Can you summarize Westergaard's approach? I know why the typical methods are bad, but I'm interested in what exactly his alternative is.

In ITT itself, Westergaard offers the following summary (p.375):

1. we can generate all the notes of any tonal piece from the pitches of its tonic triad by successive application of a small set of operations, and moreover

2. the successive stages in the generation process show how we understand the notes of that piece in terms of one another

(This, of course, is very similar to the methodology of theoretical linguistics.)

Westergaard basically considers tonal music to be a complex version of species counterpoint --- layers upon layers of it. He inherits from Schenker the idea of systematically reversing the process of "elaboration" to reveal the basic structures underlying a piece (or passage) of music, but goes even further than Schenker in completely explaining away "harmony" as a component of musical structure.

Notes are considered to be elements of lines, not "chords". They operations by which they are generated within lines are highly intuitive. They essentially reduce to two: step motion, and borrowing from other lines.

A key innovation of Westergaard is to unify pitch-operations and rhythmic operations. Every operation on pitch occurs in the context of an operation on rhythm: segmentation, delay, or anticipation of a timespan. This is arguably implicit in Schenker (and even in species counterpoint itself) but Westergaard makes it explicit and systematic. Hence he arrives at his "theory of tonal rhythm" which is the core of the book (chapters 7-9).

The table of contents, at the level of chapters, should give you an idea of how different Westergaard's book is from other texts:

Part I. Problems and Assumptions

1. What are we talking about?
2. Notes
3. Lines

Part II. A First Approximation: Species Counterpoint

4. Species counterpoint
5. Simple species
6. Combined species

Part III A little closer to the real thing -- a theory of tonal rhythm

7. Notes, beats and measures
8. Phrases, sections, and movements
9. Performance

Appendix: Constructing a pitch system for tonal music

EDIT: 1,2,3 under Part II and Part III should be 4,5,6 and 7,8,9 respectively, which is what I typed. I mostly like the comment formatting system here, but that is one hell of a bug.

EDIT2:: fixed.