Gallery of Interesting Music Notation
Introduction: Music Notation and Music Representation
It's well-known in the knowledge-representation (artificial intelligence, etc.) community that choosing a representation for anything inevitably introduces bias (Davis et al 1993). Given a representation, choosing a notation, i.e., a way to show the information graphically, inevitably introduces more bias, and music is no exception (Wiggins et al 1993). Consider CMN (Conventional Music Notation) -- or, more precisely, "CWMN" (Conventional Western Music Notation). CMN is among the most successful notations ever devised, but it's enormously complex and subtle. What are its limits, and what are its biases? It's not always obvious: some bizarre-looking CMN poses no real problems for representation, while some rather ordinary looking CMN poses very difficult problems.
Below is a little "gallery" of unusual music notation from the works of respectable composers and publishers, most of them very well-known. I also present examples of unusual music notation, and discuss the issues of music notation and music knowledge representation and their relation to knowledge representation in general, on the IU Music Informatics website, as well as in a talk entitled "Music Notation, Representation, and Intelligence" that page links to. The commentary on the current page is somewhat more for the casual observer, but its gallery of notation examples is considerably larger. A related webpage is my Extremes of Conventional Music Notation, a list of "extreme" values I've observed for many aspects of music expressed in conventional Western notation: shortest and longest note durations, most complex tuplet, slowest and fastest tempo marks, earliest use of fff, etc.
The Gallery
To
clarify the difference between notation (essentially, graphical) issues and representation
issues, this
extraordinary slur,
by far the most complex I know of, is from Sorabji's Opus Clavicembalisticum (1930), IX [Interludium B]
(Curwen ed., pp. 175-176). It has a total of 10(!) inflection points; it spans three systems,
repeatedly crosses three staves (this is also the most staves in a single system for any slur I
know of), and goes slightly backwards--i.e., from right to left--several times. However,
the complexity is almost entirely graphical: its implications for representation are minimal.
A very different example of the difference is this
excerpt from a Schubert Impromptu.
Notice that, while the right hand has triplets throughout, there are no triplet markings after the first
measure. If the (invisible) triplets are not represented explicitly, serious problems are likely to result,
e.g., in synchronization between the two hands.
A
phenomenon that -- in terms of music representation -- is much deeper than the shape of a
slur appears in a passage from a Chopin Nocturne (Op. 15 no. 2) in which
one
notehead is a triplet in one voice, but normal duration in another. Consider the last note
of the measure, and notice that both "versions" of the note end at the barline; therefore, in
the upstemmed voice, it begins earlier than in the downstemmed one!
But surely Chopin didn't intend it to be sounded twice, and pianists never play it that way.
How should it be represented? It's not easy to say.
And this "impossible" rhythm, where a single notehead that's part of two
voices has interpretations in the voices that are inconsistent, isn't as rare as you might
think: Julian Hook has found dozens of examples in the works of Brahms,
Chopin, Rachmaninov, etc.
The Brahms Capriccio for piano, Op. 76 no. 1, is in 6/8. A dotted half note lasts a full measure of 6 eighths, or 12 16ths; but this passage has a a dotted half note that lasts only eleven 16ths (on the top staff, in the second measure of the excerpt). Why? Notating a duration of 11 16ths "correctly" would have required four tied notes, but the fact that the dotted half note actually used really lasts to the end of the measure and no longer is obvious -- so obvious that, surely, few people even notice the inconsistency. Clearly, it's written the "shorthand" way it is to avoid the clutter of four notes and three ties.
This notation is much like the well-known "variable dot" of Baroque music: a dot may increase the duration of a note by more or less than the standard amount. For example, the D-major Fugue in Book I of the Well-Tempered Clavier has several instances of a quarter-note duration filled by a dotted eighth and three 32nds. Either the 32nds form an unmarked triplet, or the dotted eighth is shorthand for an eighth tied to a 32nd; all experts agree on the latter interpretation. (Cf. Rastall (1982), p. 214.)
This passage from Debussy's La Danse de Puck has a clef in mid-air, applying only to the note to its immediate right, while a different clef appears on the staff they belong to. Thus, it's bizarrely obvious that two clefs are simultaneously active on the staff. On the other hand, a subtle way to have two simultaneous clefs on a staff appears in the fourth measure on the lower staff of this excerpt, from Scarbo in Ravel's Gaspard de la Nuit. The passage is in 3/8 time, so the bass and treble clefs are both in effect for this entire measure! The obvious reason in these and other cases of two clefs simultaneously active on a staff (in music by Brahms and others, as well as other works of Debussy) is simply to save space by avoiding a third staff.
A passage from J.S. Bach's Goldberg Variations, no. 26,
changes time signature
in the middle of a measure
(at the beginning of the excerpt, the lower staff is in 3/4, the upper in 18/16).
Since a time signature describes the total duration of the measure,
what could this possibly mean? Actually, a time signature describes the metric structure of
the measure as well as its duration; this change occurs on a beat, and it goes
from a simple triple (3/4) to a compound triple meter (18/16), while the other change in
the excerpt does the opposite.
The only reasonable interpretation is that -- with each meter change -- an
equivalent tempo change keeps beats the same (real-time) duration, and
what's happening is simply a change from duple to triple subdivision of the beat.
Indeed, the passage is invariably performed that way.
So Bach (or his editor) could just have written the 18/16 parts in 3/4, but with
continuous sextuplets.
Double sharps and double flats aren't too unusual, especially in passages in "remote keys" with many sharps or flats in the key signature. But -- while they're very rare -- triple sharps and flats have appeared in print! One example is this F triple-sharp (used as a lower neighbor between two G double-sharps) near the end of the last movement of Reger's Clarinet Sonata, Op. 49 no. 2, piano part (1904; Universal ed.); it's in the right hand, in the last measure of the excerpt. Of course, MIDI doesn't even let you distinguish between sharps and flats, a distinction that most classically-trained musicians probably consider very important, even if they play an instrument like the piano that doesn't let them make the difference audible. Double-sharps and -flats allow finer distinctions, and triple-sharps and -flats such fine ones that very few people have ever bothered with them.
Other Examples
Briefly, here are a few more examples. All are interesting as examples of notation, but most aren't of much interest in terms of representation except to suggest the lengths to which composers and publishers have pushed every parameter.
Bach's Jesu, Joy of Man's Desiring, in the well-known piano arrangement by Dame Myra Hess (Oxford), has single notes (not part of a chord) on the "wrong" side of the stem almost from beginning to end. Why? Simply to show three independent voices on one staff.
By far the shortest notated duration I know of appears in this page of Anthony Phillip Heinrich's Toccata Grande Cromatica from The Sylviad, Set 2, m. 16 (c. 1825). At the very end of the page--the end of the last measure on the lower staff of the bottom system--there are some 1024th and even two 2048th(!) notes. However, the context shows clearly that these notes have one beam more than intended, so they should really be 512th and 1024th notes, respectively. The passage--in 2/4, marked "Grave"--also contains many 256th notes. How reasonable these durations are can be inferred from the fact that even at a tempo as slow as M.M. eighth = 40 (quarter = 20), a 1024th note would last only about 1/85 sec. (The next shortest notated durations seem to be 256ths in works of several composers going back as far as Vivaldi.)
Here's a passage from Marcello's Stravaganze in which inconsistent note spellings are carried to an extreme. The passage is actually rather simple, but it begins with the voice in the bizarre key of A-sharp minor while the keyboard part is in its (far more normal) enharmonic equivalent, B-flat minor! By measure 3, the two parts have swapped keys; then they repeatedly swap back and forth.
Finally, this passage from a keyboard piece by Johann Kuhnau illustrates a very unusual way to notate unison (dotted) whole notes in two voices on a staff: one note is inside of the other! It's on the downbeat of the 2nd measure of the 2nd system, top staff. This is largely a curiosity, but notice that the standard way of notating this, namely two dotted whole notes side-by-side, presents the problem of where to put the second augmentation dot.
Comments to: donbyrd(at)indiana.edu
Copyright 2006-09, Donald Byrd