Yahoo Tuning Groups Ultimate Backup tuning That little formula

I'm still intrigued with Joe Monzo's little formula 3^(-1,0,1). [TD
523:6]

Now, 3^(-1) = -3
3^(0) = 0
and 3^(1) = 3

So, couldn't we just say -3, 0, 3 and be done with it?? It's certainly a
"threethy" kind of thing...

I suppose that wouldn't give the picture of the 3-limit ratio, though (??)
Which I imagine, knowing Joe, goes off in three dimensions (! maybe 5!)

In any case, I had *NO* idea that 3^(-1,0,1) could "sum up" the entire
"common practice" period of Western music, both "classical" and "popular."
Whew. What a revelation!

Since I've garnered this, I've been humming and singing 3^(-1,0,1) all day!

Joseph Pehrson

On Tue, 15 Feb 2000, Joseph Pehrson wrote:
> Now, 3^(-1) = -3
> 3^(0) = 0
> and 3^(1) = 3

Er, no. Any nonzero number to the 0 power is 1.

--pH, whose dad is a mathematician
<manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

On Tue, 15 Feb 2000, Paul Hahn wrote:
> On Tue, 15 Feb 2000, Joseph Pehrson wrote:
>> Now, 3^(-1) = -3
>> 3^(0) = 0
>> and 3^(1) = 3
>
> Er, no. Any nonzero number to the 0 power is 1.

For that matter, 3 to the negative 1 is 1/3 anyway, not -3.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

Sorry, Joe: I was inspired by your little post to put together
this rambling response. Hope you and others find it useful.

> [Joseph Pehrson, TD 533.25]
>
> Now, 3^(-1) = -3
> 3^(0) = 0
> and 3^(1) = 3
>
> So, couldn't we just say -3, 0, 3 and be done with it??
> It's certainly a "threethy" kind of thing...
>
> I suppose that wouldn't give the picture of the 3-limit
> ratio, though (??)

Sure, -3,0,3 explains it perfectly for someone who likes to
analyze in terms of odd-limits. You're correct there.

But I (almost always) prefer the prime-limit orientation,
in which case 3^(-1,0,1) works better because it leaves room
for expansion into other prime dimensions. For example...

The 'basic' or 'primary' mediant relationships are expressible
as 5^(-1,0,1), which are equivalent in my theory to
8/5 : 1/1 : 5/4.

The secondary mediant relationships are more complicated in
this theory, because they are what cause the triangulation of
the triangular lattice formula generally used here by others
(and less often by me). What I mean is: in the rectangular
lattice formula (which resembles my formula more closely),
these secondary mediant relationships do not appear as
perfectly symmetrical to the primary mediant relationships,
but rather are offset because they contain exponents of 3.
(more on this below...)

These secondary mediant relationships (which I call 'secondary'
precisely because they include factors of 3 as well as 5,
whereas the 'primary' or 'basic' ones only have 5 as a factor)
are 5/3 : 1/1 : 6/5. Because there are more than one factor,
it's easier to notate the factors and exponents for these
relationships in a matrix than on a single line:

3^ 5^
6/5 == | 1 -1 |
1/1 == | 0 0 |
5/3 == | -1 1 |

(The '==' means 'equivalent to' because it recognizes
'octave'-equivalence: 3^1*5^-1 really equals 3/5, an
'octave' below 6/5.)

The only reason 3^(-1,0,1) and 5^(-1,0,1) work so well as
a convenient notation for those relationships is because
they make use of only a single prime-factor.

So going back to what I just said about the lattice diagrams,
we can represent the secondary mediant relationships on the
rectangular lattice (with missing ratios as placeholders) as:

3^-1*5^1
5/3 ----- -----
| | |
| | |
n^0
----- 1/1 -----
| | |
| | |
3^1*5^-1
----- ----- 6/5

On the triangular lattice the portrayal is direct:

5/3
\
\
1/1
\
\
6/5

Perhaps the difference is clearer if I also include the
Pythagorean and primary mediant relationships:

Rectangular formula:

5/3 ---- 5/4 ----
| | |
| | |
4/3 ---- 1/1 ---- 3/2
| | |
| | |
---- 8/5 ---- 6/5

Triangular formula:

5/3 5/4
/ \ / \
/ \ / \
4/3 - 1/1 - 3/2
\ / \ /
\ / \ /
8/5 6/5

It should be obvious that each approach offers its own
advantages and disadvantages.

Then, of course, to make things more complicated, *my* actual
lattice formula is neither rectangular nor triangular, eschewing
_a priori_ vector-angles and lengths, in favor of angles derived
from a circular representation of the cents-value of the
prime-factor itself, where 360 degrees represents the 'octave':

prime-angle [in DEGREES!] = MOD(((LOG(prime))*(360/LOG(2))),360)

(Actual calculations are made in radians rather than degrees.)

and vector-lengths calculated as units of the prime itself.

So, for example, in my lattice formula,

prime ~angle (degrees)
3 211
5 116
7 291
11 165
13 252

etc.; the 12-EDO semitone [= 2^(1/12)] is of course exactly
30 degrees [= 360/12].

And the vector-lengths are simply 3 units for 3,
5 units for 5, etc.

(It gets even more complicated: in order to portray the
symmetry displayed by the negative and positive exponents,
I have to use two different starting points as 0 degrees,
180 degrees apart. But I'm not going to complicate things
here with that...)

The great advantage of this approach (as I see it) is that,
in theory, no matter how complex the JI description becomes,
each ratio can be given an absolutely unique plot on the
2-dimensional face on which the lattice must be drawn.

Of course, in practice there are bound to be lattice-points
that overlap, because we can not see or print things on an
infinitely-small-enough scale to keep everything separate.

But for my purposes (i.e., the very broad historical overview
of JI theory presented in my book and on my webpages, and
my use of these ideas in my own compositions), my formula
works better for me than any others, except when the usually
simpler requirements of Tuning List postings prompt me to
use an easier-to-draw triangular or rectangular lattice.

Examples and a wordier explanation of my formula are at
http://www.ixpres.com/interval/monzo/lattices/lattices.htm

> [Joe Pehrson]
> Which I imagine, knowing Joe, goes off in three dimensions
> (! maybe 5!)

No, Joe, not at all - I think you may just be confused a
little about this.

A string of '4ths' and '5ths', which means the same as 'a
Pythagorean system' or '3-limit system', is always going to
be a linear system. That is, if one uses a formula to construct
a lattice diagram, where all exponents of 3 will be placed
in their proper position in the series, it will always be a
1-dimensional (i.e., linear) lattice.

So 3^(-1,0,1) is simply:

4/3 --- 1/1 --- 3/2

using either the triangular or rectangular lattices.

Except for the angle, it looks the same using my formula too.
If there is only one prime-factor in every ratio, it always
makes a linear 'lattice'.

> [Joe Pehrson]
> In any case, I had *NO* idea that 3^(-1,0,1) could "sum up"
> the entire "common practice" period of Western music, both
> "classical" and "popular." Whew. What a revelation!

Well, there are plenty of theorists who would argue that it
can't. But IMO, Riemann certainly looked deeply and found
something that others before him had missed.

In fact, about 150 years before Riemann, Rameau said that
the proportions 1:3:9 - yes, that's equivalent to 3^(-1,0,1)
- were the basic ones in music, or something to that effect;
AFAIK, that's the earliest statement of the idea.
(scholarly commentary here would be appreciated)

Daniel Wolf has already pointed out that the biggest hole
in Riemannian theory is its difficulty with codifying mediant
relationships, which are a result of prime-factor 5.

In fact, the difficulty more-or-less lies in the existence
of the Pythagorean and syntonic commas; the skhisma is pretty
much small enough that discrepancies in performance associated
with it aren't audible (that's *if* they exist at all), but
those associated with commas are very noticeable to 'in-tune'
ears (c.f. the infamous 'comma-pump' sequence).

Riemann himself considered his _Tonnetz_ lattice to represent
JI, but present-day theorists exploring it further use 12-EDO,
partly to eliminate the difficulties of dealing with the
commas, and partly because 12-EDO is still the standard tuning
for 'establishment' theory and they are interested in discovering
properties inherent in that tuning.

Without really having been conscious of Riemann's work as a
basis when I formulated my own theories, I realize now that
they can be seen as a sort of extension of his ideas into
other prime-dimensions.

It's thus an easy step to go from Riemann's 3^(-1,0,1) to
my 'each prime-factor is a unique dimension in the musical
fabric'. In other words, n^(-1,0,1).

...with the _caveat_ that carrying the list of exponents
out arbitrarily far will result in 'xenharmonic bridges'
(ratios within one prime-limit which are audibly
indistinguishable from those in a higher limit).

I don't claim to be the first one to make that statement
('each prime-factor...' etc.; the n^(-1,0,1) paradigm)
- anyone out there know the history on that one? - but it
certainly has become the basis of all my work at this point.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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joe Monzo wrote:

"...the biggest hole
in Riemannian theory is its difficulty with codifying mediant
relationships, which are a result of prime-factor 5."

and

"Riemann himself considered his _Tonnetz_ lattice to represent
JI..."

I think that these are two open questions.

The first I left as an open problem for theorists: How well does a set of
one dimensional functions (T,D,S) cover tonal motion over a two dimensional
lattice? Are additional terms needed to cover motion by thirds? Or can
motions by thirds be adequated described as modified T,D, or S functions?

The second is an ontological problem. Riemann, like his model Helmholtz,
knew well enough that music was mostly going to be projected onto a
temperament, yet it was described in terms of just intervals. Exactly where
the ratios occured -- in the air, the ear, the brain or only in some ideal
world -- is open.

Daniel Wolf

Joe Monzo wrote,

>Sure, -3,0,3 explains it perfectly for someone who likes to
>analyze in terms of odd-limits. You're correct there.

You mean 1/3, 1, 3?

On Monz's lattice formula:

>The great advantage of this approach (as I see it) is that,
>in theory, no matter how complex the JI description becomes,
>each ratio can be given an absolutely unique plot on the
>2-dimensional face on which the lattice must be drawn.

That's also the case with most of the types of lattices Erv Wilson designed.
Your formula shows a deficiency (IMO) already at the 5-limit, where
consonant relationships like 5/3:1/1 are farther away than dissonant ones
like 16/15:1/1. In the usual rectangular lattice, these relationships are
equidistant, and in the triangular lattice, the more consonant relationship
is closer than the more dissonant one.

> [Joseph Pehrson, TD 533.25]
>
> Now, 3^(-1) = -3
> 3^(0) = 0
> and 3^(1) = 3
>
> So, couldn't we just say -3, 0, 3 and be done with it??
> It's certainly a "threethy" kind of thing...

> [Paul Hahn, TD 535.9]
> Er, no. Any nonzero number to the 0 power is 1.

> [Paul Hahn, TD 535.10]
> For that matter, 3 to the negative 1 is 1/3 anyway, not -3.

Duh!

A few messages later in the Digest [TD 535.17] appears my
long response to this, and I quoted Joe and *still* let both
of those errors slip right by me! And *I* work with this
stuff every day...

So my sentence immediately following should first have noted
the correction that Paul made, then should say:

'Sure, 1/3, 0, 3 explains it perfectly...' <etc.>

-monz

> [me, monz, TD 535.17]
> ...the biggest hole in Riemannian theory is its difficulty
> with codifying mediant relationships, which are a result
> of prime-factor 5."
> ...
> Riemann himself considered his _Tonnetz_ lattice to
> represent JI...

> [Daniel Wolf, TD 535.18]
> I think that these are two open questions.
>
> The first I left as an open problem for theorists: How
> well does a set of one dimensional functions (T,D,S) cover
> tonal motion over a two dimensional lattice? Are additional
> terms needed to cover motion by thirds? Or can motions by
> thirds be adequated described as modified T,D, or S functions?
>
> The second is an ontological problem. Riemann, like his model
> Helmholtz, knew well enough that music was mostly going to be
> projected onto a temperament, yet it was described in terms
> of just intervals. Exactly where the ratios occured -- in the
> air, the ear, the brain or only in some ideal world -- is open.

Once again, thank you Daniel Wolf. And once again I admire
your erudition.

I mistook your statements about the first point to be a
recognized fault in Riemann's work; I stand corrected, and
I'm glad, because I was actually wondering what was so wrong
with it: to me, Riemann's 3^(-1,0,1) paradigm worked, say,
70 to 80 percent of the time.

The only serious difficulty I ever saw with it was: his
insistence that iii and vi always substituted for one of the
primary functions, whereas I often feel that they are quite
distinct harmonic _gestalts_.

About the second point: there is a consideration of exactly
this question of JI and ET in Riemann's theory, in the recent
_Journal of Music Theory_ articles.

I can't give a more complete explanation or reference now,
as I only scanned thru it once and am too busy (preparing
to move again) to visit the library now.

But that author stated quite unequivocally that Riemann
considered the relationships portrayed by his _Tonnetz_
diagram to be in JI.

While my memory of what I read does not exactly contradict
what you say in your post, I do think that the recognition
of JI ratios played a rather important role in Riemann's work;
IIRC, I think that perhaps he gradually moved from a JI
conception early in his career to one more assimilated to
ET later on.

If this was indeed the case, it's interesting that it parallels
his gradual abandonment of otonal-utonal dualism. (Note to
newbies: Riemann never used the terms 'otonal' or 'utonal';
those were coined by Partch.)

-monz

That little formula

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul Hahn <Paul-Hahn@library.wustl.edu>

🔗Paul Hahn <Paul-Hahn@library.wustl.edu>

🔗Joe Monzo <monz@juno.com>

🔗Daniel Wolf <djwolf@snafu.de>

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Joe Monzo <monz@juno.com>

🔗Joe Monzo <monz@juno.com>