Tuplet Music Definition Essay

This article is about the note groupings. For mathematical grouping, see tuple.

In music, a tuplet (also irrational rhythm or groupings, artificial division or groupings, abnormal divisions, irregular rhythm, gruppetto, extra-metric groupings, or, rarely, contrametric rhythm) is "any rhythm that involves dividing the beat into a different number of equal subdivisions from that usually permitted by the time-signature (e.g., triplets, duplets, etc.)" (Humphries 2002, 266). This is indicated by a number (or sometimes two), indicating the fraction involved. The notes involved are also often grouped with a bracket or (in older notation) a slur. The most common type is the "triplet".

Terminology[edit]

The modern term 'tuplet' comes from a mistaken splitting of the suffixes of words like quintu(s)-(u)plet and sextu(s)-(u)plet, and from related mathematical terms such as "tuple", "-uplet" and "-plet", which are used to form terms denoting multiplets (Oxford English Dictionary, entries "multiplet", "-plet, comb. form", "-let, suffix", and "-et, suffix1"). An alternative modern term, "irrational rhythm", was originally borrowed from Greek prosody where it referred to "a syllable having a metrical value not corresponding to its actual time-value, or ... a metrical foot containing such a syllable" (Oxford English Dictionary, entry "irrational"). The term would be incorrect if used in the mathematical sense (because the note-values are rational fractions) or in the more general sense of "unreasonable, utterly illogical, absurd".

Alternative terms found occasionally are "artificial division" (Jones 1974, 19), "abnormal divisions" (Donato 1963, 34), "irregular rhythm" (Read 1964, 181), and "irregular rhythmic groupings" (Kennedy 1994). The term "polyrhythm" (or "polymeter"), sometimes incorrectly used of "tuplets", actually refers to the simultaneous use of opposing time signatures (Read 1964, 167).

Besides "triplet", the terms "duplet", "quadruplet", "quintuplet", "sextuplet", "septuplet", and "octuplet" are used frequently. The terms "nonuplet", "decuplet", "undecuplet", "dodecuplet", and "tredecuplet" had been suggested but up until 1925 had not caught on (Dunstan 1925,[page needed]). By 1964 the terms "nonuplet" and "decuplet" were usual, while subdivisions by greater numbers were more commonly described as "group of eleven notes", "group of twelve notes", and so on (Read 1964, 189).

Triplets[edit]

The most common tuplet (Schonbrun 2007, 8) is the triplet (Ger.Triole, Fr.triolet, It.terzina or tripletta, Sp.tresillo), shown at right.

Whereas normally two quarter notes (crotchets) are the same duration as a half note (minim), three triplet quarter notes total that same duration, so the duration of a triplet quarter note is ​23 the duration of a standard quarter note. Similarly, three triplet eighth notes (quavers) are equal in duration to one quarter note. If several note values appear under the triplet bracket, they are all affected the same way, reduced to ​23 their original duration. The triplet indication may also apply to notes of different values, for example a quarter note followed by one eighth note, in which case the quarter note may be regarded as two triplet eighths tied together (Gherkens 1921, 19).

Tuplet notation[edit]

If the notes of the tuplet are beamed together, the bracket (or slur) may be omitted and the number written next to the beam, as shown in the second illustration.

For other tuplets, the number indicates a ratio to the next lower normal value in the prevailing meter. So a quintuplet (quintolet or pentuplet (Cunningham 2007, 111)) indicated with the numeral 5 means that five of the indicated note value total the duration normally occupied by four (or, as a division of a dotted note in compound time, three), equivalent to the second higher note value; for example, five quintuplet eighth notes total the same duration as a half note (or, in 3
8 or compound meters such as 6
8, 9
8, etc. time, a dotted quarter note). Some numbers are used inconsistently: for example septuplets (septolets or septimoles) usually indicate 7 notes in the duration of 4—or in compound meter 7 for 6—but may sometimes be used to mean 7 notes in the duration of 8 (Read 1964, 183–84). Thus, a septuplet lasting a whole note can be written with either quarter notes (7:4) or eighth notes (7:8). To avoid ambiguity, composers sometimes write the ratio explicitly instead of just a single number, as shown in the third illustration; this is also done for cases like 7:11, where the validity of this practice is established by the complexity of the figure. A French alternative is to write pour ("for") or de ("of") in place of the colon, or above the bracketed "irregular" number (Read 1964, 219–21). This reflects the French usage of, for example, "six-pour-quatre" as an alternative name for the sextolet (Damour, Burnett, and Elwart 1838, 79; Hubbard 1924, 480).

There are disagreements about the sextuplet (pronounced with stress on the first syllable, according to Baker 1895, 177)—which is also called sestole, sestolet, sextole, or sextolet (Baker 1895, 177; Cooper 1973, 32; Latham 2002; Shedlock 1876, 62, 68, 87, 93; Stainer and Barrett 1876, 395; Taylor & 1879–89; Taylor 2001). This six-part division may be regarded either as a triplet with each note divided in half (2 + 2 + 2)—therefore with an accent on the first, third, and fifth notes—or else as an ordinary duple pattern with each note subdivided into triplets (3 + 3) and accented on both the first and fourth notes. Some authorities treat both groupings as equally valid forms (Damour, Burnett, and Elwart 1838, 80; Köhler 1858, 2:52–53; Latham 2002; Marx 1853, 114; Read 1964, 215), while others dispute this, holding the first type to be the "true" (or "real") sextuplet, and the second type to be properly a "double triplet", which should always be written and named as such (Kastner 1838, 94; Riemann 1884, 134–35; Taylor & 1879–89, 3:478). Some go so far as to call the latter, when written with a numeral 6, a "false" sextuplet (Baker 1895, 177; Lobe 1881, 36; Shedlock 1876, 62). Still others, on the contrary, define the sextuplet precisely and solely as the double triplet (Stainer and Barrett 1876, 395; Sembos 2006, 86), and a few more, while accepting the distinction, contend that the true sextuplet has no internal subdivisions—only the first note of the group should be accented (Riemann 1884, 134; Taylor & 1879–89, 3:478; Taylor 2001).

In compound meter, even-numbered tuplets can indicate that a note value is changed in relation to the dotted version of the next higher note value. Thus, two dupleteighth notes (most often used in 6
8meter) take the time normally totaled by three eighth notes, equal to a dotted quarter note. Four quadruplet (or quartole) eighth notes would also equal a dotted quarter note. The duplet eighth note is thus exactly the same duration as a dotted eighth note, but the duplet notation is far more common in compound meters (Jones 1974, 20). A duplet in compound time is more often written as 2:3 (a dotted quarter note split into two duplet eighth notes) than 2:​1 12 (a dotted quarter note split into two duplet quarter notes), even though the former is inconsistent with a quadruplet also being written as 4:3 (a dotted quarter note split into two quadruplet eighth notes) (Anon. 1997–2000).

In drumming, "quadruplet" refers to one group of three sixteenth-note triplets "with an extra [non-tuplet eighth] note added on to the end", thus filling one beat in 4
4 time (Peckman 2007, 127–28), with four notes of unequal value.

Usage and purpose[edit]

Tuplets can produce rhythms such as the hemiola, or may be used as polyrhythms when played against the regular duration. They are extrametricrhythmic units.

Traditional music notation favors duple divisions of a steady beat or time unit. A whole note (semibreve) divides into two half notes, a half note into two quarters, etc. and other notes are made by tying these together.

An irrational rhythm (by definition) is one that uses exact time points or durations that lie outside the scope of the duple system.

The n-tuplet notation shows the proportional increase or decrease of tempo needed for the bracketed notes, relative to the prevailing tempo. For example, a bracket labeled "5:4" (read five in the space of four) could group together durations (notes or rests) with a total of five sixteenth notes. A tempo ​54 faster than usual then compresses these events into the space of four sixteenth notes.

The actual duration can be found by dividing the notated duration by the indicated tempo increase (​516 ÷ ​54 = ​14), in this example).

Normally, the total duration of the bracketed notes is chosen to be exactly equal to the duration of one of the duple divisions. For the example of a 5:4 bracket, this is possible if the total bracketed duration has a 5 in its numerator, ​516 in the example.

Sometimes though that requirement is dropped to create total durations not exactly expressible in the duple system. For example, one might have only three of the usual five sixteenth notes grouped by a bracket marked "3 of 5:4".

Counting[edit]

Tuplets may be counted, most often at extremely slow tempos, using the lowest common multiple (LCM) between the original and tuplet divisions. For example, with a 3-against-2 tuplet (triplets) the LCM is 6. Since 6 ÷ 2 = 3 and 6 ÷ 3 = 2 the quarter notes fall every three counts (overlined) and the triplets every two (underlined):

1 2 345 6

This is fairly easily brought up to tempo, and depending on the music may be counted in tempo, while 7-against-4, having an LCM of 28, may be counted at extremely slow tempos but must be played intuitively ("felt out") at tempo:

1 2 3 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20 2122 23 24 25 26 27 28

To play a half-note (minim) triplet accurately in a bar of 4
4, count eighth-note triplets and tie them together in groups of four. With a stress on each target note, one would count:

1-2-3 | 1-2-3 | 1-2-3 | 1-2-3

The same principle can be applied to quintuplets, septuplets, and so on.

See also[edit]

Look up tuplet in Wiktionary, the free dictionary.

References[edit]

  • Anon. 1997–2000. "Music Notation Questions Answered". Graphire Corporation, Graphire.com (Accessed 10 May 2013).
  • Baker, Theodore (ed.). 1895. A Dictionary of Musical Terms. New York: G. Schirmer.
  • Baker, Theodore, Nicolas Slonimsky, and Laura Dine Kuhn. 1995. Schirmer Pronouncing Pocket Manual of Musical Terms. New York: Schirmer Books. ISBN 0-8256-7223-6.
  • Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach. New York: Dodd, Mead. ISBN 0-396-06752-2.
  • Cunningham, Michael G. 2007. Technique for Composers. Bloomington, Indiana: AuthorHouse. ISBN 1-4259-9618-3.
  • Damour, Antoine, Aimable Burnett, and Élie Elwart. 1838. Études élémentaires de la musique: depuis ses premières notions jusqu'à celles de la composition: divisées en trois parties: Connaissances préliminaires. Méthode de chant. Méthode d’harmonie. Paris: Bureau des Études élémentaires de la musique.
  • Donato, Anthony. 1963. Preparing Music Manuscript. Englewood Cliffs, NJ: Prentice-Hall, Inc. Unaltered reprint, Westport, Conn.: Greenwood Press, 1977 ISBN 0-8371-9587-X.
  • Dunstan, Ralph. 1925. A Cyclopædic Dictionary of Music. 4th ed. London: J. Curwen & Sons, 1925. Reprint. New York: DaCapo Press, 1973.
  • Gehrkens, Karl W. 1921. Music Notation and Terminology. New York and Chicago: The A. S. Barnes Company.
  • Hubbard, William Lines. 1924. Musical Dictionary, revised and enlarged edition. Toledo: Squire Cooley Co. Reprinted as The American History and Encyclopedia of Music. Whitefish, Montana: Kessinger Publishing, 2005. ISBN 1-4179-0200-0.
  • Humphries, Carl. 2002. The Piano Handbook. San Francisco, CA: Backbeat Books; London: Hi Marketing. ISBN 0-87930-727-7.
  • Jones, George Thaddeus. 1974. Music Theory: The Fundamental Concepts of Tonal Music Including Notation, Terminology, and Harmony. New York, Hagerstown, San Francisco, London: Barnes & Noble Books. ISBN 0-06-460137-4.
  • Lobe, Johann Christian. 1881. Catechism of Music, new and improved edition, edited and revised from the 20th German edition by John Henry Cornell, translated by Fanny Raymond Ritter. New York: G. Schirmer. (First edition of English translation by Fanny Raymond Ritter. New York: J. Schuberth 1867.)
  • Kennedy, Michael. 1994. "Irregular Rhythmic Groupings. (Duplets, Triplets, Quadruplets)". Oxford Dictionary of Music, second edition, associate editor, Joyce Bourne. Oxford and New York: Oxford University Press. ISBN 0-19-869162-9.
  • Köhler, Louis. 1858. Systematische Lehrmethode für Clavierspiel und Musik: Theoretisch und praktisch, 2 vols. Leipzig: Breitkopf und Härtel.
  • Latham, Alison (ed.). 2002. "Sextuplet [sextolet]". The Oxford Companion to Music. Oxford and New York: Oxford University Press. ISBN 0-19-866212-2.
  • Marx, Adolf Bernhard. 1853. Universal School of Music, translated from the fifth edition of the original German by August Heinrich Wehrhan. London.
  • Peckman, Jon. 2007. Picture Yourself Drumming: Step-by-Step Instruction for Drum Kit Setup, Reading Music, Learning from the Pros, and More. Boston, MA: Thomson Course Technology. ISBN 1-59863-330-9.
  • Read, Gardner. 1964. Music Notation: A Manual of Modern Practice. Boston: Alleyn and Bacon, Inc. Second edition, Boston: Alleyn and Bacon, Inc., 1969., reprinted as A Crescendo Book, New York: Taplinger Pub. Co., 1979. ISBN 0-8008-5459-4 (cloth), ISBN 0-8008-5453-5 (pbk).
  • Riemann, Hugo. 1884. Musikalische Dynamik und Agogik: Lehrbuch der musikalischen Phrasirung auf Grund einer Revision der Lehre von der musikalischen Metrik und Rhythmik. Hamburg: D. Rahter; St. Petersburg: A. Büttner; Leipzig: Fr. Kistnet.
  • Schonbrun, Marc. 2007. The Everything Music Theory Book: A Complete Guide to Taking Your Understanding of Music to the Next Level. The Everything Series. Avon, Mass.: Adams Media. ISBN 1-59337-652-9.
  • Sembos, Evangelos C. 2006. Principles of Music Theory: A Practical Guide, second edition. Morrisville, NC: Lulu Press, Inc. ISBN 1-4303-0955-5.
  • Shedlock, Emma L. 1876. A Trip to Music-Land: An Allegorical and Pictorial Exposition of the Elements of Music. London, Glasgow, and Edinburgh: Blackie & Son.
  • Stainer, John, and William Alexander Barrett. 1876. A Dictionary of Musical Terms. London: Novello, Ewer and Co.
  • Taylor, Franklin. 1879–89. "Sextolet". A Dictionary of Music and Musicians (A.D. 1450–1883) by Eminent Writers, English and Foreign, 4 vols, edited by Sir George Grove, 3:478. London: Macmillan and Co.
  • Taylor, Franklin. 2001. "Sextolet, Sextuplet." The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
Wikimedia Commons has media related to Tuplet.
Irrational rhythm ( Play (help·info)): triplet above second beat features three rather than the usual two equal divisions of the beat, while the four sixteenth notes (semiquavers) above the third beat are rational, four being a multiple of two
Sextuplet ( Play (help·info)), or six notes. As the extra brackets show: six notes in the time of four = three notes in the time of two × 2
"True sextuplet": in order to contrast with the above "false sextuplet", the 1st, 3rd, and 5th notes of a sextuplet must be stressed rather than the 1st and 4th (Baker, Slonimsky, and Kuhn 1995, 208)( Play (help·info)).
Tuplet: a standard triplet; a triplet denoted without a bracket; a tuplet denoted as a ratio  Play (help·info)
Septuplet rhythm: seven against four (more frequent) and seven against eight (sometimes found) ( Play (help·info)).
Duplet and quadruplet notated in 6
8 Play (help·info). Two duplets or four quadruplets equal three regular eighth notes or a dotted quarter note.
Sextuplet in quintuple time: six against five ( Play (help·info)).
Four eighth-note triplets = one half-note triplet.

Gallery of Interesting Music Notation

Donald Byrd, Indiana University Bloomington - revised early November 2017

Introduction: Music Notation and Music Representation

Knowledge representation -- describing knowledge of some subject in a way that a computer can handle -- is a discipline that's important in several areas of computer science, for example, artifical intelligence. It's well-known in the knowledge-representation 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: some aspects of the phenomenon simply cannot be represented. Now, 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 presents 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. Nearly all of the examples here also appear in one or the other of two related webpages of mine: (1) 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.; and (2) More Counterexamples in Conventional Music Notation, a list of instances of CMN that break the supposed rules, many in very dramatic ways.

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). In addition, Byrd & Isaacson (2016) contains a lengthy table of features of CMN, and where possible, items below are cross-referenced to entries in that table. Such references are notated as, e.g., "{B&I 8.4}".

Why piano music? It will be noticed that almost all of the examples below are from piano music. The main reason is that several factors result in music for keyboard instruments, and especially idiomatic piano music from the early 19th century on, tending to place greater demands on notation than notation for any other instrument I'm familiar with. Byrd & Simonsen (2015) coined the term pianoform for this type of notation, and they discuss the features, almost unique to the piano, that are involved. The harp and guitar have some of the features in question, and the notation of some of their music is undoubtedly pianoform, but I'm far less familiar with their repertoires.

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. {B&I 17.17} 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. {B&I 11.9}

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). {B&I 17.17} 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 CMN 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. But the best explanation might simply be 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 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. {B&I 8.4} 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. {B&I 8.4}

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. {B&I 10.8}

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 at least five works containing triple sharps or 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. For comparison, MIDI (the standard "Musical Instrument Digital Interface") 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 much 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. {B&I 4.8}

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. Even at a tempo as slow as M.M. eighth = 40 (quarter = 20), a 1024th note would last only about 1/85 sec., too short to be heard even in the unlikely event that the piano mechanism responded quickly enough. However, any pianist would undoubtedly play this passage with a tremendous ritardando, slowing the tempo by the last notes enough to make these notes last long enough to be heard! (The next shortest notated durations seem to be 256ths in works of several composers going back as far as Vivaldi.) {B&I 4.5}

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; Extremes of Conventional Music Notation lists 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.) {B&I 4.6, 6.7}

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. Also notice the time signatures: a quarter note for the voice equals a half for the keyboard! Why? It seems that 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.

Acknowledgements

I'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 and Douglas Hofstadter for many discussions of these and other examples of the subtleties of CWMN.

References

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. 17–20; retrieved (in scanned form) February 20, 2011, from the World Wide Web: http://www.informatics.indiana.edu/donbyrd/Papers/MusNotSoftware+Intelligence.pdf .

Byrd, Donald, & Isaacson, Eric (2016). A Music Representation Requirement Specification for Academia. Based on our 2003 Computer Music Journal paper of the same title. Available at: http://www.informatics.indiana.edu/donbyrd/Papers/MusicRepReqForAcad.doc .

Byrd, Donald, & Simonsen, Jakob Grue (2015). Towards a Standard Testbed for Optical Music Recognition: Definitions, Metrics, and Page Images. Journal of New Music Research 44, 3, pp. 169–195.

Davis, Randall; Shrobe, Howard; & Szolovits, Peter (1993). What is a Knowledge Representation? AI Magazine, 14(1), pp. 17–33. 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. Martin’s 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. 31–42.



Comments to: donbyrd(at)indiana.edu
Copyright 2006-16, Donald Byrd

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