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<div>Hi Dave,</div><div><br></div><div>Always good to hear from you, even when we disagree.</div><div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size:14.0px"></span></font></blockquote><br><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size:14.0px">Rather than make neo-Pistonian lists of root motions, I suggest Matthew Brown’s book, Explaining Tonality, chapter 3 (indeed, hello) as a good place to start for trying to understand tonal sequences. Steve Laitz’s textbook is also a place to find a useful compendium of such phenomena.</span></font></blockquote><div><br></div><div>I think I share your aversion to "neo-Pistonian lists," though perhaps we understand the term differently. Recently I've been really interested in questions and models that unite the harmonic and contrapuntal domains. (e.g. Q: "Which diatonic triads can be linked by single-step voice leading." A: "Triads whose roots are a third apart.") By dealing with both at once, we can sometimes come up with new insights. For instance, I find it interesting that most of our examples of minor-third sequences do not use the more efficient stepwise descending voice leading, but instead ascending parallel voice leading; one reason for this may be that the stepwise sequence cycles through all its inversions in turn, leading to 6/4 chords. I also think it's cool when voice-leading facts can help explain unusual root progressions -- as in "Hey Joe," perhaps.</div><div><br></div><div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size: 14px; ">I haven’t counted them all yet, but check out Coltrane’s “Giant Steps” for 7th chords in M3 cycles (albeit mediated by V-Is, ii-V-Is) and Messiaen’s Quartet for the end of Time V for triads in minor 3rd cycles.</span></font></blockquote><div><br></div><div>Good examples! A few points:</div><div><br></div><div><span class="Apple-tab-span" style="white-space: pre; "> </span>1. Definitely, there are some examples of M3-related dominant sevenths and m3-related triads. My claim is statistical, though.<br></div><div><span class="Apple-tab-span" style="white-space: pre; "> </span>2. In the twentieth century, dramatic harmonic juxtapositions appear frequently, often without efficient voice leading. In nineteenth-century music, "odd" harmonic shifts very often exploit efficient voice leading. This is one interesting difference between, say, Schubert and Prokofiev: both love juxtaposing M3-related chords, but Schubert more often emphasizes the smooth voice leading between them.</div><div><span class="Apple-tab-span" style="white-space: pre; "> </span>3. As you probably know, "Giant Steps" never juxtaposes M3-related chords directly; instead, it juxtaposes m3- (or tritone-) related seventh chords: for example, Bmaj7 progresses directly to D7, which can perhaps be interpreted using the semitonal voice leading (B, D#, F#, A#)->(C, D, F#, A). (Note that the bass plays chord-roots -- the "Giant Steps" of the title -- and doesn't participate in the schema.) This is an example of what I call the "minor-third system." If you actually look at Coltrane's original solo, you periodically see echoes of this voice leading in his playing.</div><div><span class="Apple-tab-span" style="white-space: pre; "> </span>It's worth comparing this piece to the modulating E-minor passage in the exposition of the first movement of Beethoven's second piano sonata (Op. 2, no. 2, I, mm. 58ff). There, m3-related dominant chords (diminished sevenths or V7 chords) give rise to minor third related tonics. Here, you find a similar juxtaposition of m3-related sevenths, but since they're both dominants, you get different root relations among the tonics If you just look at the tonic chords in these two passages, you miss out on the interesting relationships between the two sorts of modulatory procedure.</div><div><br></div><div>DT</div></div><div><br></div></div><div> <span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div>Dmitri Tymoczko</div><div>Associate<span class="Apple-converted-space"> </span>Professor of Music</div><div>310 Woolworth Center</div><div>Princeton, NJ 08544-1007</div><div>(609) 258-4255 (ph), (609) 258-6793 (fax)</div><div><a href="http://music.princeton.edu/~dmitri">http://music.princeton.edu/~dmitri</a></div><div><br class="khtml-block-placeholder"></div><div><br class="khtml-block-placeholder"></div><br class="Apple-interchange-newline"></span><br class="Apple-interchange-newline"> </div><br></body></html>