It’s February 29 again, which means it’s time to announce that this blog is being resurrected. Stay tuned for a post on Schubert (and Schenker).
Two blog posts — one recent, the other less so — that have me jumping up and down in excited agreement:
- You’re Calling *Who* a Cult Leader? — in which Eliezer Yudkowsky (one of my favorite bloggers) points out that that it’s okay to be really enthusiastic about something:
Behold the following, which is my true opinion:
“Gödel, Escher, Bach” by Douglas R. Hofstadter is the most awesome book that I have ever read. If there is one book that emphasizes the tragedy of Death, it is this book, because it’s terrible that so many people have died without reading it.
I know people who would never say anything like that, or even think it: admiring anything that much would mean they’d joined a cult (note: Hofstadter does not have a cult)[…]
But I’m having trouble understanding this phenomenon, because I myself feel no barrier against admiring Gödel, Escher, Bach that highly.
You know, there might be some other things that I admire highly besides Gödel, Escher, Bach, and I might or might not disagree with some things Douglas Hofstadter once said, but I’m not even going to list them, because GEB doesn’t need that kind of moderation. It is okay for GEB to be awesome. In this world there are people who have created awesome things and it is okay to admire them highly! Let this Earth have at least a little of its pride!
Yes! As I have noted before, there is an inhibition in our culture against expressing strong feelings. Away with this!
I’m embarrassed to admit that I still haven’t read GEB, even though everybody raves about it and it’s got both a mathematician and a composer in the title. Well, it’s now (higher) on my to-do list. But anyone who has ever visited this blog will know that I harbor a similar level of enthusiasm for Westergaard’s ITT. And that does not make me some kind of crazed fanatic.
- What’s in a number? — in which a member of the Texas Tech music theory faculty correctly explains the meaning of figured-bass symbols (link added by me):
I often tell my students that figured bass evolved as a shorthand notation for species counterpoint.That is, figured bass actually suggests lines, not chords. Consider the example below:
If you look at those examples without worrying about vertical sonorities, the figured bass makes quite a lot of sense. Once you begin trying to assign Roman numerals, the task becomes a bit muddier. In the first example, we can easily understand the E-F motion in the soprano as some kind of neighbor motion or perhaps as the beginning of a passing motion. I prefer that interpretation to one which says the first chord is a root-position tonic and the second chord is a first-inversion submediant.
And rightfully so. In fact even to speak of this measure as being composed of two “chords” is a misleading distortion. If there are two entities into which this measure is divisible, they are a second-species line on the one hand, and a complex of three first-species lines on the other. (As a minor quibble, I will point out that “worrying about vertical sonorities”, which species counterpoint does just fine, is not to be confused with “assigning Roman numerals”, the discredited province of harmonic theory.)
In short: figured bass tells us diatonic intervals above the bass and nothing else. If notes are to be altered, the accidentals will appear in the figured bass. Figured bass is simply a shorthand for linear motion.
So true! Whether or not it is technically accurate that figured bass evolved in connection with species counterpoint per se, this is much closer to the truth than to suppose, as many still do, that it indicates some sort of preexisting awareness of Rameauvian “harmonic” concepts on the part of Baroque-era musicians. (Of course anyone who thinks that hasn’t read Schenker, but that’s for another time…)
(Not too long ago it finally occurred to me why this confusion exists. The reason for it is that people mistake figured basses, which are a form of musical notation, for some sort of analysis of the music. When you look in a treatise and see a figured bass at the top of a page, say, followed by a realization below, perhaps it’s natural to suppose that the figured bass on top is in some sense a “more primitive” structure, from which the realization is derived. But this is a misunderstanding. Figured bass was a performance practice; it was not the purposes of such treatises to engage in music theory as we know it, of the sort practiced by Westergaard — the subject had not yet come into existence as an explicit discipline. So one is by no means obliged to think of a passage in terms of some underlying figured bass. Quite the contrary, in fact: the figured bass is but a shorthand for the realization, and thus if anything the latter “explains” the former, rather than the other way around.)
(Boulez, for non-musical readers, is pretty much the leading figure — or at any rate the leading European figure — in the art music of our era [i.e. post-WWII]. For mathematical non-musical readers, an approximately equivalent person would be Jean-Pierre Serre.)
The interesting part of the site is, of course, the series of pages devoted to Boulez’s 1984 work Dérive I, which is described as:
an elegant, shimmering and vibrating eight minute work which explores harmony and texture from a chordal starting point using material which “derives” from three earlier pieces, Répons (1980), Messagesquisse (1976) and Éclat (1965).
(Obviously, “harmony” here is to be understood in the sense of vertical pitch collections — nothing to do with harmonic theory! I would probably have used the word “sonority” here instead.)
It seems the excerpts included comprise virtually the entire piece, so go have a listen. The commentary hardly constitutes a detailed analysis, but overall the quality is pretty good for something on the Internet.
After reading the recent anecdote at Texas Tech Music Theory about the music student who didn’t know the meaning of “Adagio”, I was amused to find the rather strange marking “Adagio agitato” in the score of Beethoven’s Christ on the Mount of Olives (p.11).
You can hear the passage in question (which, of course, features Jesus in agony) beginning at 8:54 or so in this clip:
I admit, this could easily be a misprint for “Allegro agitato” (though the tempo in the above performance doesn’t strike me as quite fast enough for that; unfortunately I don’t remember the other recordings I’ve heard well enough to compare). Still, I can’t resist indulging, at least for a moment, in the thought that Beethoven is seeking some mysterious nuance here. He did after all quite deliberately create a surreal atmosphere by opening the oratorio in the highly unusual (at least in 1802) key of E-flat minor — a stroke that has permanently endeared this piece to me, whatever its flaws.
It occurs to me that it might be nice to post solutions to miscellaneous mathematical exercises at least once in a while.
I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).
Problem: Let be such that every real-valued continuous function on is bounded. Prove that is compact.
Well, as requested, at any rate…
Here is an analysis of the first two measures of the B-flat major Prelude from WTC I.
Warning: this analysis breaks some of the rules (well, one in particular) of strict Westergaardian theory as expounded in ITT. In fact, it does so twice (at two distinct stages). Exercise: see if you can identify the rule that is broken, and give a convincing rationale for relaxing it.
1. The basic structure:
2. Segment the final two beats of the first span with: an incomplete neighbor in the bass, a complete neighbor in the soprano, a borrowing from the bass in the alto, and a rearticulated suspension in the tenor:
3. Anticipate the G in the soprano:
4. Borrow from these structural lines to create the texture of the passage:
5. Delay the fourth half-note:
6. Elaborate further (the operations being, I hope, clear):
7. Elaborate still further to obtain the passage as Bach gave it to us:
Happy New Year!