getting it. 1/4
comma
> meantone, for example, is virtually just for 5-limit sonorities and
> 1/6th comma is off some for the 5-limit sonorities, yes?

1/6-comma is a bit better for the fourths and fifths, but 1/4-comma
is better for the major and minor thirds and sixths.

> But basically both systems are going to "imply" the same 5-limit
> intervals, yes?

Yes.

> Only one is a bit more "distorted" than the other

Yes.

> and maybe approximates (say in 1/6th comma) *more* 5-limit
intervals
> in various keys, yes?

How do you mean?

> But they are all the same intervals that are being approximated, no?

In my opinion, yes.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Hi Paul (and Gene),
>
>
> > From: Paul Erlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, December 03, 2001 2:31 PM
> > Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-
limit JI
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > and these rational implications follow
> > > a general linear trend across the 2-dimensional 5-limit
> > > ratio-space plane of the lattice, with the meantone itself
> > > at the center.
> >
> > Try doing this for LucyTuning or Golden Meantone or 55-tET and
you'll
> > soon see the weakness of this approach.
>
>
> I anticipated you saying something like that... that's why
> I was careful to specify that this latticing method is for
> the fraction-of-a-comma types of quasi-meantones. The other
> types you mention simply aren't calculated as fractions of
> a comma, and therefore don't have anything to do with the
> 5-limit lattice _per se_.
>
> So if something describes oranges well but apples poorly,
> what's wrong with using it to describe oranges?

Because there is no way, in this case, anyone can tell the difference
between an "apple" and an "orange" by just listening!

> > > So obviously, my theory is that each type of meantone has
> > > a certain sound/feeling, which can be modeled as the set of
> > > implied 5-limit ratios and their general trend.
> >
> > To me, the implied 5-limit ratios are exactly the same from one
> > meantone to another, and they each have a different sound/feeling
> > because of how they distribute the syntonic comma error
differently
> > among the consonances.
>
>
> But those error distributions are going to be reflected in the
> harmonic relationships by the types of lattice-movement that
> can be seen in the "general trend" to which I referred. In other
> words, for example, as a piece in 1/6-comma quasi-meantone
> modulates to the flat or sharp end of the circle-of-5ths, the
> implied JI proportions will tend to be those latticed along the
> vicinity of the linear axis graphed by the 1/6-comma pitches.
> Likewise, in 1/4-comma meantone, the relationships will tend
> to follow the 5-axis, etc.

Totally meaningless, musically speaking. All meantone temperaments
bend the 5-limit lattice into a cylinder. All these movements will
look exactly the same on the cylinder.

> Paul, please elaborate on why you disagree with me on this.
> It seems to me that the elegant mathematics portrayed on my
> lattice would convince you of the validy of my line of reasoning.
> Since Gene is also tending to disagree with me, I feel like
> I must be missing something significant.

Year after year, it seems as if you're failing to see meantone
temperament for what it is (and thus missing out on its true
excellence, it seems to me).

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I guess the main point I'm making with these lattices is that
> there is a particular set of JI pitches associated with each
> meantone chain, which I would think could be modeled as a
> periodicity-block.
>
> For instance, the 1/6-comma variety might have a 55-tone PB,
> since that is the point in the 1/6-comma chain where a small
> interval appears which would be a good candidate for a
> unison-vector.

Yes -- but this is seen directly in 1/6-comma meantone, without
making any reference to the powers of 45/32 than 1/6-comma meantone
happens to express justly.

> So what I'm thinking is that a composer with sensitive ears
> would know intuitively that meantone or a particular
> quasi-meantone would imply a distinctive set of JI pitches,

Let's be clear on what you're claiming. You're claiming that it's
musically significant that 1/6-comma meantone gives you just powers
of 45:32; that 1/7-comma meantone gives you just powers of 135:128;
that 7/26-comme meantone gives you just powers of ________, etc.

🔗dmb@sgi.com

Joe Monzo wrote:
> I thought some readers would appreciate a little more explanation.
>
>
> First, the JI lattice plots the meantone pitch-classes precisely
> according to their fractional prime-factorization. Thus, with
> C = n^0 at the center as a reference, it can be easily seen that
>
> G = 3^(1/3) * 5^(1/6),
> D = 3^(2/3) * 5^(1/3),
> A = 3^1 * 5^(1/2), etc.
>
>
> To check the accuracy of this, use vector subtraction to
> deduct 1/6 of a syntonic comma (comma = 81:80 ratio =
> 3^4 * 5^-1 in "8ve"-equivalent prime-factor notation):
>
> 3^(6/6) * 5^(0/6) = 3:2 ratio = 3^1
> - 3^(4/6) * 5^(-1/6) = 1/6 of a syntonic comma
> --------------------
> 3^(2/6) * 5^(1/6) = 1/6-comma meantone "5th"
>
> Compare this result with the value for "G" plotted on the lattice.

Joe,

I know that most of the time we ignore the factors of two in our
ratios, since we assume octave equivalence. But octave equivalence
doesn't apply when we have fractional powers of two. So you really
ought to say that G = 2^(-1/3) * 3^(1/3) * 5^(1/6) and

2^(-6/6) * 3^(6/6) * 5^(0/6)
- 2^(-4/6) * 3^(4/6) * 5^(-1/6)
-------------------------------------
2^(-2/6) * 3^(2/5) * 5^(1/6)

unless you really want me to believe that 1.5 = 1.88. Sorry for the
delay in responding. I'm a Digest subscriber and Yahoo was having some
problems earlier this morning.

David Bowen

🔗monz <joemonz@yahoo.com>

Hi Paul,

re: /tuning/files/monz/6th-cmt-lattice.gif

Thanks for you many comments on this. I'd like to keep
discussing it with you until the two of us reach some kind
of rapport... let me know if you think our conversation
should go private.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, December 04, 2001 11:29 AM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
>
> > So what I'm thinking is that a composer with sensitive ears
> > would know intuitively that meantone or a particular
> > quasi-meantone would imply a distinctive set of JI pitches,
>
> Let's be clear on what you're claiming. You're claiming that it's
> musically significant that 1/6-comma meantone gives you just powers
> of 45:32; that 1/7-comma meantone gives you just powers of 135:128;
> that 7/26-comme meantone gives you just powers of ________, etc.

Well, that's a limited way of saying what I'm thinking. More to
the point, is that the entire meantone or quasi-meantone chain
will imply a whole set of specific 5-limit JI ratios *better*
than any other 5-limit ratios.

The *trend* for 1/6-comma is along the axis which includes just
powers of 45:32, the trend for 1/4-comma is along the axis which
includes just powers of 5:4 (which happens to be easier to describe
simply as the 5-axis), the trend for 1/3-comma is along the axis
which includes just powers of 5:3, etc.

So, for one example, let's examine the JI ratios implied by
various meantone/quasi-meantone "A"s ("comma" in every case
refers to the syntonic comma 81:80):

The 1/3-comma quasi-meantone "A" is exactly the 5:3 ratio.
Doesn't do such a great job of implying 27:16.

The 1/4-comma meantone "A" is 1/4-comma sharper than 5:3
and 3/4-comma flatter than 27:16. Therefore, it implies
5:3 much more strongly (or better) than 27:16.

The 1/6-comma quasi-meantone "A" is exactly midway between
these two ratios, 1/2-comma sharper than 5:3 and 1/2-comma
flatter than 27:16. Therefore, it implies either ratio
equally well (or badly).

Perhaps I'm wrong in claiming that there's some significance
to this, or perhaps I'm simply not explaining myself clearly.
It may be easier to grasp the full gist of what I'm saying after
I've made other lattices portraying other quasi-meantones and
comparing them.

But it seems to me that the ears of someone *listening* to music
performed in a particular meantone/quasi-meantone will be the
recipients of a barrage of intervals which strongly imply *certain*
5-limit ratios, to the exclusion of others -- unless, of course,
the musical context strongly implies other ratios which are not
acoustically implied well by the tuning.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗monz <joemonz@yahoo.com>

Hello David,

> From: <dmb@sgi.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, December 04, 2001 1:25 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> I know that most of the time we ignore the factors of two in our
> ratios, since we assume octave equivalence. But octave equivalence
> doesn't apply when we have fractional powers of two. So you really
> ought to say that G = 2^(-1/3) * 3^(1/3) * 5^(1/6) and
>
> 2^(-6/6) * 3^(6/6) * 5^(0/6)
> - 2^(-4/6) * 3^(4/6) * 5^(-1/6)
> -------------------------------------
> 2^(-2/6) * 3^(2/5) * 5^(1/6)
>
> unless you really want me to believe that 1.5 = 1.88. Sorry for the
> delay in responding. I'm a Digest subscriber and Yahoo was having some
> problems earlier this morning.

Yes, you're exactly correct about this. In order to have the
mathematics work out properly, prime-factor 2 and its exponents
must be included.

But the lattice I made of 1/6-comma quasi-meantone does not
include 2 as a factor, and so it is safe to ignore powers of 2
in my discussion of the lattice. One could imagine that
powers of prime-factor 2 are latticed along a "z"-axis which
would run from front-to-back and therefore would be irrelevant
when viewing the lattice from the front.

(Robert Walker, in case you're reading... *this* is something
I'd like you to help me make in VRML!, so that the 2-axis
would be viewable by rotating the diagram.)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> But it seems to me that the ears of someone *listening* to music
> performed in a particular meantone/quasi-meantone will be the
> recipients of a barrage of intervals which strongly imply *certain*
> 5-limit ratios, to the exclusion of others -- unless, of course,
> the musical context strongly implies other ratios which are not
> acoustically implied well by the tuning.

As you know, the only 5-limit ratios that I feel can be
psychoacoustically "implied" are the 5-*odd*-limit ratios, and
perhaps also other simple ratios like 9:4, 9:2, 9:1, 15:2, and 15:1
that fall outside the scope of an octave, and perhaps 9:5 as well.
The only way you can "imply" or "fail to imply" some more complex
ratio is if the listener is already trained in JI musical interval
recognition. Otherwise, dissonant intervals are merely dissonant
intervals, and don't imply anything -- they're constructed from
consonant intervals, have melodic tendencies followed or thwarted in
the music, etc. . . . but they're not implying some ridiculously high
members of the overtone series.

Even if I'm wrong about this, you have to understand how extremely
exacting the tuning would have to be for your claims to make logical
sense. Let's say (though I personally wouldn't believe this applies
to many listeners) that someone's ear is so sensitive that when they
hear 590 cents, they perceive it as 45:32, and not as 7:5 (which is
583 cents). Now since 7:5 is a much simpler ratio than 45:32, it will
have a much larger range of allowable mistuning than 45:32. So 45:32
will have an allowable mistuning of at most two or three cents before
it is heard as some simpler ratio. Given that the highest quality
performers today have errors in intonation typically about 12-20
cents (which do not, by the way, represent an attempt to approximate
ratios more closely), and given that Mozart's keyboards were tuned to
well-temperaments and not to meantones at all, I think the
plausibility of Mozart implying certain 5-limit ratios but not others
is . . . well I was going to mention Santa Claus, but Johnny stole
that one before I could use it . .

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_30953.html#31034

> Hi Paul,
>
> re: /tuning/files/monz/6th-cmt-
lattice.gif
>
>
> Thanks for you many comments on this. I'd like to keep
> discussing it with you until the two of us reach some kind
> of rapport... let me know if you think our conversation
> should go private.
>

Whyzzat? I've been enjoying the discussion, myself... Isn't that
what this list is for, to "critique" the stuff that we throw out
there??

>
> So, for one example, let's examine the JI ratios implied by
> various meantone/quasi-meantone "A"s ("comma" in every case
> refers to the syntonic comma 81:80):
>
> The 1/3-comma quasi-meantone "A" is exactly the 5:3 ratio.
> Doesn't do such a great job of implying 27:16.
>
> The 1/4-comma meantone "A" is 1/4-comma sharper than 5:3
> and 3/4-comma flatter than 27:16. Therefore, it implies
> 5:3 much more strongly (or better) than 27:16.
>
> The 1/6-comma quasi-meantone "A" is exactly midway between
> these two ratios, 1/2-comma sharper than 5:3 and 1/2-comma
> flatter than 27:16. Therefore, it implies either ratio
> equally well (or badly).
>
>
> Perhaps I'm wrong in claiming that there's some significance
> to this, or perhaps I'm simply not explaining myself clearly.
> It may be easier to grasp the full gist of what I'm saying after
> I've made other lattices portraying other quasi-meantones and
> comparing them.
>
> But it seems to me that the ears of someone *listening* to music
> performed in a particular meantone/quasi-meantone will be the
> recipients of a barrage of intervals which strongly imply *certain*
> 5-limit ratios, to the exclusion of others -- unless, of course,
> the musical context strongly implies other ratios which are not
> acoustically implied well by the tuning.
>

Maybe I'm missing something crucial, or misunderstanding something
crucial, but wouldn't the compositional significance of going to
the "smaller" meantones, all the way to 1/11th comma
simply "encourage" a composer to do more modulation and use more keys
that "approximate" overtone series intervals, even though the
approximations get worse as the comma gets cut up into smaller
segments and distributed more widely?? ??

So the compositional change could be evaluated, if one cared to, in a
scientific study where the *statistical* use of certain intervals
throughout *all* of the possible keys could be compared?? ??

Yes, no??

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Maybe I'm missing something crucial, or misunderstanding something
> crucial, but wouldn't the compositional significance of going to
> the "smaller" meantones, all the way to 1/11th comma
> simply "encourage" a composer to do more modulation and use more
keys
> that "approximate" overtone series intervals, even though the
> approximations get worse as the comma gets cut up into smaller
> segments and distributed more widely?? ??

I think you're exactly right.

But I think Monz is missing the fact that more distant points on the
lattice _come from_ stacking simpler intervals, and it is only the
quality of approximation in the simpler intervals, _not_ in those
that result from the stacking, that is perceptually relevant --
unless the perceiver had most spent most of his/her life listening to
and making music in extended JI systems.
>
> So the compositional change could be evaluated, if one cared to, in
a
> scientific study where the *statistical* use of certain intervals
> throughout *all* of the possible keys could be compared?? ??

Well, Joseph, I suppose Monz is not simply focusing on fixed 12-tone
subsets of these meantone tunings, as you are.

🔗monz <joemonz@yahoo.com>

Hi Paul and Joe,

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, December 09, 2001 5:58 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Maybe I'm missing something crucial, or misunderstanding something
> > crucial, but wouldn't the compositional significance of going to
> > the "smaller" meantones, all the way to 1/11th comma simply
> > "encourage" a composer to do more modulation and use more keys
> > that "approximate" overtone series intervals, even though the
> > approximations get worse as the comma gets cut up into smaller
> > segments and distributed more widely?? ??
>
> I think you're exactly right.

I don't follow Joe's logic, and therefore don't understand
why Paul agrees. Clarification would be appreciated.

What's plain to see on my lattices -- at least for me -- is that,
as one progresses from 1/4-comma thru 1/5- and 1/6- to 1/11-comma
meantones, the *Pythagorean* pitches are approximated better and
better.

I haven't even considered drawing the "xenharmonic bridges"
which would connect the meantone pitches to higher-prime implied
ratios, but that's a good idea too. It would make the lattices
more cluttered, too...

> But I think Monz is missing the fact that more distant points on the
> lattice _come from_ stacking simpler intervals, and it is only the
> quality of approximation in the simpler intervals, _not_ in those
> that result from the stacking, that is perceptually relevant --
> unless the perceiver had most spent most of his/her life listening to
> and making music in extended JI systems.

But that's not entirely relevant to my lattices, because a few
composers and theorists from several centuries did (and do) experiment
with more-than-12-tone meantone chains, thus including the implied
ratios I've pictured on my diagrams... and in some cases, ratios
beyond the ones I pictured (i.e., the many 31-toners, Vicentino,
Saveur, Bosanquet, etc.).

> >
> > So the compositional change could be evaluated, if one cared to,
> > in a scientific study where the *statistical* use of certain
> > intervals throughout *all* of the possible keys could be compared?? ??
>
> Well, Joseph, I suppose Monz is not simply focusing on fixed 12-tone
> subsets of these meantone tunings, as you are.

Right, I'm not. The only meantone chains on my webpage which
have enough pitches in the diagram to form a closed system are
1/3-comma (~= 19-EDO) and 1/11-comma (~= 12-EDO), the others
have an arbitrary limit of +/- 13 meantone generators. And
even in those two cases, I carried the diagram out to my arbitrary
limits, because, for example, in the 1/11-comma case, skhismatic
equivalents become significant.

All I'm saying with these lattices is that 1/4-comma meantone
and its related quasi-meantones carry with them a built-in set
of implied 5-limit JI ratios, based on the proximity of the
frequencies of those JI ratios to those of the pitches actually
heard in the meantone.

Of course, a composer's particular compositional practice may
emphasize other kinds of tuning relationships. But this is a
good place to begin an exploration of exactly what types of
harmonic relationships may have been intended or implied in
the music of composers who have utilized meantone tunings.

Given the fact that a particular form of meantone, even if it
only has 12 different pitch-classes, will imply a particular
set of JI ratios, I no longer believe that one should simply
assume that a composer intended a "typical" JI harmonic template,
such as a diatonic major scale of

4:3 ._
/ '-._
/ '- 5:3
1:1 ._ /
/ '-._ /
/ '- 5:4
3:2 ._ /
/ '-._ /
/ '- 15:8
9:8

to be the basis of his harmonic practice. While this particular
example fits perfectly into the 1/4-, 1/5-, and 1/6-comma meantone
lattices, it doesn't fit as well into 1/11-comma, and even less
well into the 2/7- and 1/3-comma varieties, as my lattices show.

So, for an example, my position is that analysis of work by a
composer who utilized 2/7-comma (Zarlino was the first prominent
advocate of that tuning, even tho he stipulated the major scale
given in the above lattice) should begin with the assumption that
the diatonic major scale is:

10:9
4:3 ._ /
/ '-._ /
/ '- 5:3
1:1 ._ /
/ '-._ /
/ '- 5:4
3:2 ._ /
'-._ /
'- 15:8

REFERENCE
---------

Monzo, Joe. 2001.
_Lattice diagrams comparing rational implications
of various meantone chains_.
http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Hi Paul and Joe,

> > But I think Monz is missing the fact that more distant points on
the
> > lattice _come from_ stacking simpler intervals, and it is only
the
> > quality of approximation in the simpler intervals, _not_ in those
> > that result from the stacking, that is perceptually relevant --
> > unless the perceiver had most spent most of his/her life
listening to
> > and making music in extended JI systems.
>
>
> But that's not entirely relevant to my lattices, because a few
> composers and theorists from several centuries did (and do)
experiment
> with more-than-12-tone meantone chains, thus including the implied
> ratios I've pictured on my diagrams... and in some cases, ratios
> beyond the ones I pictured (i.e., the many 31-toners, Vicentino,
> Saveur, Bosanquet, etc.).

Joe, this makes my point _even more_ relevant to your lattices!
Please read it again.

> All I'm saying with these lattices is that 1/4-comma meantone
> and its related quasi-meantones carry with them a built-in set
> of implied 5-limit JI ratios, based on the proximity of the
> frequencies of those JI ratios to those of the pitches actually
> heard in the meantone.

But how are you going to "imply" or "fail to imply" a JI ratio that
is way beyond the ear's natural capacity to "understand ratios"? I'm
arguing that what you're saying with these lattices is far less
perceptually relevant than, say, what a "traditional" meantone
lattice (such as one of Barbour's) is saying.

> Of course, a composer's particular compositional practice may
> emphasize other kinds of tuning relationships. But this is a
> good place to begin an exploration of exactly what types of
> harmonic relationships may have been intended or implied in
> the music of composers who have utilized meantone tunings.

Sorry, I have to believe, instead, that the harmonic ratios intended
or implied were the ones that could be directly _perceived_, error or
no, and that the more complex ratios are mere mathematical
curiosities. The reasons for using meantone tunings have been laid
down by various musicians for almost 500 years now, and I have to
believe what _they_ have said, rather than your far-fetched
mathematical theory (you know I love you anyway, Monz).

> Given the fact that a particular form of meantone, even if it
> only has 12 different pitch-classes, will imply a particular
> set of JI ratios, I no longer believe that one should simply
> assume that a composer intended a "typical" JI harmonic template,
> such as a diatonic major scale of
>
> 4:3 ._
> / '-._
> / '- 5:3
> 1:1 ._ /
> / '-._ /
> / '- 5:4
> 3:2 ._ /
> / '-._ /
> / '- 15:8
> 9:8
>
> to be the basis of his harmonic practice.

I see that fact that the second scale degree is consonant with the
sixth scale degree as an important feature of meantone, not shown in
this template. A good meantone lattice would either roll up this
template, or repeat it infinitely, so that this feature is shown.

> So, for an example, my position is that analysis of work by a
> composer who utilized 2/7-comma (Zarlino was the first prominent
> advocate of that tuning, even tho he stipulated the major scale
> given in the above lattice) should begin with the assumption that
> the diatonic major scale is:
>
> 10:9
> 4:3 ._ /
> / '-._ /
> / '- 5:3
> 1:1 ._ /
> / '-._ /
> / '- 5:4
> 3:2 ._ /
> '-._ /
> '- 15:8

This is an exact mirror-image of your diagram above and so should fit
any lattice or tuning exactly as well as that one does. If you don't
agree, then I have an additional criticism of your theory, which is
that you're focusing too narrowly on _pitches_ at the expense of
_intervals_. If Zarlino plays you just one pitch, are you going to
place it on your lattice according to its fractional prime powers? I
would argue that no ratio-implications take place at all until the
first _interval_ (harmonic or melodic) is sounded.

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_30953.html#31235

> >
> > > Maybe I'm missing something crucial, or misunderstanding
something
> > > crucial, but wouldn't the compositional significance of going
to
> > > the "smaller" meantones, all the way to 1/11th comma simply
> > > "encourage" a composer to do more modulation and use more keys
> > > that "approximate" overtone series intervals, even though the
> > > approximations get worse as the comma gets cut up into smaller
> > > segments and distributed more widely?? ??
> >
> > I think you're exactly right.
>
>
> I don't follow Joe's logic, and therefore don't understand
> why Paul agrees. Clarification would be appreciated.
>

Hello Joe! It's Joe!

Well... there really wasn't much to that. That was simple "textbook"
stuff... that the more the comma is distributed through the scale,
the more the tendency for composers to modulate to "distant" keys.
There's no news in that. I was just saying that, compositionally,
that could be the most significant aspect between the various
meantones, again, I believe, standard textbook stuff... which is
probably why Paul agreed with it.. :) I wish I had contributed
something a bit more "original..."

> What's plain to see on my lattices -- at least for me -- is that,
> as one progresses from 1/4-comma thru 1/5- and 1/6- to 1/11-comma
> meantones, the *Pythagorean* pitches are approximated better and
> better.
>

Well, I don't believe that's exactly going to make headline news
either... And the thirds become more distorted, but more "equal" on
the average leading to more "usable" keys.

You must be thinking about this in some "deeper" way than *I* am,
because I'm just regurgitating standard stuff. Fun to do, though.

>
> But that's not entirely relevant to my lattices, because a few
> composers and theorists from several centuries did (and do)
experiment
> with more-than-12-tone meantone chains, thus including the implied
> ratios I've pictured on my diagrams... and in some cases, ratios
> beyond the ones I pictured (i.e., the many 31-toners, Vicentino,
> Saveur, Bosanquet, etc.).
>

But, Monz... did *Mozart* use that many extra notes? I believe you
were particularly concerned with Mozart and 1/6th comma meantone, if
I'm not mistaken...

But, that's not exactly banner news either, is it? 1/4 has the
*best* approximation for certain keys, of course. I must be missing
something... :(

>
> Of course, a composer's particular compositional practice may
> emphasize other kinds of tuning relationships. But this is a
> good place to begin an exploration of exactly what types of
> harmonic relationships may have been intended or implied in
> the music of composers who have utilized meantone tunings.
>
>
> Given the fact that a particular form of meantone, even if it
> only has 12 different pitch-classes, will imply a particular
> set of JI ratios, I no longer believe that one should simply
> assume that a composer intended a "typical" JI harmonic template,
> such as a diatonic major scale of
>
> 4:3 ._
> / '-._
> / '- 5:3
> 1:1 ._ /
> / '-._ /
> / '- 5:4
> 3:2 ._ /
> / '-._ /
> / '- 15:8
> 9:8
>
> to be the basis of his harmonic practice. While this particular
> example fits perfectly into the 1/4-, 1/5-, and 1/6-comma meantone
> lattices, it doesn't fit as well into 1/11-comma, and even less
> well into the 2/7- and 1/3-comma varieties, as my lattices show.
>
>
> So, for an example, my position is that analysis of work by a
> composer who utilized 2/7-comma (Zarlino was the first prominent
> advocate of that tuning, even tho he stipulated the major scale
> given in the above lattice) should begin with the assumption that
> the diatonic major scale is:
>
> 10:9
> 4:3 ._ /
> / '-._ /
> / '- 5:3
> 1:1 ._ /
> / '-._ /
> / '- 5:4
> 3:2 ._ /
> '-._ /
> '- 15:8
>

But, these aren't really too different, are they? These lattices got
ruined, by the way, as all the others have been. I can only see the
original when I view the posting screen. This is a shame.

Don't mind me, Monz... I'm just confused. I was trying to figure out
what you were saying...

best,

Joe

🔗monz <joemonz@yahoo.com>

Hi Joe,

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 10, 2001 7:05 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> But, Monz... did *Mozart* use that many extra notes? I believe you
> were particularly concerned with Mozart and 1/6th comma meantone, if
> I'm not mistaken...

According to the notes left behind by Mozart's student Thomas Atwood,
Mozart's teaching of intonation was based on a 20-tone ~1/6-comma quasi-
meantone chain of 21 "5ths", a symmetrical system from -10 generator "Ebb"
to +10 generator "A#", without the -7 generator "Cb".

Anyway, my original post and diagram from last week concerned only
1/6-comma and its acoustical rational implications, but now I've
made a webpage comparing a variety of different quasi-meantones
<http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm>,
which was my original intention.

To help clarify your confusion a bit, Joe, I'll add that I agree
to some extent with many of Paul's criticisms of my diagrams, but
I do still feel that there is *some* revelance to understanding the
ratios that are *acoustically* implied by these meantones.

The implications Paul discusses are those that are implied by
compositional practice, which certainly are very relevant, but
it seems to me that what a listener *actually hears* when listening
to a meantone must have *something* to do with the harmonic
maneuvers in that piece.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

Hi Paul,

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, December 11, 2001 6:47 PM
> Subject: [tuning] Re: Monzo lattice comparing 1/6-c MT and 5-limit JI
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > The implications Paul discusses are those that are implied by
> > compositional practice, which certainly are very relevant, but
> > it seems to me that what a listener *actually hears* when listening
> > to a meantone must have *something* to do with the harmonic
> > maneuvers in that piece.
>
> How does that differ from anything I said?

If that's your position, it didn't come thru clearly to me
in the posts you've sent to the list. I'm understanding you
to view meantones as representing a basic core 5-limit lattice
for the simplest intervals, and then an extension in any
arbitrary direction by compounding those intervals. ...???

My postion is that the tempering involved in the meantone
can be modeled as a linear axis on the 5-limit lattice,
based on a the acoustically most closely approximated 5-limit
ratios, causing it to have a certain distinctive "flavor"
or "mood".

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> If that's your position, it didn't come thru clearly to me
> in the posts you've sent to the list. I'm understanding you
> to view meantones as representing a basic core 5-limit lattice
> for the simplest intervals, and then an extension in any
> arbitrary direction by compounding those intervals. ...???

Not at all -- directions which involve compounding 81:80 or multiples
thereof will be _favored_ by composers working in meantone media,
because such motions will be associated with a minimum of chromatic
alteration. I'm confident that a study of virtually any piece of
music from the meantone era (Pachelbel's Canon being a rare
exception) will bear this out.

> My postion is that the tempering involved in the meantone
> can be modeled as a linear axis on the 5-limit lattice,
> based on a the acoustically most closely approximated 5-limit
> ratios, causing it to have a certain distinctive "flavor"
> or "mood".

And my position is that the "flavor" and "mood" is only a function of
how the melodic and harmonic intervals happen to come out in the
meantone in question. There's no distinct "flavor" or "mood"
associated with 1/6-comma or 2/7-comma meantone -- these are merely
arbitrary stopping points along a "flavor/mood" continuum.