One response to that article touches on the rising line and so I continue comment here, with the caveats that (1) I am not at all knowledgeable about jazz repertoire and practices; but (2) I am wary of the ideological work being done by any applications of Schenkerian or Neo-Schenkerian models to this music.
That said, the other scholar who was engaged with Schenkerian analysis of jazz over a long period of time is the late Steve Larson. Martin and Larson seem to me to have carried over the 1970s' era differences with respect to tonal analysis between Princeton and the New York/Yale axis, the one more methodologically liberal and composition-oriented, the other more methodologically conservative and musicology-oriented. As that survives into the present here, it is mostly about different attitudes towards the Schenkerian background constructs and the rules or heuristics for their derivation.
In Music Theory Online (18n3, September 2012--link to the issue), a memorial issue for Larson, Mark McFarland describes the basic differences well: Larson, he says, "strove as much as possible to approach jazz using as orthodox a form of Schenkerian theory as possible" (¶1), where Martin was willing to entertain a "list of modifications to Schenkerian theory" based on the view that conventional readings may be unconvincing and thus alternatives should be developed that could "provide superior readings” (¶2). The core of the difference in practice is that Larson insisted on reducing dissonances, including the most characteristic dissonances of jazz, to traditional consonances, where Martin insisted on the inviolability of the tune, which not only preserved dissonances but provoked readings with non-traditional backgrounds.
McFarland's critique of Martin's analysis of "Four" is in ¶¶5-10 of his article, and I refer the reader to those paragraphs for details. McFarland starts by noting that the analysis "is, in some ways, the least controversial graph in Martin’s study as it ultimately reduces to the ascending Urlinie." Further, "While I have no problem with the ascending Urlinie, . . . I question the scale degrees at which [this] Urlinie is interrupted, the number of appearances of interruption within this opening chorus, and the radically different reading of the two halves of this antecedent-consequent period" [¶5].
Note that, although McFarland uses the term "ascending Urlinie," he has notated the alto voice with the background's open notes and the "4-zug" with beamed closed notes. Thus, in fact the ascending cadence gesture, according to him, belongs to the middleground, not the background. This is pretty much what William Rothstein, whom McFarland cites, said in 1991. Note also that the alto is given priority at bar 9 -- this reminds me of Channon Willner's privileging of the alto in Baroque music (see an earlier post on this blog), but where I found that plausible, I have trouble seeing the justification here.
Finally, I have isolated the same elements from McFarland's transcription (it is Example 1 in the article examples file on MTO). The boxes indicate the specific elements included in my "short" version of McFarland's graph (the first example in this post). The asterisks mark the only two chords that are the same. That alone suggests to me that the "underlying consonances" approach threatens to distort not only the surface of the music but its foundations. All this connects, not surprisingly, to the theoretical problem of variations: does the structure of the theme underlie all the variations, or is it the musical task of the variations to "rewrite" the theme? That can't be answered with any simple pronouncement.
Further citations:
William Rothstein 1991. “On Implied Tones.” Music Analysis 10, no. 3: 289–328.
Steve Larson 1998. "Schenkerian Analysis of Modern Jazz: Questions About Method," Music Theory Spectrum 20/2, 209-241.
Steve Larson 2009. Analyzing Jazz: A Schenkerian Approach. Hillsdale, NY: Pendragon Press.
Henry Martin. 2011. “More Than Just Guide Tones: Steve Larson’s Analyzing Jazz—A Schenkerian Approach.” Journal of Jazz Studies 7, no. 1: 121–44.