Science 313

I think Julian Hook taught my freshman music theory class
at IU. He also wrote the 'editorial' on Dmitri's article
in vol. 313 of Science.

Dmitri, your article contains the statement,

"Western music typically uses only a discrete lattice of
points in this space."

Which is is an unfortunate simplification. One might
say "Western keyboard instruments and fretted strings"
instead of "Western music".

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I think Julian Hook taught my freshman music theory class
> at IU. He also wrote the 'editorial' on Dmitri's article
> in vol. 313 of Science.
>
> Dmitri, your article contains the statement,
>
> "Western music typically uses only a discrete lattice of
> points in this space."
>
> Which is is an unfortunate simplification. One might
> say "Western keyboard instruments and fretted strings"
> instead of "Western music".
>
> -Carl

I also notice you use a non-standard meaning of "inversion".
This term is usually used to describe subtraction from a fixed
value in the context of thematic transformations used in
counterpoint, or perhaps in the literature of "diatonic set
theory". 99.9% of the time it refers to octave-equivalent
transformations of a chord.

-Carl

Also, the article makes it seem as though the symmetries
involved in transposition make for efficient voice leading.
It's somewhat misleading because, if 5-limit triads are
our consonances, and we're using octave-equivalence, the
biggest interval we can cut is the 4th. So the worst-case
voice leading step is the half-fourth. But then we're
assuming 12, in which the fourth can't be divided evenly.
So the worst-case step is only a whole tone.

Furthermore, there's no psychoacoustic basis for "inversion"
being audible without extensive training (even in its
original context, applied to melodies). Try it on chords
bigger than a triad (an 11-limit hexad, for example) and see
if it still produces "fairly similar" -sounding chords.

There is evidence that chords sharing common tones sound
good, since the brain gets to re-interpret the harmonic-series
position of one or more pitches. They also lead to smaller
voice leading steps for obvious reasons, but this is probably
less important than the harmonic re-assignment.

-Carl

Replies to Carl Lumma:

>"Western music typically uses only a discrete lattice of
>points in this space."
>
>Which is is an unfortunate simplification. One might
>say "Western keyboard instruments and fretted strings"
>instead of "Western music".

Please remember that the context of this statement was my declaration that I was *NOT* limiting myself to a discrete lattice, but rather considering the full, continuous case. I wanted to the fact that I was doing something unusual, namely thinking about continuous pitch spaces.

Personally, I feel reasonably comfortable with the assertion that the vast majority of Western composers have conceived their music with regard to a discrete lattice of some kind. Notation provides us a single sharp and flat symbol.

If you're worried, then feel free to help yourself to a very large lattice -- it's still going to be discrete. The surprising thing is that understanding voice leading requires us to go beyond this discrete space to the continuous one.

>I also notice you use a non-standard meaning of "inversion".
>This term is usually used to describe subtraction from a fixed
>value in the context of thematic transformations used in
>counterpoint, or perhaps in the literature of "diatonic set
>theory". 99.9% of the time it refers to octave-equivalent
>transformations of a chord.

Sorry, I don't follow you there. I'm not exactly sure what the worry is, but I'm quite sure I used a standard music theoretical definition of pitch and pitch class inversion.

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

>Also, the article makes it seem as though the symmetries
>involved in transposition make for efficient voice leading.

Yes, indeed. This is one of the major points of the article.

>It's somewhat misleading because, if 5-limit triads are
>our consonances, and we're using octave-equivalence, the
>biggest interval we can cut is the 4th. So the worst-case
>voice leading step is the half-fourth. But then we're
>assuming 12, in which the fourth can't be divided evenly.
>So the worst-case step is only a whole tone.

The smoothest voice leading between (C, E, G) and (F#, A#, C#) involves a minor third. Voice leadings between tritone-related triads are found throughout the entire tonal corpus, for instance in the resolution of the Neapolitan sixth to the dominant.

Now you're quite right that triads are very special because all of them can be connected by very efficient voice leading. By contrast clusters such as {C, C#, D} and {F, F#, G} cannot always be connected by efficient voice leading. The whole point of my article is to explain which properties of the harmonies allow for the voice leading.

In the triads' case, the fact that they divide the octave nearly evenly allows you to find efficient voice leadings between any two transpositionally-related chords. In the case of the clusters, the fact that they divide the octave unevenly allows you to find efficient voice leadings between the chord and itself.

>Furthermore, there's no psychoacoustic basis for "inversion"
>being audible without extensive training (even in its
>original context, applied to melodies). Try it on chords
>bigger than a triad (an 11-limit hexad, for example) and see
>if it still produces "fairly similar" -sounding chords.

First of all, major and minor triads sound audibly more similar to each other than either does to, say, the fourth chord C-F-G or the augmented triad C-E-G#. You can easily elicit such similarity judgments from listeners without extensive training.

Second of all, music theory does several things. Modelling listeners' perceptions is one of them. Modelling composers' conceptions is another. The fact is, that in many musical styles, major and minor triads were thought to be similar -- they were the two consonant triads. So the task of finding efficient voice leadings between them becomes an interesting one.

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

> >"Western music typically uses only a discrete lattice of
> >points in this space."
> >
> >Which is is an unfortunate simplification. One might
> >say "Western keyboard instruments and fretted strings"
> >instead of "Western music".
>
> Please remember that the context of this statement was my
> declaration that I was *NOT* limiting myself to a discrete
> lattice, but rather considering the full, continuous case.
> I wanted to the fact that I was doing something unusual,
> namely thinking about continuous pitch spaces.

That's true, and you deserve credit for that. There's a lot
of action in continuous pitch spaces in music that's all
around us that, as you say, (almost) nobody thinks about.

> Personally, I feel reasonably comfortable with the assertion
> that the vast majority of Western composers have conceived
> their music with regard to a discrete lattice of some kind.
> Notation provides us a single sharp and flat symbol.

And double and triple sharps and flats (at least, double sharp
has its own symbol), but yes, you're right.

> >I also notice you use a non-standard meaning of "inversion".
> >This term is usually used to describe subtraction from a fixed
> >value in the context of thematic transformations used in
> >counterpoint, or perhaps in the literature of "diatonic set
> >theory". 99.9% of the time it refers to octave-equivalent
> >transformations of a chord.
>
> Sorry, I don't follow you there. I'm not exactly sure what
> the worry is, but I'm quite sure I used a standard music
> theoretical definition of pitch and pitch class inversion.

I don't think so. C E G is a root position triad. Its
first inversion is E G C', its second inversion is G C' E'.
In Bach's more exotic music, themes are sometimes inverted
by subtracting them from a fixed value. I think the
serialists borrowed this language (but unfortunately, not
much else) from counterpoint study, but I wasn't aware that
even they used it to describe simultaneities. In Partchian
theory, we have the otonal/utonal terminology for this. If
I were you, I'd just call it reflection.

-Carl

> >It's somewhat misleading because, if 5-limit triads are
> >our consonances, and we're using octave-equivalence, the
> >biggest interval we can cut is the 4th. So the worst-case
> >voice leading step is the half-fourth. But then we're
> >assuming 12, in which the fourth can't be divided evenly.
> >So the worst-case step is only a whole tone.
>
> The smoothest voice leading between (C, E, G) and (F#, A#, C#)
> involves a minor third.

By worst-case, I meant 'the worst we have to accept if we're
allowed any transposition we want'.

> The whole point of my article is to explain which properties
> of the harmonies allow for the voice leading.

I'm still reading it in between taking care of my 7-month-old,
so I shouldn't really be commenting yet. :)

> >Furthermore, there's no psychoacoustic basis for "inversion"
> >being audible without extensive training (even in its
> >original context, applied to melodies). Try it on chords
> >bigger than a triad (an 11-limit hexad, for example) and see
> >if it still produces "fairly similar" -sounding chords.
>
> First of all, major and minor triads sound audibly more similar to
> each other than either does to, say, the fourth chord C-F-G or the
> augmented triad C-E-G#. You can easily elicit such similarity
> judgments from listeners without extensive training.

What's the significance of C-F-G and C-E-G# here? Or are you
just pulling chords out of a hat? Inversion means a pair of
chords will have the same component dyads. That means something,
but not a whole lot, as my example (roughly C E G Bb D F# ->
F# Bb D F Ab C) shows.

> Second of all, music theory does several things. Modelling
> listeners' perceptions is one of them. Modelling composers'
> conceptions is another.

That's true.

-Carl

> > >It's somewhat misleading because, if 5-limit triads are
> > >our consonances, and we're using octave-equivalence, the
> > >biggest interval we can cut is the 4th. So the worst-case
> > >voice leading step is the half-fourth. But then we're
> > >assuming 12, in which the fourth can't be divided evenly.
> > >So the worst-case step is only a whole tone.
> >
> > The smoothest voice leading between (C, E, G) and (F#, A#, C#)
> > involves a minor third.
>
> By worst-case, I meant 'the worst we have to accept if we're
> allowed any transposition we want'.

Sorry, this doesn't make any sense. :(

> > The whole point of my article is to explain which properties
> > of the harmonies allow for the voice leading.
>
> I'm still reading it in between taking care of my 7-month-old,
> so I shouldn't really be commenting yet. :)

I'm done now. Very interesting stuff. Thanks for
participating!

-Carl

Carl wrote:

> I don't think so. C E G is a root position triad. Its
> first inversion is E G C', its second inversion is G C' E'.
> In Bach's more exotic music, themes are sometimes inverted
> by subtracting them from a fixed value. I think the
> serialists borrowed this language (but unfortunately, not
> much else) from counterpoint study, but I wasn't aware that
> even they used it to describe simultaneities. In Partchian
> theory, we have the otonal/utonal terminology for this. If
> I were you, I'd just call it reflection.

In Czech, we use "inversion" to mean subtraction from a fixed value. So if
you used "reflection" for this, then you'd discontinue the term
similarities. I'm not sure how this is called in Italian or German but I
think this is quite a strong decision.

For the "chord inversion", we use something which I can hardly find an
English word for -- maybe "reversion" or something like that, I'm pretty
unsure about how to translate this. - Some people, when wishing to
distinguish the two in English, make difference between "chord inversion"
and "mirror inversion". As I've studied most music theory in Czech, the term
"inversion" is very strongly tied with "mirror inversion" in my mind.

Petr

> > Sorry, I don't follow you there. I'm not exactly sure what
>> the worry is, but I'm quite sure I used a standard music
>> theoretical definition of pitch and pitch class inversion.
>
>I don't think so. C E G is a root position triad. Its
>first inversion is E G C', its second inversion is G C' E'.
>In Bach's more exotic music, themes are sometimes inverted
>by subtracting them from a fixed value. I think the
>serialists borrowed this language (but unfortunately, not
>much else) from counterpoint study, but I wasn't aware that
>even they used it to describe simultaneities. In Partchian
>theory, we have the otonal/utonal terminology for this. If
>I were you, I'd just call it reflection.

Oh, I get it now. Unfortunately, music theory in English uses the term "inversion" in two senses. There's "registral inversion," which is what you're talking about here. And then there's "pitch class inversion" (or "pitch space inversion") which is, as you say, reflection. Both senses of the term are common.
--
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