[Smt-talk] Classical Form and Recursion

[Fred Lerdahl]

Dmitri concludes his last communication with:

> My one regret about ending this discussion here is that I was hoping 
> was to get you to explain your particular responses to these kinds of 
> issues.  I suspect that you have lots to say about these specific 
> criticisms -- that you've thought about them carefully, and have more 
> to offer than simple generalities about science or cognitive science.  
> What I've been trying to do, not totally successfully, is get you to 
> share these responses.

The tenor of these exchanges has been general from the start; I thought 
that this was expected. But I do not like to disappoint someone of 
Dmitri’s intelligence and boldness, so let me leave aside all the other 
issues (which I could answer!) and try to address in a little detail 
one topic that has been touched upon.
 
The topic is how the mind/brain registers scales, chord progressions, 
and key relationships. Begin with the fact that it is able, as a matter 
of course, to sort into coherent auditory objects the spectral smear 
that is the auditory input. (This is the subject of Bregman’s 
research.) It ought to be astonishing that the brain is able to 
construct sensations of pitches and chords when processing a Beethoven 
symphony, much less that a given line is played by an oboe and even 
that it is a line. The auditory system does this by sorting out subtle 
features and patterns in the changing waveform. All of this happens 
online, automatically and unconsciously. Signal processing algorithms, 
despite much ingenious effort, are still quite poor at pitch extraction 
and line identification. In the welter input data, it is difficult to 
determine which data points and which patterns really matter.

The perceptual step from auditory signal to pitches and chords is 
perhaps more complex than the cognitive step from pitches and chords to 
the inference of scales, chord progressions, and key relationships. If 
this seems counterintuitive, it is because hearing pitches and chords 
is so automatic that it is assumed as given, whereas levels of musical 
structure are supposedly learned in music classes. But here again 
arises the distinction between explicit and implicit knowledge. 
Cognitive scientists are largely concerned with explicating implicit 
structures and processes. These structures and processes are as 
automatic and unconscious as are the perception of pitches and chords, 
although phenomenologically they may not feel so immediate. Even those 
music theorists with a tabula rasa bent, however, take scale degrees to 
be as "real" as pitches. Gjerdingen (Music in the Galant Style, Oxford, 
2007) treats galant schemas not only with reference to pitches but also 
to scale degrees. To hear a scale degree is to be aware, however 
implicitly, of a scale and a tonic.

Now consider TPS’s basic space. A particular configuration of the basic 
space expresses a scale and the relative stability of each scale 
degree. The most stable degree is the tonic. A basic-space 
configuration also represents any given chord by the relative stability 
of its root, fifth, and third. Such a representation is supported at 
the psychoacoustic level through a combination of spectral and virtual 
pitches (harmonics and subharmonics). In this connection, Parncutt 
(Harmony: A Psychoacoustical Approach, Springer, 1989) tried, with 
mixed success, to account for harmonic and even key perception directly 
from the psychoacoustics, bypassing levels of mental representation 
that are normally employed in cognitive science and taken for granted 
in most music theories. In comparison, TPS’s basic space can be seen as 
a cognitive abstraction built on a psychoacoustic foundation. It is a 
schema that makes regular and predictable the variables in 
psychoacoustic inputs.

A steady-state auditory signal is often represented with frequency on 
the x-axis and amplitude on the y-axis. As the signal changes, the 
frequency columns bob up and down in accordance with the energy of the 
spectral pitches. Similarly, the heights of the columns of pitch 
classes in varying states of the basic space increase or decrease in 
response to progressions in a phrase. (For a demo, see Kent Williams’ 
pedagogical TPS website at http://music.uncg.edu/etps/, password 
“torus”; play around with Figure 4.2 [ignoring a few glitches].) Just 
as the amplitude of a given frequency in a waveform equals the additive 
combination of its spectral components, so the strength (or stability) 
of a given pitch class equals the additive combination of the 
basic-space levels at which it appears. The abstraction into chord 
progressions and key relationships is possible because the basic space 
has distinct levels that transform in specific ways; these are aspects 
of the schema.

This analogy suggests that configurations of the basic space might be 
treated by Fourier analysis. The idea is not to use Fourier analysis to 
address tension directly, as Krumhansl attempted a few years ago, but 
to focus on transformations of the basic space, which is but one input 
to tonal tension. If successful, this approach would help explain how 
higher levels of musical structure emerge more or less continuously 
from the auditory input. Thus we return to one of the starting points 
of these exchanges.

On this admittedly sketchy note, a full cadence--

Fred Lerdahl
Columbia University
awl1 at columbia.edu