Yahoo Tuning Groups Ultimate Backup tuning complexity, ambiguity, and the lattice

Think of Keenan's "musical complexity" as it relates
to the model of musical pitch resources on a
multi-dimensional harmonic lattice:

9
/
11 /
| / _13
| 3 _.-'
| / _.-'
5 _ | / _.-'
'-._ |/_.-'
1------------7

This work is getting at the essence of what I've
always felt in my theory of prime-factors, but
couldn't explain adequately.

We'll start with some basic assumptions,
so that I don't get criticized on those points.

Assume first of all that in harmonic music,
we will be guided in our listening by the
comparison of pitches as small-integer ratios.
This is not to deny the validity of using
equal-temperaments, meantones, or other more
bizarre tunings, but "just" to say that it is
pretty much accepted now that we hear harmony
in terms of the musical gestalts of intervals
of or near the size of the small-integer ratios. [1]

Assume that in harmonic music, that is, with more
than two tones, which will be our subject matter
here, Parncutt's theory of "tonalness" has significant
validity. This theory states that a collection
of tones heard together will strongly imply segments
of a harmonic series and its fundamental, even if
the actual pitches belonging to that set are not
present. In a case where the the ratios of the
tones in the chord imply one series, and the
combination (summation and difference) tones imply
another, a sort of "best fit" fundamental will be
perceived. The lattice is a good generalized model
of his proposition.

Assume also that I am not taking a stance here in
the prime/odd dichotomy, and that when I say
"prime" it can just as well stand for "odd" with
the mathematics adjusted. I *do* believe that
prime-factoring is important in our listening
habits, but that's tangential to my main point
here, which is the mental process itself and the
ambiguity involved in it; what follows will work
similarly for non-prime odd-factors; I will stick
to a description in terms of prime-factors.

The last assumption is that of "octave"-equivalence.
Though scales can be constructed which do not
have "octaves", and good music can be written
with them, most harmonic systems take "octave"-
equivalence for granted, and I will do so here
also. This simply ignores factors of 2, and
if necessary, 2 can be included in the theory
in the same way as the other primes.

Now down to the matter at hand.

Start with a centrist view. Our hearing mechanism
allows us to recognize sounds from roughly 20 Hz to
roughly 20,000 Hz. There are sounds at either end
which are audible to some other animals but not to
we humans. So we can't reference to any external
absolute "C", for example. (perhaps we can do this
in notation, but not in actual listening)

There has to be one specific point from within our
range of hearing which we use as a reference
and from which we will recognize all other pitches.

To a large extent, we've already produced "middle-C"
as this reference, and there is also widespread use of
tuning to "A" at or somewhere near 440 Hz. Regardless
of how well-defined it is, we take some pitch as our
reference 1/1. Were the "octaves" represented on the
lattice, they would simply be straight vertical lines:

upper limit of hearing

/|\
|
4:1
|
|
2:1
|
__|__
| 1:1 |
|_____|
|
1:2
|
|
1:4
|
\|/

lower limit of hearing

Ignoring the "octaves", and modelling our understanding
of the musical harmony as a prime-factor lattice,
we tend to think of harmonic expansion from our
reference 1/1 in two main ways. One model is a chain
or cycle of powers of 3, or "steps" of a "5th", on a
few low-prime axes:

1)
====

3-axis alone
27:16
/
/
/
9:8
/
/
/
3:2
/
/
/
1:1
/
/
/
4:3
/
/
/
16:9
/
/
/
32:27

- this example is a type of Pythagorean "major" scale,
just a chain of 3/2s, or powers of 3. All intervals
will also be 3-limit intervals: 3/2s, 9/8s, 27/16s,
and their complements and combinations.

2)
======

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

- this is the standard 5-limit "just major" scale.
It has a chain of three 3/2s on the 3-axis, and two
3/2s on the 5-axis, giving the following intervals:

five 3/2s
three 5/4s
three 9/8s
two 10/9s
two 6/5s
one 40/27 (9:8 to 5:3)
one 32/27 (9:8 to 4:3)

and so on.

The other way of modelling an expansion from 1/1 is
assuming the notes have various higher-prime factors
with the lowest possible exponents along all prime axes.

9:8
/
11:8 /
| /
| 3:2
| /
5:4 | /
'-._ |/
1:1------------7:4

- this example is an 11-limit Otonal, or "major", hexad,
giving identities 1-2-5-7-9-11. It provides two 3/2s,
one 9/8, one 5/4, one 7/4, one 11/8, and the intervals
resulting from their combinations. The factors of the
collection are the primes 3, 5, 7, and 11, each to the
first power, and the composite odd number 9, which is
3 to the second power.

Actually, the way we perceive harmony is usually a
combination of these two models, the chain of exponents
and the prime-factoring, each one being employed
to varying degrees for different people and circumstances.
The chain model tends to have a lateral orientation,
while the prime model is overwhelmingly centric.

With the recognition of the xenharmonic bridges, the
interchange between the two is dynamic and flexible.
It's largely how both classical and Partchian tonality work.
(Yes, Partch took advantage of xenharmonic bridges too.)

When we imagine that the ratios we're hearing wander
out among the axes of the higher primes, say 11, 13,
or 19, their exponents along the higher-primes axes
stay as closely as possible to 1/1, usually right
along the "harmonic spiral" [2] - in other words,
whatever prime-factor, to the 1st power; that is,
just the prime (or odd) harmonics themselves.

This is a vivid description on the lattice of Parncutt's
"tonalness". The "best fit" case can be thought of as
a whole series of xenharmonic bridges connecting to
the central harmonic spiral of the "best fit" fundamental,
from high among outlying axes, all pulling in different
directions as they force our perception of the sound
into the center of the lattice, as it were.

Shown in stages, the expansion of harmonic resources
from 1 to 3:

3:2
/
/
/
1:1

to 5:

3:2
/
5:4 /
'-._ /
1:1

to 7:

3:2
/
5:4 /
'-._ /
1:1------------7:4

to 11:

11:8
|
| 3:2
| /
5:4 | /
'-._ |/
1:1------------7:4

to 13:

11:8
| 13:8
| 3:2 _.-'
| / _.-'
5:4 | / _.-'
'-._ |/_.-'
1:1------------7:4

etc.

These diagrams of the harmonic chords remind
me of molecules that are open-ended and ready
to bond with others. Each identity of the chord
is ready to assume a new identity of another
chord if one comes by which has the right connectors
open.

This is certainly part of the reason why the development
of higher identities in harmony about a century ago led
to the "breakdown of tonality". A harmonic module is
rife with a chameleon-like potential to change its
"appearance" and conform to a new host harmonic block.

Paradoxically, the harmonic modules also have a very
centrist aspect themselves. Their vectors all meet
at 1/1, or whatever happens to be functioning as the
local 1/1. There is therefore much ambiguity built
right into this structure.

When our perception of the ratios has their lattice
points wandering farther away from this centrist model,
deviating off the prime axes themselves, they
are usually still imagined as being close by,
on a composite axis. Thus the higher-prime axis
is replicated at the distance of one or more of
the lower-primes, and so on, for however many
factors the ratio has. For example,

45:32
/ '-._
/ 9:8
/ /
15:8----------105:64
/ '-._ / / '-._
/ 3:2---/-------21:16
/ / / /
5:4-----------35:32 /
/ '-._ / / '-._ /
/ 1:1----/-------7:4
/ / / /
5:3-----------35:24 /
/ '-._ / / '-._ /
/ 4:3---/--------7:6
/ / / /
10:9-----------35:18 /
'-._ / '-._ /
16:9-----------14:9

This example shows two new chains of 3/2s: one
on the 7-axis and another along the 7*5 composite
axis (the chain with 35/18, 35/24, etc.).
This is simply the result of applying traditional
ideas of "major" and "minor" chords and scales
(i.e., up to 5- and possibly 7-limit) to the new
higher-prime axes.

Even along the higher-prime axes, we still tend to
think in terms of a chain of powers of 3. For example,
while avoiding all ratios having 5 as a factor, La Monte
Young, in the lattice of his tunings, still deploys
chains of powers of 3 among the 7-limit ratios.
Using the same lattice structure we've been developing
so far, here is La Monte's current tuning for
"The Well-Tuned Piano": [3]

567:512
/
/
/
189:128------1323:1024
/ /
/ /
/ /
9:8------------63:32--------441:256
/ / /
/ / /
/ / /
3:2------------21:16---------147:128
/ / /
/ / /
/ / /
1:1-------------7:4-----------49:32

Even in many very large just-intonation systems, such
as the one devised on a paper keyboard model by Henry
Poole [4], with 100 different pitches to the "octave",
the ratios are in only 5 chains (the 1-, 5-, and 7-axes
and their combinations) of up to 20 powers of 3 - a good
example of what Partch called "Poly-pythagoreanism". [5]

Of course, by modelling the harmonic system on the
lattice, it's very easy to see that we could build
a chain of 5/4s, or 7/4s, or any other series of
exponents along any of the prime axes.

And when those higher-prime ratios become "roots"
of chords themselves, other tones in the chord tend
to stay as close as possible in lattice-distance,
along the 3-, 5-, or sometimes 7- axes.

Think of the primary prime vectors as being in
bold on the lattice diagram, with all other composite
axes a thinner weight. I use double lines here in
ASCII to emulate bold along the primary 3, 5, and 7
vectors:

45:32
/ '-._
/ 9:8
/ // '-._
15:8 // 9:5
/ '-._ // /
12:7----/--------3:2-----------21:16
/ / // '-._ / /
/ 5:4 -._ // 6:5 /
/ / '-._'- // / /
8:7====/========1:1'============7:4
/ / // '-._'- / /
/ 5:3_ // 8:5 /
/ / '-._ // / /
32:21---/--------4:3------------7:6
/ // '-._ /
10:9 // 16:15
'-._ // /
16:9 /
'-. _ /
64:45

The *pattern* of the primary axes is replicatable
at any other point on the lattice, and ordinarily
(holding to the JI view) this is how modulation
and transposition both work - keeping the same
tonal pattern, but moving it to another area of
the lattice. For example, here is a shift in tonal
center from 1/1 to 5/3:

45:32
// '-._
// 9:8
// / '-._
15:8 / 9:5
// '-._ / /
12:7----//-------3:2-----------21:16
/ // / '-._ / /
/ 5:4 / 6:5 /
/ // '-._ / / /
25:24 8:7---//--------1:1------------7:4
'-._'-.// / '-._ / /
/ '5:3=========35:24 _ 8:5 /
/ // '-._'-/ '-._ /
32:21---//--------4:3-._---------7:6
// / '-._ /
10:9 / 16:15
'-._ /
16:9

- I see Schoenberg is once again a subject of
discussion on the List. Good - he was already
a part of this essay. - Note right here that his
"Theory of the Regions" [6] is entirely centrist
and monophonic in the Partchian sense, and in
the sense of my harmonic lattice diagrams.

In his theory, a modulation into a new key
never negated the original key - it was felt
rather as a region with a distinct relationship
to the original key. Very much like moving
around on the lattice.

In a more profound sense, this is also the basic
idea behind the technique for composing with Erv
Wilson's Combination Product Sets, such as the
Hexany, which tend to eliminate a central "fundamental"
and replace centric tonality with a kind derived
from the combined intervallic relationships among
all tones in a specific set but without referring
to a center. [7]

Now we're approaching the more interesting stuff.

A few weeks back there was discussion here of
what truly new things could be done in music.
Perhaps this is where the new directions in harmony
and tonality are going - imagining rational
harmonic implications along different prime axes
other than 3 and 5.

This is *exactly* what Schoenberg had in mind
[I have references!], but he was willing to accept
the amount of error in the 12-Eq scale and make the
*utmost use of its ambiguities*.

He said in 1911 that "5 is already far behind us",
in a polemic against Schenker's mystical speculations
about keeping a 5-limit in harmony. [8]

Schoenberg explicitly stated that "in 12-tone compostion
...almost everything that used to make up the ebb and
flow of harmony [is], as far as possible, avoided".
In other words, his aim was to discard all the 3- and
5-limit rational implications - and "to use only the
new resources" of 7, 11, and 13.

Schoenberg defined the 12-Eq scale as a conglomeration
of the rational implications of the first 13 partials
of the "tonic", "dominant", and "subdominant" tones,
so he kept the chain of 3/2s only in its most basic
sense: [9]

33:32
/|
/ | _39:32
/ | _.-' /
11:8 | _.-' /
/| | _.-' /
/ |15:8 | _.-' _13:8
/ /| '-._ | _.-' _.-' /
11:6 / | 3:2--------_.-'21:16/
| / | / _.-' / /
|5:4 | / _.-' /13:12
/| '-._ |/_.-' _.-/'
/ | 1:1--------_.-'-7:4
/ | / _.-' /
5:3 | / _.-' /
'-._ |/_.-' /
4:3-------------7:6

This is my diagram of the famous analysis of the
rational implications of the 12-Eq scale by
Schoenberg which was ridiculed by Harry Partch. [10]

Compared to the lower-prime lattices, look at
the beautiful new geometries that come into play
with Schoenberg's scheme!

So his intention was to stop implying ratios with
3 and 5 as factors, while emphasizing the implication
of those of 7, 11, and 13. Of course he wasn't trying
to stick to this rule dogmatically, but his invention
of "pantonality" (the term he himself preferred to
the more current "atonality") was a way to make
use of these previously unexplored implications
in the 12-Eq scale. As he put it, "at the root
of this is the unconscious urge to try out the
new resources independently". [11]

Of course, from an acoustical standpoint Partch's
criticisms are entirely valid. Schoenberg's
"bridges" are far larger intervals than any that
I would call by that term, with the most absurd
pair of "equivalents" being 33:32 and 13:12, tones
which are separated by 85 cents! But there was a
very sound basis to Schoenberg's retention of 12-Eq,
as we will find in a moment.

So taking out of our lattice not only the 5-axis
(as in La Monte Young's tuning), but also the
3-axis, leaves us with, as our basic harmonic
building-blocks:

11:8
| 13:8
| _.-'
| _.-'
| _.-'
| _.-'
1:1------------7:4

(I should also mention that Schoenberg may have
felt 9 to be a legitimate new identity, as do the
odd-limit proponents, rather than simply another
power of 3. He does not explicity say one way or
the other. His actual model what the "possibly
uncertain" overtone series, implying mainly an
integer-limit approach.)

Let's build part of a lattice with only these elements,
"just" to see what we get:

121:64
| 143:128
| _.-' |
| _.-' |
| _.-' |
| _.-' |
11:8----------77:64 |
| | 13:8-----------91:64
| | _.-' _.-'
| _.-|' _.-'
| _.-' | _.-'
| _.-' | _.-'
1:1------------7:4-----------49:32

You get the idea. I leave it to you, dear reader,
to expand the lattice further, and to calculate the
cents values of the ratios in your scale.

With so many composers, performers, theorists,
*and mathematicians, psychologists, therapists, etc.*,
interested in tunings these days, we're bound to
discover many of the sounds and feelings Schoenberg
could only imagine.

One of the most exciting new ideas in music to me
today is polymicrotonality - using many different
tunings in the same piece. This means that
we are *not* keeping the same pattern from
one key to another, but indeed hearing different
keys *and* different patterns all the time.
They give the opportunity to compose new
patterns among their differences. This is
also exactly what Schoenberg had in mind, but he
certainly *tempered* the idea in practice.

Now this is getting to the crux of the matter.

This is exactly the kind of thing Schoenberg was thinking
when he decided to adhere to the 12-Eq scale and use
each of its tones independently. The overwhelming,
even unimaginable, multiplicity of relationships and
ambiguities in the "lattice" he imagined was too much
to try to deal with in a practical way.

It was suggested to Schoenberg that he try the
53-Eq scale, as it had such good approximations of so
many "important" ratios. But he did not accept it,
and no wonder - he could already imagine so many uses
for the ambiguity of rational implication in 12-Eq that
he either didn't want the precision of 53-Eq or he
didn't feel that the cumbersomely large new scale would
be worth it, or both.

But today we have all kinds of new tunings at our disposal,
and a harmonic theory that is becoming ever more
sophisticated and precise. I believe that Erlich's work,
and particularly the latest developments by Keenan on
"musical complexity", are very important as they apply to
the ideas I've discussed here. We may be leaving
Schoenberg's ambiguities behind, but it's opening up a
whole new world of sound for us to play in.

- Monzo

===================
NOTES

[1] see Paul Erlich's "Harmonic Entropy" at
http://www.ixpres.com/interval/td/entropy.htm
for an interesting new development of this idea.

[2] There is an excellent diagram of the harmonic
spiral on the cover of Xenharmonikon 6, by Erv Wilson
- "the harmonic series expressed as a logarithmic
spiral". This particular diagram of Wilson's has
more in common with my lattice diagrams than any
other musical lattice diagram I've seen, although
it does not express prime-factoring.

It does express odd-factoring and octave-equivalence
very well, and certainly portrays one kind of
"musical complexity" - perhaps something already
formulated by Wilson mathematically, and if not,
something else for Keenan to add to his spreadsheet!

[3] see Kyle Gann, "The Outer Edge of Consonance",
in _Sound and Light_, p 167.

[4] See Ellis in Helmholtz, _On the Sensations of Tone_,
appendix 20, p 474-479. I will be putting up a webpage
about Poole's keyboard when I return to San Diego.

[5] Another example can be seen in my analysis of
Indian tuning:
http://www.ixpres.com/interval/monzo/indian.htm

Although my point in that paper was to emphasize
that the extended chain of 3/2s implied a 5-limit
system, that 5-limit system itself is still based
on long chains of 3/2s.

[6] Schoenberg, _Structural Functions of Harmony_.

[7] see "The Wilson Archives" at:
http://www.anaphoria.com

[8] Schoenberg, _Harmonielehre_, quoted from English
translation _Theory of Harmony_, p 318.

[9] Schoenberg, "Problems of Harmony" in
_Style and Idea_, p 271.

[10] Partch, _Genesis of a Music_.

[11] Schoenberg, _Style and Idea_, p 207.

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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complexity, ambiguity, and the lattice

🔗Joseph L Monzo <monz@xxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗David Beardsley <xouoxno@xxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>