jazz root movement

Hey Monz...in digest 1349, you mentioned that the roots of chords in
a 2-5-1 jazz progression usually move in cycles of 4th/5ths, while the
other voices try to move as little as possible. Just a note...often, in
these types of progressions, the bassline will move down chromatically,
to avoid that cycle of 5ths sound. In C, we would have a Dmin7 as the 2,
then we would use the flat 5th of the G7 chord (Db), to resolve to the
1, CMA7...one of the fun things about jazz is trying to see how many
ways one can avoid the obvious...there's a zillion of them...Hstick

--- In tuning@y..., "Neil Haverstick" <STICK@U...> wrote:

/tuning/topicId_23618.html#23618

> Hey Monz...in digest 1349, you mentioned that the roots
> of chords in a 2-5-1 jazz progression usually move in cycles
> of 4th/5ths, while the other voices try to move as little as
> possible. Just a note...often, in these types of progressions,
> the bassline will move down chromatically, to avoid that cycle
> of 5ths sound. In C, we would have a Dmin7 as the 2, then we
> would use the flat 5th of the G7 chord (Db), to resolve to the
> 1, CMA7...one of the fun things about jazz is trying to see
> how many ways one can avoid the obvious...there's a zillion
> of them...Hstick

Hi Neil. Boy, I am glad you mentioned this! I definitely
should have included something about it in my post, because
it's a *very* important aspect of jazz harmony!

For those unfamiliar, what Neil is talking about here is
referred to as "tritone substitution". It's one of the
most interesting things one finds in jazz harmony.

Two chords with roots a tritone apart will both have in
common as a pair of chord-members notes which themselves
form the same tritone interval.

Neil's example illustrates an inversion of a G-dominant-7th
chord with a "flat 5", where the "flat 5" is in the bass,
for instance:

C B B
A G G
F F E
D Db C

But note that if one substitutes a Db-dominant-7th-flat-5
for the G, it's exactly the same chord with different
enharmonic spelling!:

C Cb B
A Abb G
F F E
D Db C

This idea of tritone substitution *probably* originated
in late-1800s German/Austrian theory, but the earliest
recognition of it that I am absolutely sure of is by...
once again, my man Schoenberg.

This is a prime example that Paul Erlich would use to
argue that many typical jazz harmonies are only possible
in 12-EDO.

... Except that Paul would probably call it "12-tET", which,
BTW, brings up *that* issue. In jazz, my first thought
would be definitely to almost always call the tuning 12-EDO,
because I don't see it as a temperament of any implied
JI sonorities. But then again, maybe it is... ?

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

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

> This is a prime example that Paul Erlich would use to
> argue that many typical jazz harmonies are only possible
> in 12-EDO.

"Tritone substitution" is also possible in 22-tET (though diatonic
harmony isn't). Tritone substitution in 12-tET makes use of the fact
that the third and seventh of a dominant seventh chord are the same
as the seventh and third of a dominant seventh chord a tritone lower
(or higher). In 22-tET, the considerably more consonant "dominant
seventh chords" still have this feature, since 7:5 and 10:7 are both
represented by the same half-octave (i.e., the 50:49 comma vanishes).
In fact, the symmetrical decatonic scales in 22-tET are _identical_
when transposed by a half-octave, so they're inherently _bitonal_
scales (in the sense of Stravinsky simultaneously using two
tonalities a tritone apart).
>
> ... Except that Paul would probably call it "12-tET", which,
> BTW, brings up *that* issue. In jazz, my first thought
> would be definitely to almost always call the tuning 12-EDO,
> because I don't see it as a temperament of any implied
> JI sonorities.

That seems a complete about face from the extreme JI view of jazz
harmony I read about in your early _JustMusic: A New Harmony_
edition, where you analyze every possible chord extension as an
overtone of the root.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23640

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > ... Except that Paul would probably call it "12-tET", which,
> > BTW, brings up *that* issue. In jazz, my first thought
> > would be definitely to almost always call the tuning 12-EDO,
> > because I don't see it as a temperament of any implied
> > JI sonorities.
>
> That seems a complete about face from the extreme JI view of jazz
> harmony I read about in your early _JustMusic: A New Harmony_
> edition, where you analyze every possible chord extension as an
> overtone of the root.

Yes, well... I've been hanging out on the tuning list for
the last 3 years, arguing stuff like this with you a lot. :)

The bottom line is that I'm far more interested in temperaments
and other kinds of tunings now than I used to be. When I first
issued my book in 1995, I was definitely a "Partch moonie"
(to use McLaren's term), or at least something still not too
far evolved from that.

But I did add:

> But then again, maybe it is... ?

at the end of my post, and you chose to snip it out.
This reflects my older view, and to a large extent I suppose
I probably do still believe this way.

I dunno... is it possible to hold both of these opposing
opinions at the same time? Boy, Schoenberg's influence
is *really* rubbing off on me!...

(It was very typical of him to consider opposite extremes as
being the endpoints of a continuum, and therefore as being
equally valid.)

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

Hey Monz,

What I would have hoped you might have gotten from me after 3 years
on the tuning list, is this.

I believe approximating small-integer ratios is very important for
consonance and rootedness.

I don't believe that any tuning which exploits this can necessarily
be represented in JI.

**************************************************************
Non-JI tunings take the small-integer ratio approximations and
connect them in a way that is not possible in JI.
**************************************************************

This is the main point that seemed to be absent in your book, and
continued to be absent in your work as recently as
your "rationalization" of the MIRACLE scale (yes, I understand that
you did that for _practical_ reasons, but as a matter of
principle . . .).

Even if adaptive JI is used to make the chords pure, most music will
still tend to have "pitch classes" which cannot be expressed with a
single JI ratio.

I think I've spend 30% of my time on this list just making this one
point, since it's so often ignored.

So if you take anything from your interaction with me, let this be it.

-Paul

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23645

> **************************************************************
> Non-JI tunings take the small-integer ratio approximations and
> connect them in a way that is not possible in JI.
> **************************************************************
>
> This is the main point that seemed to be absent in your book,
> and continued to be absent in your work as recently as your
> "rationalization" of the MIRACLE scale (yes, I understand that
> you did that for _practical_ reasons, but as a matter of
> principle . . .).
>
> Even if adaptive JI is used to make the chords pure, most
> music will still tend to have "pitch classes" which cannot
> be expressed with a single JI ratio.

Right, Paul... I understand this and agree. My whole concept
of finity (and xenharmonic bridges) is based on this idea.

It's very true that I have not yet added anything substantial
about finity to my book, other than the definition on the webpage
(which actually is pretty hefty). At this point I may not,
and may just have to put it in a subsequent major treatise.

As for the rationalized Canasta scale, please focus on the
fact that the *only* reason I did this was becuase I *had*
to have it in rational form to feed it into my software.
This is currently the only way I can hear it.

Otherwise I would gladly have used 72-EDO, or perhaps better
still, a variety of non-2/1 equal tunings based on the actual
calculated optimum generators.

Hey, that brings up a question I may have missed if it was
addressed in the MIRACLE discussions, and which at any rate
I never thought of before: was 2:1 ever included as one of
the intervals in the optimalization calculations? and what
about 1:2?

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

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

> Hey, that brings up a question I may have missed if it was
> addressed in the MIRACLE discussions, and which at any rate
> I never thought of before: was 2:1 ever included as one of
> the intervals in the optimalization calculations?

No -- the 2:1 was assumed just, and all the other intervals assumed
to be octave-equivalent interval classes (that is, 5:3 same as 6:5,
etc.). A non-octave-equivalent optimization would be possible, with
5:3 and 6:5 separate, etc., and including 2:1, 4:1, etc. as intervals
allowed to deviate from JI.

BTW, we've also mentioned non-octave variants of these scales, which
simply take the 72-tET pattern 2 5 or 2 2 3 and repeat them
indefinitely, not stopping to repeat at the octave. These are very
rich in consonant chords too.

> and what
> about 1:2?

Now how could you optimize 2:1 without simultaneously optimizing 1:2?
That's kind of silly, isn't it?

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_23618.html#23645
>
>
> > **************************************************************
> > Non-JI tunings take the small-integer ratio approximations and
> > connect them in a way that is not possible in JI.
> > **************************************************************
> >
> > This is the main point that seemed to be absent in your book,
> > and continued to be absent in your work as recently as your
> > "rationalization" of the MIRACLE scale (yes, I understand that
> > you did that for _practical_ reasons, but as a matter of
> > principle . . .).
> >
> > Even if adaptive JI is used to make the chords pure, most
> > music will still tend to have "pitch classes" which cannot
> > be expressed with a single JI ratio.
>
>
> Right, Paul... I understand this and agree. My whole concept
> of finity (and xenharmonic bridges) is based on this idea.

I see it as different. For example, I don't hear the 256:243
(Pythagorean limma) as a "bridge" in the sense that two pitches
separated by this interval are the same pitch class. But when the
limma is spread over many consonant intervals, the two pitches become
identical, and thus each can participate in a larger number of
consonant relationships with other notes. This happens, for example,
in Herman Miller's 15-tET music.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23659

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Hey, that brings up a question I may have missed if it was
> > addressed in the MIRACLE discussions, and which at any rate
> > I never thought of before: was 2:1 ever included as one of
> > the intervals in the optimalization calculations?
>
> No -- the 2:1 was assumed just, and all the other intervals
> assumed to be octave-equivalent interval classes (that is,
> 5:3 same as 6:5, etc.).

Aha. That's what I thought.

> A non-octave-equivalent optimization would be possible,
> with 5:3 and 6:5 separate, etc., and including 2:1, 4:1,
> etc. as intervals allowed to deviate from JI.

Given that the MIRACLE generators we've already calculated
would not give 2/1s if cycled infinitely, I'm *very* interested
in seeing what results we get when we include the "octaves"
in the optimization process. Could you or Dave (or someone)
please do that? Thanks.

My guess is that we'd end up with a generator that does give
better "octaves" than the ones we've already got.

>
> BTW, we've also mentioned non-octave variants of these scales,
which
> simply take the 72-tET pattern 2 5 or 2 2 3 and repeat them
> indefinitely, not stopping to repeat at the octave. These are very
> rich in consonant chords too.
>
> > and what
> > about 1:2?
>
> Now how could you optimize 2:1 without simultaneously optimizing
1:2?
> That's kind of silly, isn't it?

At first that's what I thought, and wasn't going to add that bit.
But isn't it possible that there could be a difference?
When you say above, "optimization would be possible,
with 5:3 and 6:5 separate, etc.,", it makes me think so.
Why can't 1:2 and 2:1 be separate? Guess it's *because*
they are the definition of "octave-equivalent" intervals?
Circular definition? Am I on the right track here?

What about being able to include overtone ratios as part of
the optimization calculation? Timbre and "octave"-register
influence how we perceive sonance in very perceptible ways.
In that case, it seems to me that there *would* be a difference
between the 1:2 and 2:1 optimizations.

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

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

> > A non-octave-equivalent optimization would be possible,
> > with 5:3 and 6:5 separate, etc., and including 2:1, 4:1,
> > etc. as intervals allowed to deviate from JI.
>
>
> Given that the MIRACLE generators we've already calculated
> would not give 2/1s if cycled infinitely, I'm *very* interested
> in seeing what results we get when we include the "octaves"
> in the optimization process. Could you or Dave (or someone)
> please do that? Thanks.
>
> My guess is that we'd end up with a generator that does give
> better "octaves" than the ones we've already got.

I think you're a little confused. The size of the generator will not
have anything to do with the size of the octave. These are two
_independent_ parameters. In neither optimization will the octave be
expressed in terms of powers of the generator. If you want that,
though, you're probably interested in one of the non-octave variant
of the MIRACLE scales, that I described here:
> >
> > BTW, we've also mentioned non-octave variants of these scales,
> which
> > simply take the 72-tET pattern 2 5 or 2 2 3 and repeat them
> > indefinitely, not stopping to repeat at the octave. These are
very
> > rich in consonant chords too.

So is that really what you're interested in?

> >
> > > and what
> > > about 1:2?
> >
> > Now how could you optimize 2:1 without simultaneously optimizing
> 1:2?
> > That's kind of silly, isn't it?
>
> At first that's what I thought, and wasn't going to add that bit.
> But isn't it possible that there could be a difference?
> When you say above, "optimization would be possible,
> with 5:3 and 6:5 separate, etc.,", it makes me think so.
> Why can't 1:2 and 2:1 be separate? Guess it's *because*
> they are the definition of "octave-equivalent" intervals?

No, it has nothing to do with that. Forget about octave equivalence.
Now, how would you make 5:4 and 4:5 separate? It makes no sense.

> What about being able to include overtone ratios as part of
> the optimization calculation?

Aren't we doing that already?

> Timbre and "octave"-register
> influence how we perceive sonance in very perceptible ways.

Yes.

> In that case, it seems to me that there *would* be a difference
> between the 1:2 and 2:1 optimizations.

I don't see that. Mathematically, it's six of one, half a dozen of
the other.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23662

> > > [Paul]
> > >
> > > Even if adaptive JI is used to make the chords pure, most
> > > music will still tend to have "pitch classes" which cannot
> > > be expressed with a single JI ratio.
> >
> >
> > [monz]
> >
> > Right, Paul... I understand this and agree. My whole concept
> > of finity (and xenharmonic bridges) is based on this idea.
>
> [Paul]
>
> I see it as different. For example, I don't hear the 256:243
> (Pythagorean limma) as a "bridge" in the sense that two pitches
> separated by this interval are the same pitch class. But when
> the limma is spread over many consonant intervals, the two
> pitches become identical, and thus each can participate in
> a larger number of consonant relationships with other notes.
> This happens, for example, in Herman Miller's 15-tET music.

Hmmm... are you saying that in Herman's 15-tET pieces the
limma is a vanishing unison-vector?

I'd still like to do a lot more thinking about xenharmonic
bridges, and hone my definition(s) of what they are. But
generally, the limma is way too big for me to consider it
to be a bridge. Pitches that far apart are generally not
perceived as the same... with some exceptions of course.

(A related example that comes to mind is one I've discussed
before which just popped up today in Neil's post: the
substitution of the flat-5 for the regular "perfect 5th" in
a chord in jazz. I say "related" because of course this
difference is not a Pythagorean limma = ~90 cents, but a
12-EDO semitone = 100 cents.)

But usually I think of bridges as really small intervals
where we really don't perceive the unique JI identites
as they are actually tuned, but rather as some other ratio
which is either implied by the harmonic context or is
more familiar because of an individual's listening experience.

As you pointed out to me, my concept of bridges is not
quite the same as Fokker's unison-vectors. I get the sense
that you're really thinking in terms of Fokker's theory
in your reference to Herman Miller's work.

Dunno... can't think about it more now. Time to go to Starr Labs.

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

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23664

> I think you're a little confused. The size of the generator
> will not have anything to do with the size of the octave.
> These are two _independent_ parameters. In neither
> optimization will the octave be expressed in terms of powers
> of the generator.

Right, Paul, I understand all that. (Thanks anyway.)

> If you want that, though, you're probably interested in one
> of the non-octave variant of the MIRACLE scales, that I
> described here:
> > >
> > > BTW, we've also mentioned non-octave variants of these
> > > scales, which simply take the 72-tET pattern 2 5 or 2 2 3
> > > and repeat them indefinitely, not stopping to repeat at
> > > the octave. These are very rich in consonant chords too.
>
> So is that really what you're interested in?

Nope... that's why I left that bit out of my quote.

I understand this too, and it would be trivial for me to
create scales this way. I'm interested in seeing what results
when we include the 2:1 in the optimization, and I can't
do that calculation.

> > Why can't 1:2 and 2:1 be separate? Guess it's *because*
> > they are the definition of "octave-equivalent" intervals?
>
> No, it has nothing to do with that. Forget about octave
> equivalence. Now, how would you make 5:4 and 4:5 separate?
> It makes no sense.

OK - got it now. Thanks.

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Hmmm... are you saying that in Herman's 15-tET pieces the
> limma is a vanishing unison-vector?

Yes!
>
> I'd still like to do a lot more thinking about xenharmonic
> bridges, and hone my definition(s) of what they are. But
> generally, the limma is way too big for me to consider it
> to be a bridge. Pitches that far apart are generally not
> perceived as the same... with some exceptions of course.

Such as when they _are_ the same! See, I think you're a little too
stuck in JI thinking.
>
> But usually I think of bridges as really small intervals
> where we really don't perceive the unique JI identites
> as they are actually tuned, but rather as some other ratio
> which is either implied by the harmonic context or is
> more familiar because of an individual's listening experience.

This is a fine way of thinking about it . . . as long as you're stuck
in JI. But the musical possibilities beyond JI are tremendous. You
can make limmas vanish and nothing awkward happens!
>
> As you pointed out to me, my concept of bridges is not
> quite the same as Fokker's unison-vectors. I get the sense
> that you're really thinking in terms of Fokker's theory
> in your reference to Herman Miller's work.

Not really. I'm just trying to get at a basic point you seem to
consistently be missing. In fact, about a year ago (it seems) I wrote
a long post to you about how my philosophical point of view differs
from yours, and you responded that you failed to see anything that
would differ from your point of view. I felt completely dried up, in
that I had attempted to be as clear as possible but still did not
appear to get through to you. One other person (I believe it was
David Finnamore) was involved and understood what I was saying.

Anyway, have fun at work today, and I'm looking forward to resuming
discussions later . . .

Monz wrote,

> I'm interested in seeing what results
> when we include the 2:1 in the optimization, and I can't
> do that calculation.

OK -- we now have to do a two-parameter optimization, where the two
parameters are the size of the generator (let's call it G), and the
size of the approximate 2:1 (let's call it O). So this requires
multivariate calculus rather than the freshman univariate kind. Also,
we'll now be minimizing the sum-of-squares of the errors of all the
intervals in an integer limit, rather than an odd limit. This integer
limit can be 7, 8, 9, 10, 11, or 12. Pick one and I'll try to work it
out (it _will_ be hairy). If you will allow me to use MATLAB, I can
perform the optimization numerically (that is, the computer will make
repeated guesses and converge on the correct solution), which will at
least reduce _somewhat_ the amount of work I have to do.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
>
> Such as when they _are_ the same! See, I think you're a
> little too stuck in JI thinking.

It's true that I was thinking entirely in terms of JI
(and possibly higher-odd-limit RI) when I formulated the
finity and xenharmonic bridge concepts. And so now when
I invoke those terms I usually *am* thinking "rationally".
:) <groan...>

>
> > <monz snipped>
>
> This is a fine way of thinking about it . . . as long as
> you're stuck in JI. But the musical possibilities beyond JI
> are tremendous. You can make limmas vanish and nothing
> awkward happens!

OK, Paul, I don't want you to feel like you're beating a
dead horse. I do understand that equal divisions and
temperaments in general have an entirely different set of
parameters concerning sonority from JI.

>
> > <monz snipped>
>
> I'm just trying to get at a basic point you seem to
> consistently be missing. In fact, about a year ago (it seems)
> I wrote a long post to you about how my philosophical point
> of view differs from yours, and you responded that you failed
> to see anything that would differ from your point of view.
> I felt completely dried up, in that I had attempted to be as
> clear as possible but still did not appear to get through to
> you. One other person (I believe it was David Finnamore) was
> involved and understood what I was saying.

I remember that. I'm sorry to be such a blockhead. I really
do try hard to understand different perspectives on tuning
theory, especially yours.

Considering that perhaps in the intervening time I have
acquired data that will now allow me to understand your
point of view, I'd like to resume the discussion from last
year if others on the list are interested. Otherwise,
should we continue it privately?

Thanks always for your help in clarification.

(And Dave F., if you remember this and have links, please
post. Thanks.)

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

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Monz wrote,
>
> > I'm interested in seeing what results
> > when we include the 2:1 in the optimization, and I can't
> > do that calculation.
>
> OK -- we now have to do a two-parameter optimization, where the two
> parameters are the size of the generator (let's call it G), and the
> size of the approximate 2:1 (let's call it O). So this requires
> multivariate calculus rather than the freshman univariate kind.

WARNING - MATHEMATICAL BUZZWORDS!

Better put the rest of this discussion on the tuning-math list.

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

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

> OK, Paul, I don't want you to feel like you're beating a
> dead horse. I do understand that equal divisions and
> temperaments in general have an entirely different set of
> parameters concerning sonority from JI.

It's much more about _scale structure_ and _chord progression_ than it is about sonority. You
could take Herman Miller's example and adaptively make all the vertical sonorities JI, without
greatly altering its effect. But in strict JI, where every note is a ratio, the progression would end
up a limma away from where it began, or else you'd have to shift some tied notes by a limma
mid-stream -- in either case, clearly not a musically viable move.

> Considering that perhaps in the intervening time I have
> acquired data that will now allow me to understand your
> point of view, I'd like to resume the discussion from last
> year

We should.

> (And Dave F., if you remember this and have links, please
> post. Thanks.)

That would help, since finding it in the archives might be like finding a needle in a haystack, to
quote a recent post by Gary Morrison.

Paul Erlich wrote:

>
>
> "Tritone substitution" is also possible in 22-tET (though diatonic
> harmony isn't). Tritone substitution in 12-tET makes use of the fact
> that the third and seventh of a dominant seventh chord are the same
> as the seventh and third of a dominant seventh chord a tritone lower
> (or higher). In 22-tET, the considerably more consonant "dominant
> seventh chords" still have this feature, since 7:5 and 10:7 are both
> represented by the same half-octave (i.e., the 50:49 comma vanishes).
> In fact, the symmetrical decatonic scales in 22-tET are _identical_
> when transposed by a half-octave, so they're inherently _bitonal_
> scales (in the sense of Stravinsky simultaneously using two
> tonalities a tritone apart).

I can successfully negotiate a very jazz blues minor ii V i, ie, ii(some form of half
diminished), v7(very altered) to i minor, say landing on A minor on the 22 tet guitar. It's a
lovely improvisation and if I can persuade a gullible keyboard player I know to tune his posh
Roland synth to 22 tet I hope to hit Wes Montgomery heaven in 22.

Regards

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23645

> Hey Monz,
>
> What I would have hoped you might have gotten from me after 3 years
> on the tuning list, is this.
>
> I believe approximating small-integer ratios is very important for
> consonance and rootedness.
>
> I don't believe that any tuning which exploits this can necessarily
> be represented in JI.
>
> **************************************************************
> Non-JI tunings take the small-integer ratio approximations and
> connect them in a way that is not possible in JI.
> **************************************************************
>
> This is the main point that seemed to be absent in your book, and
> continued to be absent in your work as recently as
> your "rationalization" of the MIRACLE scale (yes, I understand that
> you did that for _practical_ reasons, but as a matter of
> principle . . .).
>

I still think that was pretty funny (humorous) ... Monz, lighten up,
I don't mean to be derogatory in any way. That was funny... Monz'
program is SO INTENT on Just Intonation that it's missing the "forest
for the trees" so to speak!

> Even if adaptive JI is used to make the chords pure, most music
will still tend to have "pitch classes" which cannot be expressed
with a single JI ratio.
>
> I think I've spend 30% of my time on this list just making this one
> point, since it's so often ignored.
>
> So if you take anything from your interaction with me, let this be
it.

Well, I think this was probably well worth your time, since I believe
it is the most important thing I have learned in my entire time on
this Tuning List!

As far as *I* am concerned, 72-tET and the MIRACLE subsets *ARE* Just
Intonation, only a Just Intonation of a VERY PRACTICAL sort!

Practical microtonality!

__________ _________ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> That was funny... Monz'
> program is SO INTENT on Just Intonation that it's missing
the "forest
> for the trees" so to speak!

That reminds me of Daniel Wolf's way of putting something similar. He
remarked that the consonant triads of common-practice music lead many
to believe that it inhabits a "flat" JI realm, much as most ancient
cultures believed that the world was flat and perpendicular to the
direction they were standing. However, looking at the structure of
most works of common-practice music on a larger scale reveals that
the global topology is not flat at all, but curved -- the lattice
closes back on itself.

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

> Considering that perhaps in the intervening time I have
> acquired data that will now allow me to understand your
> point of view, I'd like to resume the discussion from last
> year if others on the list are interested. Otherwise,
> should we continue it privately?

You have three choices:

1 Privately
2 On this list
3 On the tuning-math list

I'll leave it up to you.
>
> Thanks always for your help in clarification.
>
> (And Dave F., if you remember this and have links, please
> post. Thanks.)

I found the links:

Message #s 16148 and 16255.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23618.html#23740

> --- In tuning@y..., jpehrson@r... wrote:
>
> > That was funny... Monz' program is SO INTENT on
> > Just Intonation that it's missing the "forest
> > for the trees" so to speak!
>
> That reminds me of Daniel Wolf's way of putting something
> similar. He remarked that the consonant triads of
> common-practice music lead many to believe that it
> inhabits a "flat" JI realm, much as most ancient cultures
> believed that the world was flat and perpendicular to the
> direction they were standing. However, looking at the
> structure of most works of common-practice music on a
> larger scale reveals that the global topology is not flat
> at all, but curved -- the lattice closes back on itself.

Guys, please understand that I *intend* for my software to
eventually be able to lattice any type of structure and any
type of tuning.

It just hasn't gotten past stage 1 yet, which is 13-limit JI
scales and Monzo and Planetary lattices.

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