Re: Tuning Lattices (II)

Moved from tuning.

Dmitri Tymoczko wrote:

>>Normally, when Gene talks about lattices, I think he assumes the norm
>>measures harmonic distance. That's a lot woolier, but it's also one of
>>the main things you get from a group of intervals that you don't get
>>from a continuous, one-dimensional pitch space.
> > Well, yes and no. (Note, by the way: a continuous pitch space is > going to have a continuous group of intervals associated with it.) > It's remarkable that the configuration space of a series of points in > a one-dimensional pitch space is going to get you surprisingly close > to familiar notions of harmonic distance. It's not clear to me that > we need traditional tuning lattices for this purpose.

Configuration space? You can use a continuous dissonance function. Harmonic lattices are a simple way of specifying the basic consonances.

> The orbifolds I talk about in my paper are just configuration spaces > of one-dimensional pitch and pitch classes. Each point in one of my > orbifolds represents a configuration of points on the one dimensional > spaces. It's a very cool fact that many common tonal notions of > "harmonic distance" fall out of these spaces automatically.
> > For example: if you look at the 12 diatonic scales in the > 7-dimensional space, they are naturally ordered by voice-leading > proximity into a circular chain, where each scale shares 6 notes with > its neighbors. This is just the circle of fifths.

That isn't something that falls out naturally from the geometry. Diatonic scales are very unusual musical objects in that they're related by fifths in this way. In general terms, the optimal interval for transposing an MOS scale is the generator. For an octave period, that always means that one note changes. But you can generate a scale from any interval you like. There's nothing about fifths that makes them unusually good intervals for generating a scale unless you consider that they're also strong consonances.

It's certainly a good thing that the properties of a diatonic scale are in accord with harmonic principles. If you go out looking for alternative diatonics this is something you need to think about. And that's the business we're in -- alternative tunings and alternative harmonies to back them up. If it's good for modulation distance to tie in with harmonic distance then we'd better go out looking for scales with that property. That means an MOS generated by a consonance. Or, more generally, a regular temperament with a low complexity.

Possibly this is why tuning theorists need to be aware of you work even if it doesn't touch explicitly on tuning. Ideally, considerations of melody or voice leading will lead to archetypal consonances falling out of a scale. But finding the right scales to do that is difficult and it helps to know what properties are important. Currently the alternative diatonics project is stalled because we don't have a large enough data set. We know the diatonic scales are good, but what else? More research into what properties of diatonic scales are useful in practice will be useful. Not as useful as more music in alternative diatonics, but every little helps.

Incidentally, relating scales by the notes they have in common is similar to Rothenberg's efficiency. I think his work is more closely related to yours than tuning lattices are, although it's not what this thread's about. He's generally in favor among tuning theorists but ignored by academics. The diatonic set theorists are slowly reinventing his ideas. But don't let that put you off.

> An exactly analogous structure exists in three-dimensional space. If > you use the diatonic metric on that space, you find that voice > leading proximity groups the seven diatonic triads into a circular > chain whose members share two notes with each of their neighbors. In > C major:
> > C - a -F - d - b dim - G - e - [C]
> > The analogy between this "circle of triads" and the traditional > "circle of fifth-related major scales" is very, very deep. (I talk > about this in Appendix I of my "Scale Networks and Debussy," on my > website.)

In that a triad's a third-generated MOS in 7-equal by the looks of it. Follow this logic and you get neutral third scales (or "dicot" I think the temperament's called).

What's this "diatonic metric"? It looks suspiciously like a distance in a lattice defined on scale steps.

For anybody else without Flash, that Debussy paper's at:

http://music.princeton.edu/~dmitri/scalesarrays.pdf

> It's quite possible to construct a very useful notion of tonal > harmonic distance out of these two structures. Again, I think it's > really interesting to ask whether these continuous orbifold > structures can do some of the work we've traditionally used tuning > lattices to do.

Where harmonic lattices are used to show relationships between chords, then yes, there is an overlap. Lattices as arrays of notes on paper are quite useful for finding chord sequences with common notes.

Oh, and I think it's a general property of a reasonable harmonic lattice that, if octaves (2:1), fifths (3:2) and thirds (5:4 and 6:5) are consonances then fourths (4:3) have to be as well. That means that such lattices aren't obviously useful for explaining common practice harmony, and may explain why they aren't traditionally used for much at all. Perhaps voice leading considerations can explain this anomaly.

>>As your paper's being hyped as putting music theory
>>on a scientific footing I hope we'll be seeing more of these kind of
>>considerations.
> > Personally, I would reject any hype of this sort! I think the paper > is noteworthy because it provides a new and interesting set of models > for thinking about music. I don't think of myself as doing anything > different, methodologically, from what theorists have always done.

It's at least promising that there's a desire for a more systematic theory, and perhaps we'll see funding for statistical or experimental research in the future.

>>You may be wondering what a torsor is. I chased it down to here:
>>
>>http://math.ucr.edu/home/baez/torsors.html
>>
>>and the link near the bottom. It looks like if intervals are a group
>>then notes are a torsor.
> > This site also discusses torsors in the context of music:
> > http://www.math.ucr.edu/home/baez/week234.html
> > BTW, one thing I think is cool is that you needn't take your space of > notes to be a torsor of your group of intervals. It's perfectly > possible, for instance, to let the group of real numbers act as > intervals on circular pitch space. This allows you to group together > octave-related pitches while still distinguishing motion by an > ascending octave from motion by a descending octave.

Does it? How does that work?

Graham

Graham wrote...
>Where harmonic lattices are used to show relationships between chords,
>then yes, there is an overlap. Lattices as arrays of notes on paper are
>quite useful for finding chord sequences with common notes.
>
>Oh, and I think it's a general property of a reasonable harmonic lattice
>that, if octaves (2:1), fifths (3:2) and thirds (5:4 and 6:5) are
>consonances then fourths (4:3) have to be as well. That means that such
>lattices aren't obviously useful for explaining common practice harmony,
>and may explain why they aren't traditionally used for much at all.
>Perhaps voice leading considerations can explain this anomaly.

4:3 certainly shows up on most tuning lattices...?

-Carl