1. For full details, see Pople, `Using Complex Set Theory for Tonal Analysis', n. 1.
2. I hope, in due course, to be able to comment and stimulate debate on the
3. The later dating of the file is probably simply a case of the computer registering a
later date because the file was opened for some reason and a small change made
that was later discarded.
4. Anthony Pople, Getting Started with Tonalities, unpublished handbook available
at http://www.nottingham.ac.uk/music/tonalities/, p. 7.
5. Carol Krumhansl, Cognitive Foundations of Musical Pitch (Oxford: Oxford
University Press, 1990).
6. In preparation for working through these extracts the reader is encouraged to
download a copy of Tonalities, and of the files associated with the examples
discussed here. These are mounted on the University of Nottingham web site:
http://www.nottingham.ac.uk/music/tonalities/. It is also assumed that the reader
has access to scores of the relevant extracts.
7. Readers without access to the Tonalities web site may find it useful to inspect the
screen shot of a Common-Practice Language setting in `Using Complex Set
Theory for Tonal Analysis', Ex. 11.
8. Pople's remarks are from the `OXMAC' presentation (retained in various
subsequent drafts). Pople's original text refers to a half-diminished seventh being
reported. However, when the latest version of Tonalities is run it reports an
incomplete diminished seventh as shown in Fig. 2. The discrepancy may result
from developments in the software subsequent to the `OXMAC' presentation.
9. The `prolongation filtering' and `summary of analysis' may be inspected by
downloading and analysing this extract from the web site.
10. Pople's remarks are from unpublished notes for the `EuroMAC' presentation.
11. `Custom chords' are discussed in more detail below; Pople's analysis does not
include this chord as a custom chord.
12. `Using Complex Set Theory for Tonal Analysis'.
13. This passage is notated in D[ major, but Tonalities does not care at all for key
signatures. It is also blind to what happens in segments ahead. For this reason,
Tonalities, since it hears only F[ in the first segment, and F does not appear until
late on in the second, decides that the prolonging gamut is D[ minor, despite the
larger D[ major context.
14. These are the opening to rehearsal fig. 3; the `second theme' from figs 20+2±22
and the `F minor theme' from figs 6±8.
15. This quotation is taken from the uncompleted final section of `Using Complex
Set Theory for Tonal Analysis'.
16. Schoenberg does not use the term `octatonic', but the passage in Structural
Functions of Harmony refers back to Harmonielehre (Vienna: Universal Edition,1911), pp. 366±7 and Ex. 304, where Schoenberg describes how `one diminished
seventh chord can be interpreted as four different ninth chords' if the notes of
another diminished seventh are placed underneath. Schoenberg writes out the
aggregate formed as a scale of half and whole steps.
17. Daniel Harrison, `Supplement to the Theory of Augmented-Sixth Chords',
Music Theory Spectrum, 17/ii (1995), pp. 170±95.
18. `Using Complex Set Theory for Tonal Analysis', unfinished notes for final part.
20. Ibid. Oddly, despite Pople's remark and despite the falling whole-tone
tetrachords in the flutes (whose notation in groups of alternating flats and sharps
provides the `abandon' to which Pople refers), there are no whole-tone
prolonging or connective gamuts in this passage, only the whole-tone dominant
chord in segment two.
21. Quotations in this passage are from the `EuroMAC' notes.
22. Pople used Deryck Cooke's performing edition for this analysis. The passages
analysed here are largely, but not exclusively Mahler.
23. The prominent 13th (E[) in the violin in bar 21 was added by Deryck Cooke.
24. In a DOUTH2 progression, one dominant seventh is transformed into another
by the holding of two notes and the movement of the other two by semitones.
25. Pople discusses inclusive chord types in `Using Complex Set Theory for Tonal
26. Pople also made a `reserve' analysis of this passage with a language setting that
includes the French sixth which affects the analysis of segment $5. Instead of a
`dominant seventh [5' on A, a French sixth on E[ is produced.
27. Pople's remark from the `OXMAC' presentation (retained in various subsequent
presentations and drafts).
28. See Richard S. Parks, The Music of Claude Debussy (New Haven: Yale University
29. Pople highlighted the importance of the Dorian in his `EuroMAC' presentation.
30. These remarks come from the `EuroMAC' presentation.
31. George Perle, `Berg', in Stanley Sadie (ed.), The New Grove Dictionary of Music
and Musicians, 2nd edn (London: Macmillan, 2001).
32. Getting Started with Tonalities, p. 8.
33. `Using Complex Set Theory for Tonal Analysis'.
34. These remarks come from the `EuroMAC' presentation.
35. `Using Complex Set Theory for Tonal Analysis'. Full investigation of the
internal workings of the software will, it is hoped, make this clearer.
36. `EuroMAC' presentation.37. Ibid.
38. Part of unpublished notes at the end of `Using Complex Set Theory for Tonal
39. Analysis of the `Tristan chord' and the beginning of the first song from
Dichterliebe are examples.
40. I have in mind the kinds of processes described in Joseph N. Straus, `Uniformity,
Balance, and Smoothness in Atonal Voice Leading', Music Theory Spectrum, 25/ii
(2003), pp. 305±52.
41. `EuroMAC' Conference Proposal.