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 in the "real world" inevitably introduces bias (Davis et al 1993) and, if the phenomenon being represented is very subtle, hard limits: aspects of the phenomenon that simply cannot be represented. Given a representation, choosing a notation for it, i.e., a way to show the information graphically, inevitably introduces more bias and often, more limits; 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 (though perhaps still too technical for the layperson), but its gallery of notation examples is much larger. A related webpage is my Extremes of Conventional Music Notation, an extensive list of "extreme" values I've observed over some 35 years 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.
It will be noticed that almost all of the examples below are from keyboard music. The main reason is that several factors result in keyboard music -- especially idiomatic piano music from the early 19th century on -- tending to place greater demands on notation than notation for almost any other instrument. Harp and guitar music may be equally demanding, but I'm far less familiar with their repertoires.
My interest in this kind of thing goes way back. For background on many of these issues, see my dissertation (Byrd 1984) and a short article (Byrd 1994).
Main Gallery: Representation Issues
Note: Clicking on the small musical examples or on the links in the text will display the examples in context, and usually at higher resolution.
Representation is a matter of what information is present; notation is concerned with how the information is shown. To clarify the difference, first consider this extraordinary slur, the most complex I know of; it's 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 within a 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 is this excerpt from a Schubert Impromptu: the interesting thing here is what doesn't appear on the page. Notice that, while the right hand has triplets throughout, there are no triplet markings after the first measure. This is a serious matter because if the (invisible) triplets are not represented somehow, notes in the two hands won't be synchronized. And unmarked tuplets -- especially triplets -- are quite common. Still, significant as this is for representing the information, handling it is straightforward.
Two phenomena that -- in terms of music representation -- are each much more challenging than Schubert's invisible triplets appear in a passage from a Chopin Nocturne (Op. 15 no. 2, after the marking "Doppio movimento") in which one notehead is a triplet 16th in one (logical) voice, but a normal 16th in another. Consider the last note of the measure on the top staff, 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. This "impossible" rhythm, where a single notehead that belongs to two voices has inconsistent contexts in the voices, is much less common than unmarked tuplets; but it's not as rare as you might think: Julian Hook has found dozens of examples in the works of Schubert, Mendelssohn, Franck, Brahms, Rachmaninov, etc., as well as Chopin (Hook 2011). The second issue this passage raises is how to tell which voice or voices each notehead belongs to. The top staff certainly has at least three voices, though only one or two notes are sounding at a time; but features of the beaming in some places (e.g., m. 29) strongly suggest four!
The aforementioned Chopin Nocturne (Op. 15 no. 2) contains an extraordinary amount of interesting notation. A passage near the beginning includes one of the strangest pieces of notation I've ever seen: notes that arguably must be played in right-to-left order! The notes in question are in the upper staff of m. 9, the last measure of the second system. Notice that this measure -- in 2/4 -- appears to be a 16th note too long, and the extra 16th is obviously the first note in the upper staff. The preceding measure offers a clue. That measure appears to be an eighth note too long, the extra eighth being at the end of the measure. In each case, a grace note or grace notes separates the "extra" note or notes from the adjacent note on their staff. But the extra notes have the opposite stem direction from the adjacent notes, suggesting they're in a different (logical) voice. That in turn suggests that they should be played simultaneously with the adjacent notes, which solves the excessive duration problem -- but at the expense of requiring the grace note (or notes) to be played before the normal note to its (their) left! No, this isn't very satisfying, but neither is any other explanation I can think of. Of course, an explanation of a different sort is simply that Chopin is trying here to make music notation show more simultaneous relationships than it's capable of.
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 in this "shorthand" way 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 Bach's 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; to my knowledge, 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 very 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 suite of virtuoso piano pieces 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; this helps to minimize page turns, an important consideration since the player's hands are busy enough as it is.
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, and notated with a double-sharp followed by a normal sharp before the notehead. 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 -- even when, from the standpoint of music theory, the situation requires it -- very few people have ever bothered with them. Most composers and editors would undoubtedly write A, G sharp, A in place of Reger's G double-sharp, F triple-sharp, G double-sharp.
Annex: 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 -- and in the vast majority of cases, for purely musical purposes, not for the sake of the notation (Byrd 1994).
Bach's Jesu, Joy of Man's Desiring, in the well-known piano arrangement by Dame Myra Hess (published by Oxford), has single notes (not part of a chord) on the "wrong" side of the stem; this odd bit of notation occurs on the right-hand staff almost from beginning to end. Why? In the words of Sadie (2001), article "Notation", these are "reversed note shapes representing one strand of a complex texture". In plainer language, there are three independent voices on the staff, and the standard method of showing independence of voices on a staff, with upstemmed and downstemmed notes, is inadequate for three or more voices. Several other methods are possible, including via beaming, as in the Chopin Nocturne discussed above, and stem side, as in this publication.
Here's a measure with no less than four horizontal positions for notes that are all on the downbeat (taken from Johannes Brahms's Intermezzo, Op. 117 no. 1). As a result of the seconds in both the right-hand chord (upstemmed, so the odd note is to the right of the standard position) and the left-hand chord (downstemmed, so the odd note is to the left of the standard position), the notes in the dotted-quarter chords occupy three different positions; the first eighth-note on each staff, in yet a fourth position, is also on the downbeat.
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 (ca. 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.)
Several quadruple-dotted notes appear in the third movement of Schumann's String Quartet no. 1, Op. 41 no. 1. Double-dotted notes are pretty common, but triple-dotted notes are not. Still, even quadruple-dotted notes have been used many times; I know of six works by major composers that employ them, and no doubt there are others. Of course the quadruple dotting could have been avoided by using two or more tied notes, but the 32nd note following complements the quadruple-dotted note, filling the 4/4 measure; so it's easy to see the rhythm without counting the dots, and writing it this way is almost certainly the most readable way to describe the intended rhythm. (As for rests, I know of a few examples with triple dots, but none with quadruple.)
A wild "X"-shaped pattern appears in the very next measure of the passage cited above from Scarbo in Ravel's Gaspard de la Nuit. Here, an accompanying voice descends over a range of more than five octaves, from far above the melody to far below it; therefore, the pianist's hands swap melody and accompaniment, the notation swaps staves, and the result is this "X"-shaped pattern. But was it really necessary to write it this way? No. Even in this rare case, where one voice covers such a huge range, the music could have been notated so as to avoid the pattern. The most readable alternative would probably be adding a third staff to the system, using it for just the low notes of the accompaniment voice. That would be acceptable; pianists are used to reading three and even four staves on occasion. But, again, it's preferable to avoid extra staves in order to minimize page turns, and it's not that hard to read the music as published -- certainly not that hard in comparison to how hard it is to play it!
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. Composers of the time actually competed to see who could write the most extreme notation (Eleanor Selfridge-Field, personal communication, July 2009).
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.
AcknowledgementsI'd like to thank Perry Roland and David Lewis for their penetrating comments on the "notes must be played from right-to-left" example, and Julian Hook for many discussions of these and other examples of the subtleties of CWMN.
Byrd, Donald (1984). Music Notation by Computer (doctoral dissertation, Computer Science Dept., Indiana University). Ann Arbor, Michigan: UMI ProQuest (order no. 8506091); also available from www.npcimaging.com. Retrieved (in scanned form) December 10, 2011, from the World Wide Web: http://www.informatics.indiana.edu/donbyrd/Papers/DonDissScanned.pdf .
Byrd, Donald (1994). Music Notation Software and Intelligence. Computer Music Journal 18(1), pp. 1720; retrieved (in scanned form) February 20, 2011, from the World Wide Web: http://www.informatics.indiana.edu/donbyrd/Papers/MusNotSoftware+Intelligence.pdf .
Davis, Randall; Shrobe, Howard; & Szolovits, Peter (1993). What is a Knowledge Representation? AI Magazine, 14(1), pp. 1733. Retrieved December 20, 2011, from the World Wide Web: http://medg.lcs.mit.edu/ftp/psz/k-rep.html
Hook, Julian (2011, December). How to Perform Impossible Rhythms. Music Theory Online 17(4). Retrieved December 21, 2011, from the World Wide Web: http://www.mtosmt.org/issues/mto.11.17.4/mto.11.17.4.hook.html
Rastall, Richard (1982). The Notation of Western Music. New York: St. Martins Press.
Sadie, Stanley, ed. (2001). The New Grove Dictionary of Music and Musicians, 2nd ed. Macmillan.
Wiggins, Geraint, Miranda, Eduardo, Smaill, Alan, & Harris, Mitch (1993). A Framework for the Evaluation of Music Representation Systems. Computer Music Journal 17(3), pp. 3142.
Copyright 2006-13, Donald Byrd