Functional tunemony (had to stay on topic)

Regarding Joe Monzo's comments concerning "functional harmony" in TD.
523:6, there is a bit more complexity, I believe to the "matrix" of
functions than were described, and I guess I would have to disagree with
Paul Erlich that this is "not the list" for this kind of discussion --
since I believe our traditional harmonic practice is clearly related to
historical development of tuning systems and the implications for our
future development with tuning systems is very much linked to our
understanding of previous "functions" of harmony as well as with the
development of new "functional systems. " Regarding Joe Monzo's
description, perhaps Riemann described traditional functional harmony as
reducible to, basically, I, IV and V, but surely Allen Forte, and many
others, do not. I would say that most people referring to the traditional
term "functional harmony" also do not. There is a web of complexity that
the following chart does not, I believe, properly convey:

> VII Dominant or Subdominant
>VI Tonic or Subdominant
>V Dominant
>IV Subdominant
>III Tonic or Dominant
>II Subdominant (sometimes Dominant, I think)
>I Tonic

>So typically, in an analysis, every chord is marked either
>'T', 'S', or 'D'.

Rather than taking my word for it, let's make a few citations from Allen
Forte's _Tonal Harmony in Concept and Practice_.

"II (Supertonic) in the Major Mode -- the term supertonic indicates the
position of this triad _above_ the tonic. Functionally, however, the
supertonic triad has little relationship to the tonic triad. By virtue of
the 5th relationship II always functions as a dominant preparation."

This is, of course, your "standard" V of V relationship which is so crucial
to functional harmony. Perhaps if we were to consider functional harmony
as a string of fifths (obviously originally derived from the Pythagorean
practice) we would be getting someplace in our understanding. This is
different, however, from a "reduction" to the three primary triads.

More Forte:

"III (Mediant) in the Major Mode" [don't forget that "key" term, literally]
"It is important to notice that III does NOT serve as a dominant
preparation. On the contrary, it occurs as a SUFFIX to V. [Joey P's caps]
it also occurs as a suffix to I. THUS III IS QUITE DISTANT FROM BOTH
ELEMENTS OF THE HARMONIC AXIS, AND IS RELATED TO THEM ONLY THROUGH VI."
[NB,NB,NB!!] "The function of III in minor differs significantly from that
of III in major. Whereas in major the mediant is relatively unimportant,
in minor it is a quasi-primary triad." [This is, of course, your "garden
variety" relative major]

We've been over IV and V a bit too heavily, I believe, so let's go on to
"juicy" VI (Submediant)

Again from Forte. (Don't trust ME for a minute...):

"We have seen that VI SERVES AS A DOMINANT PREPARATION BOTH IN MAJOR AND
IN MINOR. [NB, NB!] Often it follows I immediately, providing the first
cue to progression toward V."

And, of course, our traditional gems of I-vi-ii-V-I, etc. the nourishment
of beginning harmony students everywhere, come from this Submediant.)
Note, again, however that this "functional harmony" is linked by our
Pythagorean (now "squeezed" into 12-tEt) fifth relationships.

From Forte:

"VII (Leading-Note Triad) in Major. The VII is a dissonant triad both in
major and in minor. Like II in minor, it occurs rarely in fundamental
position. When represented by its first inversion, it often stands between
I and I-6" (I 6/3 of course).

Naturally, the VII chord is a substitute for an incomplete dominant seventh
chord.

So, let's not forget that there is LOTS of "baggage" associated with
traditional functional harmony. All the traditional progressions, etc. It
is NOT reducible to three triads, but possibly could be considered
reducible to a whole string of dominant fifths. This is where the example
given by, I believe, Carl Lumma of barbershop is interesting. They are,
obviously, moving their dominant sevenths around like some kind of parallel
set theory, and are not, as Paul Erlich queried, interested in "functional"
modulation of seventh chord to seventh to seventh around the circle,
established by our 3-limit ancestors.

Here's Forte's functional harmony chart, again from _Tonal Harmony in
Concept and Practice_:
I quote:

II -- Supertonic -- Dominant preparation
III (in major)-- Mediant -- Leads to dominant preparations IV, II-6 or VI
III (in minor) -- Mediant -- Independent triad that often usurps role of
tonic
IV Subdominant -- Dominant preparation or melodic embellishment of tonic
VI (in major) Submediant -- Dominant preparation or tonic substitute
VI (in minor) Submediant -- Dominant preparation
VII (in major and harmonic minor) -- leading tone triad -- Dominant
substitute or melodic embellishment of tonic
VII (in natural minor) -- Natural VII - Secondary dominant in relation to
the mediant triad.

Unquote...

There have been various attempts to "simplify" traditional "functional
harmony" over the years. Classifying everything as I, IV or V would, to my
mind, not be a good way (!!) Others have been superior -- Heinrich
Schenker comes to mind...

Schenker was a weird bird. His book _Harmony_ is right here, and some
incredible statements "jump out" right away: "Against all traditional and
historical notions, I would go so far as to claim that even Greek music
never was real art. It can only be ascribed to its very primitive stage of
development that Greek music has disappeared without leaving any traces..."

And, even _worse_ for us here: "NO OVERTONE BEYOND THE FIFTH IN THE SERIES
HAS ANY APPLICATION TO OUR TONAL SYSTEM" "The human ear can follow Nature
as manifested to us in the overtone series only up to the major third as
the ultimate limit; in other words, up to that overtone which results from
the fifth division."

Schenker didn't even recognize the 7th partial. (No jazzer he). However,
there was SOME validity in his simplifications. Anyone who has done a
"Schenkerian analysis" can attest to this. Basically, everything is
reducible to "pitch level." Yes, the primary I, IV and V (major and minor)
were emphasized (remember, one uses BIGGER notes for the BIGGIE functional
chords and SMALLER notes for the functional chords and lines of lesser
magnitude -- like stars in the sky!), but it is always with POSITIONING or
pitch height in mind.

That kind of "positioning simplification" is a more logical dig.

Joseph Pehrson

> [Joe Pehrson, TD 525.20]
> Regarding Joe Monzo's comments concerning "functional harmony"
> in TD. 523:6, there is a bit more complexity, I believe to the
> "matrix" of functions than were described,

Joe, thanks for elaborating on what I wrote, and for quoting
Forte. As I stated (not quite this precisely) at the beginning
of my post, I have Forte's _The Structure of Atonal Music_, but
not his book on Tonal Harmony. So the quotes are much
appreciated.

However, keep in mind that I am more familiar with, and was
communicating in my post my knowledge of, the *origin* of
the concept and term 'functional harmony', with Riemann.
Forte's descriptions are a much later elaboration - or perhaps
one should call it a summary of the later elaborations of
others - of Riemann's theories.

I see nothing in any of the information you provided
that contradicts Riemann's fundamental hypothesis that all
*functions* of chords in tonal music (meaning 'common-practice'
Western-European art and popular music) are ultimately reducible
to a paradigm which places the tonic in the center as a
balancing point and the subdominant and dominant on either
side of it in opposing forces, with all other chord-roots
being categorized into one of the three 'positions'.

This is at its most basic level a recognition of the primacy
of the Pythagorean tuning scheme, or, as I said, 3^(-1,0,1).
So when you say:

> [Joseph Pehrson]
> Perhaps if we were to consider functional harmony as a string
> of fifths (obviously originally derived from the Pythagorean
> practice) we would be getting someplace in our understanding.
> This is different, however, from a "reduction" to the three
> primary triads.

I don't see the difference at all.

Certainly Riemann and Forte both mean to include other
elements in their pitch-sets, including those with 5 as
a factor, and including an extension of the Pythagorean
axis in either direction (i.e., strings of 'dominant-7th'
relationships).

(In fact, Riemann was thinking in terms of 5-limit JI, while
my guess is that Forte's assumed tuning scheme is 12-EDO/tET.)

But the fundamental dynamic in a 'string of fifths' progression
is the one portrayed in the V-I-IV paradigm.

Note that Riemann did not consider the I, IV and V simply as
'primary triads', as you called it, but as opposing forces
exhibiting a symmetrical 'pull' on the tonic.

I get the feeling that even reading Forte, your mind is more
locked into the Roman-numeral analysis than it should be.
Roman-numerals are a product of German/Austrian _stufen_-theorie.
I don't know enough about the history of this to offer further
comment, so I defer to Daniel Wolf or other experts. But
in order to understand Riemann, you need to think in terms
of the 3^(-1,0,1) paradigm.

The whole point of having the new qualifier 'functional'
added to 'harmony' is to explain the secondary chord-roots
as 'substitutes' for the three primary functions.

Some quotes you gave from Forte which support my explanation
of Riemann:

> [Forte, quoted by Joe Pehrson]
> Functionally, however, the supertonic triad has little
> relationship to the tonic triad. By virtue of the 5th
> relationship II always functions as a dominant preparation.

> [monz referring to Riemann]
> II Subdominant (sometimes Dominant, I think)

Note that the one category that's missing from my description
is that of 'tonic'. In complete agreement with Forte.

> [Forte, quoted and edited by Joe Pehrson]
> It is important to notice that III does NOT serve as a
> dominant preparation. On the contrary, it occurs as a SUFFIX
> to V. [Joey P's caps] it also occurs as a suffix to I.

> [monz referring to Riemann]
>III Tonic or Dominant

Note that V (dominant) and I (tonic) are precisely the
functions I specified.

And as I made clear in my original post, there's a whole lot
more to what Riemann had to say on this than what's in my
little abstract of it. He may himself have made many of
the points made by both Forte and yourself.

For those interested, there's been quite a resurgence of
interest in extending Riemann's theories within the last
year or two. A recent _Journal of Music Theory_ was entirely
devoted to it; the articles include lots of 12-tET adaptations
of 'our' lattice diagrams.

And also be aware that the 3^(-1,0,1) paradigm was of
fundamental importance in Schoenberg's theories. In one
place in _Harmonielehre_ he even provides an amusing little
allegory about the 'war' between the 'forces' in a simple
chord-progression.

> [Joseph Pehrson]
> There have been various attempts to "simplify" traditional
> "functional harmony" over the years. Classifying everything
> as I, IV or V would, to my mind, not be a good way (!!)

Then you simply don't agree with Riemann; or perhaps, you
don't understand that 'functional harmony' is not synonymous
with 'traditional harmony'. 'Functional harmony' refers
specifically to the simplification of everything to I, IV, or V.

> [Joseph Pehrson}
> Others have been superior -- Heinrich Schenker comes to mind...
>
> Schenker was a weird bird. ...<snip>

You can say that again! I don't find Schenker's methods
superior in any way to those of Riemann. There are certainly
very different. While I go along with recognizing the merits
of Schenker's idea of prolongation, I've been on record long
ago as disagreeing with the specifics of just about everything
else he said.

I suppose everyone whose primary interest in music is tuning
would agree that pitch is the most interesting (if not
necessarily the most important) aspect of music.

But one of the favorite games of contemporary music-theory
writing is to note how wrong it was in past theory to assume
that pitch-height held the preeminent position in a consideration
of musical analysis, and to right that wrong with new in-depth
explorations of timbre, rhythm, meter, amplitude (i.e.,
'dynamics'), and spatial location.

And of course, anyone with a serious interest in tuning will
laugh at many of Schenker's more dogmatic statements. I'm
afraid I just can't accept a lot of the Schenker-derived
modern analyses of music entirely at face value. I think
lattice diagrams explain a whole lot more, and are a lot
more 'truthful'.

-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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