Reply to Joseph Pehrson

>The idea of the just thirds piling on one another and creating the "diesis"
>and the relationship of this to
>"pumping" the pitch is pretty clear now. I also understand how this would
>be a particular problem in relation to the way Schubert structures his
>harmonies. However, if Paul Erlich would like to go into this question in
>further detail, as he suggested, I would find it very interesting...

It seems like you've got the picture now, so I don't know what you'd like me
to discuss. So I'll try to give you a sense of why meantone (and thus
Vicentino's JI scheme) works for anything in the Western repertoire without
enharmonic respellings, and what happens when you do have enharmonic
respellings. Basically, there are only two independent commas that come into
play when trying to analyze or render music of the Western conservatory
tradition in just intonation. Let's take the syntonic comma and the
Pythagorean comma as the two basic ones (any two would do). The syntonic
comma is four fifths minus a third, so we can write it as s=4v-t. The
Pythagorean comma is twelve fifths; write it as p=12v. Other important
commas:

1. schisma = -8v-t = s-p
2. diaschisma = -4v-2t = 2*s-p
3. (minor or Great) diesis = -3t = 3*s-p
4. major diesis = 4v-4t = 4*s-p

If you think there might be others, check out Mathieu's _Harmonic
Experience_, which is a thorough survey of Western harmony through JI
glasses. Mathieu does not mention the "major diesis" but he does name one
other, the "superdiesis", which is 8v-2t = 2*s. But basically, we've got it
all covered.

Structurally, the syntonic comma results in no notational change, while the
Pythagorean comma changes a flat to the enharmonic sharp (e.g., Ab->G#).
(You may want to verify this for yourself). From the equations above, we can
see that commas 1-4 above all result in a notational change opposite that of
the Pythagorean comma, i.e. they change a sharp to the enharmonic flat.

So, in any music which does not involve an enharmonic respelling, the only
comma that comes into play is the syntonic comma (and perhaps multiples such
as the "superdiesis"). As you probably know, the syntonic comma is already
an equivalence relation in meantone tuning. Therefore, this music always
"works" in meantone with absolutely no shifts or drifts. Triads in meantone
have major thirds in JI, and minor thirds and fifths 1/4-comma (5.4�) off
JI. So an adaptive JI scheme for this music could take meantone as its
starting point, tweak half of the notes by 5.4� to produce JI harmonies, and
the largest melodic shifts introduced would be 0+5.4 = 5.4�, an inaudible
melodic interval.

Once the music makes use of enharmonic spelling, however, the situation is
not so clear, but if we are to preserve the syntonic comma as a structural
equivalence in our system, Fokker's formalism tells us that we are working
with a closed 12-tone system (see my posts on Fokker periodicity blocks).
For each of the commas above, when taken with the syntonic comma as a pair
of unison vectors, define a 12-tone periodicity block:

syntonic comma & Pythagorean comma:

| 4 -1|
| | = 4*0 - (12*(-1)) = 0 + 12 = 12
|12 0|

syntonic comma & major diesis:

| 4 -1|
| | = 4*(-4) - (4*(-1)) = -16 + 4 = -12
| 4 -4|

syntonic comma & diesis:

| 4 -1|
| | = 4*(-3) - (0*(-1)) = -12 + 0 = -12
| 0 -3|

syntonic comma & diaschisma:

| 4 -1|
| | = 4*(-2) - (-4*(-1)) = -8 - 4 = -12
|-4 -2|

syntonic comma & schisma:

| 4 -1|
| | = 4*(-1) - (-8*(-1)) = -4 - 8 = -12
|-8 -1|

A geometrical way of looking at all this is to envision the infinite 2-d web
of fifths and thirds. Equivalence at the syntonic comma rolls this web into
an infinitely long cylinder, with the lines of fifths collapsed into a
single helix, and with the thirds connecting notes in adjacent turns of the
helix. This is of course the model of meantone temperament. Equivalence at
any of the other commas then rolls the cylinder into a torus, a finite
figure. The number of notes in this torus turns out to be 12.

So the only way to deal with such a pair of commas is to institute a closed
12-tone tuning; if fixed-pitch, the "wolves" of such a tuning are of course
minimized by using 12-tone equal temperament. If adaptive JI is used, one
would independently retune each chord from 12-tET to JI. 12-tET of course
has deviations of up to 16� in its triadic intervals. If you consider 2�
deviations from JI insignificant, a closed system that I proposed several
years ago on this list is the most efficient way to acheive adaptive JI that
covers all commatic situations. The scheme involves two standard 12-tET
keyboards, tuned 15� apart. For major triads, put the root and fifth on the
higher keyboard, and the third on the lower keyboard. For minor triads, put
the root and fifth on the lower keyboard, and the third on the higher
keyboard. Audible 15� melodic shifts will be quite common, but that is the
price you pay for incorporating the diesis (etc.) into the system as
intervals of equivalence. One could view this system as a modern version of
Vicentino's system, taking into account the proliferation of commas other
than the syntonic in post-18th century music.

Joseph Pehrson wrote,

>My question is perhaps a naive one... and I admit it's time to read my
>"Genesis" again, but IS THE UTONAL SERIES something in nature?? or is it
>an entire fabrication?

The short answer is, it's a fabrication -- at least as a simultaneous
phenomenon. However, in the approach to choas, oscillatory systems create
fundamental frequencies, one at a time, that are subharmonics of the normal
oscillating frequency. But the only frequencies present at a given point in
time are the overtones of the fundamental "subharmonic."

The very, very long answer is in the archives. Search for "undertones",
"subharmonic series", you get the picture.

>From what I think I am understanding, let's be moronically simplistic
>for a moment and start on "middle C." Then your UTONAL ratios would
>yield a C below then an F, then a C, then an Ab, then an F... your
>"F-minor triad..."

>But does this "inverse ratio" process really come from anything in
>nature?? Of course the "minor triad" is up the Otonal series at 6:7:9,
>but that has nothing to do with the Utonal, yes??

Right. The inverse ratio process can be said to be natural, since all notes
share a common overtone, namely the Partchian "Unity of the Utonality".

>So does the Utonal ratio series really "mean" anything?? It seems there
>is a kind of "descending series" from "difference tones," yes?? or does
>this have no relation to it???

No, difference tones tend to produce otonal series. At best, the "natural"
property of otonalities is the reinforcement of a common overtone, which you
can learn to hear, but whos relevance to music is questionable.

Complete Utonalities form a subset of the class of chords I call "saturated
chords", whose definition is closely related to the natural phenomenon of
critical band dissonance. (Partch's one footed bride is a pretty good map of
critical band dissonance for harmonic timbres [with many octave doublings],
and at the same time, a pretty good map for octave-equivalent harmonic
entropy [of dyads only]). See http://www.cix.co.uk/~gbreed/erlichs.htm for
more about utonalities and saturated chords.

>Comments are encouraged... I would like to find an "optimal" tuning for
this >piece! Joseph Pehrson

Every piece can have its own optimal tuning, and the tuning need not be
fixed-pitch -- it can be adaptive! John deLaubenfels is the man who could
maybe help you at least calculate the optimal fixed tuning, if you give him
a list of the intervals and chords you're trying to approximate.