Re: Tuning Lattices (II) (from tuning)

--- In tuning@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> >How do we know that listeners' responses are
> > > consistent with any metric at all?
> >
> >Doesn't this question cut to the heart of your own work?
>
> I would say "no." This is because in my own work (see the first four
> sections of the online materials) I take great care only to assume a
> metric that satisfies certain general features. I'm pretty confident
> that my assumptions -- for instance, that "voice crossings" don't
> decrease the size of a voice leading -- are consistent with the way
> composers thing.

Well, the original question was how listeners hear. However, I think
successful models attend to at least one of these not very distinct
factors, and you are clearly doing that. My own question is still to
what extend voice leading has to do with *distinct* voices, and that
of course really depends to a large extent on timbre and style. The
question is important for what you are doing because I think it is
important to get as much milage as possible out of R^n or R^n mod
octaves before proceeding on to orbifolds. You should at the end of
the day be prepared to show what orbifolds get you--what do orbifolds
have than R^n aint got when it comes to voice leading? R^n models
voice leading in terms of distinct voices, which I think is clearly an
important thing to do, and it does it with a much more tractable
mathematical structure, which should always be a consideration.

> I'm not sure I understand how you guys on this list use the term
> "harmonic." What does "harmonically connected" mean to you? The
> fact that the E major chord shares a common tone with a C major
> chord?

(i) E major and C major share a common tone
(ii) E--> root progression is by way of a 5-limit consonance

Formally, it also "looks like" dominant to tonic, in that instead of
3-->1 it is a 5-->1, and it also has the leading tone.

Because there are plenty of dominant substitutions that don't
> share common tones. For example, (Db, F, Ab, B)->(C, E, G, C) can
> act as a T-D progression, as can (B, D#, G#)->(C, E, G).

The first is harmonically related to C, but more remotely, sharing two
notes of the 5-limit tonality diamond around C. Moreover, it has a
leading tone.

With chords like this I think it is justifiable to lift it to the
7-limit, in which case we have a 16/15-4/3-8/5-28/15 chord. In terms
of the 7-limit tetrad lattice discussed here:

http://www.xenharmony.org/sevlat.htm

the chord is [-1,-1,-2], which has a greater harmonic distance from
[0,0,0] than the dominant tetrad [0,1,1] but not so remote as to be
utterly out of harmonic contact, I think.

Your second chord progression, (B, D#, G#)->(C,E,G) again has a
leading tone. If you spell it (Cb, Eb, Ab) instead, I think its
harmonic connection to C is clear, in terms of the tetrad lattice this
is [-2,0,-1]. In fact, the two examples are quite similar, in the
sense that both chords have a leading tone to the tonic and both share
two notes of the 5-limit tonality diamond with C major.

There are only a few 7-limit tetrads which share these two properties;
aside from the above two chords, we only have [0,-2,-1] and [0,0,-1];
in terms of septimal meantone notation these are (D, F, A, Cb) and
(Db, E, G, B), in other words the ii and iii minor tetrads. Since this
criterion is so strict it excludes the V tetrad as bening like
dominant harmony (but includes the dominant seventh chord), we might
want to relax it which nets some further chords, including a number of
major tetrads.

I have to say, these Dimitri-dominant tetrads seem interesting, and
involve both harmonic and voice-leading considerations. It may bear
further investigation.

> > I'm pretty confident
> > that my assumptions -- for instance, that "voice crossings" don't
>> decrease the size of a voice leading -- are consistent with the way
> > composers think.
>
>Well, the original question was how listeners hear.

Not my original question. As I said, music theory can do a number of things: modelling how listeners hear, modelling how composers think. I am, personally, much more interested in the composer's perspective. For me, music theory teaches you how to steal from other composers without getting caught -- instead of borrowing licks and riffs, you can borrow principles and concepts.

>My own question is still to
>what extend voice leading has to do with *distinct* voices, and that
>of course really depends to a large extent on timbre and style. The
>question is important for what you are doing because I think it is
>important to get as much milage as possible out of R^n or R^n mod
>octaves before proceeding on to orbifolds. You should at the end of
>the day be prepared to show what orbifolds get you--what do orbifolds
>have than R^n aint got when it comes to voice leading? R^n models
>voice leading in terms of distinct voices, which I think is clearly an
>important thing to do, and it does it with a much more tractable
>mathematical structure, which should always be a consideration.

I feel like we've gone over this before.

1. The advantage of the orbifolds is that they very clearly and easily model concepts that are central to our basic musical life. The elementary, Music 101, concept of a "chord" is an unordered set of pitch classes. The orbifolds represent the space of chords, considered in this sense, where distance is measured by minimal voice leading size.

2. Using the orbifolds, you can understand very obvious an simple facts about voice leading at-a-glance.

Q: What internal properties of a chord allow for efficient voice leading?
A: Near-symmetry under transposition, permutation, or inversion.

Q: If I have a two-note chord, what transposition of that chord gives the smoothest voice leading?
A: 1/2 of an octave. Example: (C, G)->(C#, F#)

Q: If I have an n-note chord, what transposition of that chord gives the smoothest voice leading?
A: 1/n of an octave. Examples: (C, E, G)->(B, E, G#) and (C, E, G, Bb)->(C#, E, G, A)

Q: How does acoustic consonance relate to efficient voice leading?
A: Consonant chords tend to divide the octave evenly and hence can be connected to their transpositions by efficient voice leading.

Q: The history of music theory has given us tons of "maps" of little portions of musical space, such as the circle of fifths, the Tonnetz, the circle of diatonic triads, etc. How do they relate?
A: They're all lattices found inside the orbifolds.

Again, I consider examples like these to be enlightening. They are what make it worthwhile to understand the orbifolds. Furthermore, you can understand individual pieces in new ways -- for example, Chopin's E-minor prelude, which moves very systematically along a lattice of 4D cubes stacked point to point, a lattice that lives in the orbifold T^4/S_4. More generally, you can uncover deep commonalities across a range of Western styles -- which from the 11th century on have exploited efficient voice-leading between structurally similar chords.

If you do not find these facts interesting, then so be it. Use R^n to your heart's content.

3. The business about independent voices is a red herring. Choose a point in the orbifold and stipulate how its pitch classes are assigned to voices. Voice leadings are represented by paths in the orbifold. Having made your initial assignment of notes to voices, the paths determine exactly how the voices move. Different voice leadings lead to different paths. There is never any confusion of voices or loss of voice independence.

> > I'm not sure I understand how you guys on this list use the term
>> "harmonic." What does "harmonically connected" mean to you? The
> > fact that the E major chord shares a common tone with a C major
>> chord?
>
>(i) E major and C major share a common tone
>(ii) E-->C root progression is by way of a 5-limit consonance
>
>Formally, it also "looks like" dominant to tonic, in that instead of
>3-->1 it is a 5-->1, and it also has the leading tone.

OK, now we're getting to the heart of the matter. Personally, I suspect that these sorts of considerations do not really explain "harmonic connectedness" or tonic-dominant function.

First, I think that chords can be harmonically connected without common tones. Take a standard I-vii6-I6 progression, for example.

Second, "dominantness" seems to be to be largely conventional -- it's a stylistic concept, not one that we're going to explain using tuning lattices.

There's a lot more to say here. But one reason I'm on this list is because I want to argue that some of the work traditionally done by these tuning lattices can be better done using the orbifolds.

The deep issue here is generalization. A lot of people want to generalize ideas found in diatonic music. One way to do this is to try to find "hyperdiatonic scales" in other tuning systems and analogous to the familiar major scale. This is, I take it, a research direction shared by many people on this list -- as well as by academic theorists such as Clough etc.

My own view is that there are musically more profitable generalizations to be had. The best analogues to the diatonic scale are familiar near-diatonic scales such as the "altered" (melodic minor ascending), octatonic, harmonic major, and so on. I think that the music based on these scales -- for instance Debussy's, Jazz, the early Stravinsky's, a lot of minimalism, etc. -- has proven to be remarkably successful. I do not think that the same can be said for "hyperdiatonic" music in other temperaments.

What I'm trying to do is explain this extended-tonal music, and to try to think about why it's been so successful.

Again, I want to emphasize that these are deep philosophical waters. I don't expect to convince anyone, really. But I do think this question is the deepest one here.

> Because there are plenty of dominant substitutions that don't
>> share common tones. For example, (Db, F, Ab, B)->(C, E, G, C) can
>> act as a T-D progression, as can (B, D#, G#)->(C, E, G).
>
>The first is harmonically related to C, but more remotely, sharing two
>notes of the 5-limit tonality diamond around C. Moreover, it has a
>leading tone.

So anything is "harmonically related to C" if it has some C major notes in it? Is there a deeper explanation here?

How about this: the chord (Db, F, Ab, B) works as a dominant substitution because it

a) contains scale degrees 4 and 7; and
b) is close to the G7 chord from a voice leading perspective.

For this reason, you can substitute it into the following progression without much disrupting the musical fabric:

(D, F, A, C)->(D, F, G, B)->(C, E, G, B)

Which, upon tritone substitution, becomes

(D, F, A, C)->(Db, F, Ab, B)->(C, E, G, B)

Again, the deeper structural mechanisms here involve the near T6-symmetry of the perfect fifth and the T-symmetry of the tritone; all of this is shown very clearly in the 2D orbifold.

>With chords like this I think it is justifiable to lift it to the
>7-limit, in which case we have a 16/15-4/3-8/5-28/15 chord. In terms
>of the 7-limit tetrad lattice discussed here:
>
>http://www.xenharmony.org/sevlat.htm
>
>the chord is [-1,-1,-2], which has a greater harmonic distance from
>[0,0,0] than the dominant tetrad [0,1,1] but not so remote as to be
>utterly out of harmonic contact, I think.

Personally, I think we can do better. How do we know these lattices have anything to do with anything? Are we doing a priori psychology here, or what?

>I have to say, these Dimitri-dominant tetrads seem interesting, and
>involve both harmonic and voice-leading considerations. It may bear
>further investigation.

Well, good. BTW, it's "Dmitri," not "Dimitri."

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

DT wrote...
>The deep issue here is generalization. A lot of people want to
>generalize ideas found in diatonic music. One way to do this is to
>try to find "hyperdiatonic scales" in other tuning systems and
>analogous to the familiar major scale. This is, I take it, a
>research direction shared by many people on this list -- as well as
>by academic theorists such as Clough etc.
>
>My own view is that there are musically more profitable
>generalizations to be had. The best analogues to the diatonic scale
>are familiar near-diatonic scales such as the "altered" (melodic
>minor ascending), octatonic, harmonic major, and so on. I think that
>the music based on these scales -- for instance Debussy's, Jazz, the
>early Stravinsky's, a lot of minimalism, etc. -- has proven to be
>remarkably successful. I do not think that the same can be said for
>"hyperdiatonic" music in other temperaments.

I'm sure I'll by now the last to point out that the scales you
mention show up as MOS of simple rank 2 temperaments. Octatonic
is 8 notes of the "diminished" temperament, which tempers out
648:625. But the "porcupine" temperament (250:243) should be at
least as good, since its associated comma is both smaller and
simpler. Listen to this piece by Herman Miller

http://lumma.org/music/theory/tctmo/mizarian.mp3

for an example. The biggest difference between porcupine and
diminished seems to be that the former isn't possible in 12-
or meantone-based temperaments.

(This discussion adapted from:
http://www.lumma.org/music/theory/tctmo/
)

>What I'm trying to do is explain this extended-tonal music, and to
>try to think about why it's been so successful.

Thus your work is of considerable interest to us here.

>Again, I want to emphasize that these are deep philosophical waters.
>I don't expect to convince anyone, really.

I don't think I'm particularly hard to convince. Just dense!

If you could rank all 2- or 3-chord progressions (return the
best 10 and worst 10, say) by the distance you're using, and show
you can do this in, say, 15-tET also, and there are no obvious
anomalies in the list, I'll convert in a heartbeat.

>How about this: the chord (Db, F, Ab, B) works as a dominant
>substitution because it
>
> a) contains scale degrees 4 and 7; and
> b) is close to the G7 chord from a voice leading perspective.

In the 7-limit harmonic lattice, this chord is rooted graph distance
2 away from the tonic, verses distance 1 for G7.

>Personally, I think we can do better. How do we know these lattices
>have anything to do with anything? Are we doing a priori psychology
>here, or what?

They seem to be somewhat free of obviously wrong predictions,
they capture common-tone relationships, and lots of popular scales
(historically) show up as compact, convex regions on them (in
fact, regions bounded by simple commas).

-Carl