The two-part Inventions.
There is an interesting part in number 13 that has always intrigued me (measures 14 to 17).
The exact chords can be interpreted several ways, because the arpeggio sequences in both hands do not define an irrefutable root. Analyzing the harmony with chord names, you may end up with something like:
Bar 14 A7b9
Bar 15 G9, G7b9
Bar 16 F#m7b5b9, F#dim7
Bar 17 E7b9.
The problem is:
A. are these even the proper names for the "chords" in question?
B. how do you account for the presence of the non-scale degrees?
C. is there any reason that one should lead or "resolve" to the next?
Analyzing the harmony in LCC-terms, to me, yeilds more interesting and insightful results.
In this piece, I found it easy to identify the LC scale and tonal order for each measure, but not so easy to identify the modal genre, since the downbeats are moving around in thirds, and the thing seems to "morph" rather seamlessly between different MG's.
The majority of the invention has F as it's tonal centre (F LCS). Applying the "Lydian #2" scale to the main theme's V-i resolution works well (FL#2 VII-III).
As Russell has pointed out, Bach seems to have exploited the 9-tone order in many circumstances, and measures 14-17 is just such an example.
My interpretation of these measures is:
Bar 14 Bb 9-tone (MG VII)
Bar 15 F 9-tone (MG II)
Bar 16 C 9-tone (MG +IV)
Bar 17 F 9-tone (MG VII)
The one exception that does not fit neatly into this scheme is the C# in measure 16 - but this note is on a very weak beat, and does not contribute to the passage's overall tonal colour. It's just part of a descending fifth that Bach uses to anchor the last beat of these measures into the downbeat of the next.
I listed the tonal order rather than a specific scale, because, those specifics depend on what other LCS degrees are present at the moment (the Lydian 2nd or 3rd will determine whether Lydian Diminished or Lydian #2 is your scale, but both represent the 9-tone order).
As for the modal genre ... perhaps you may offer another view. To my musical senses, the note on the fourth downbeat of each of these measures seems to supply a satisfactory root, but that's just my opinion.
Anyway, regardless of what modal genres are seen to be expressed here, knowing about tonal orders beyond the 7-note barrier offers an insight into music that is difficult or impossible to analyze successfully using traditional music theory and concepts. With LCC, you can at least see the overall movement of tonal centres, and account for the presence of non-scale tones; even understanding their relationship to one another and to the peice as a whole. To me, that's way cool.