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The time signature (also known as meter, metre, and measure signature) is a convention in Western music notation to specify how many beats (stress points) of a particular note value are contained in each measure (bar). The time signature is a notational device representing the meter, an auditory feature of the music.
In a music score the time signature appears at the beginning as a time symbol or stacked numerals, such as or 4
4 (read common time or four-four time, respectively), immediately following the key signature (or immediately following the clef symbol if the key signature is empty). A mid-score time signature, usually immediately following a barline, indicates a change of meter.
Most time signatures are either simple (the note values are grouped in pairs) or compound (grouped in threes). Less-common signatures correspond to complex, mixed, additive, and irrational meters.
Time signature notation
Most time signatures consist of two numerals, one stacked above the other:
- The lower numeral indicates the note value that the signature is counting. This number is always a power of 2, usually 2, 4, or 8. 2 corresponds to the half note (minim), 4 to the quarter note (crotchet), 8 to the eighth note (quaver).
- The upper numeral indicates how many such note values constitute a bar.
For instance, 2
4 means two quarter-notes (crotchets) per bar, while 4
8 means four eighth-notes (quavers) per bar. The most common time signatures are 2
4, and 4
By convention, two special symbols are sometimes used for 4
4 and 2
- The symbol is sometimes used for 4
4 time, also called common time or imperfect time.
- The symbol is typically used in place of 2
2 and is called alla breve or, colloquially, cut time or cut common time.
These symbols derive from Mensural time signatures, described below.
Frequently used time signatures
Simple versus compound
Simple meters are those whose upper number is 2, 3, or 4, sometimes described as duple meter, triple meter, and quadruple meter respectively.
In compound meter, the note values specified by the bottom number are grouped into threes, and the upper number is a multiple of 3, such as 6, 9, or 12. The lower number is most commonly an 8 (an eighth-note or quaver): as in 9
8 or 12
Other upper numbers correspond to irregular meters.
Beat and subdivision
Musical passages commonly feature a recurring pulse, or beat, usually in the range of 60-100 beats per minute. Depending on the tempo of the music, this beat may correspond to the note value specified by the time signature, or to a grouping of such note values. Most commonly, in simple time signatures, the beat is the same as the note value of the signature, but in compound signatures, the beat is usually a dotted note value corresponding to three of the signature's note values. Either way, the next lower note value shorter than the beat is called the subdivision.
On occasion a bar may seem like one singular beat. For example, a fast waltz, notated in 3
4 time, may be described as being one in a bar. Conversely, at slow tempos, the beat might even be a smaller note value than the one enumerated by the time signature.
Mathematically the time signatures of, e.g., 3
4 and 3
8 are interchangeable. In a sense all simple triple time signatures, such as 3
2, etc.—and all compound duple times, such as 6
16 and so on, are equivalent. A piece in 3
4 can be easily rewritten in 3
8, simply by halving the length of the notes.
Other time signature rewritings are possible: most commonly a simple time-signature with triplets translates into a compound meter.
The choice of time signature in these cases is largely a matter of tradition. Particular time signatures are traditionally associated with different music styles—it would seem strange to notate a conventional rock song in 4
8 or 4
2, rather than 4
In the examples below, bold denotes the primary stress of the measure, and italics denote a secondary stress. Syllables such as "and" are frequently used for pulsing in between numbers.
4 is a simple triple meter time signature that represents three quarter notes (crotchets), usually perceived as three beats. In this case the subdivision would be the eighth note (quaver). It is felt as
4: one and two and three and ...
Compound: Most often, 6
8 is felt as two beats, each being a dotted quarter note (crotchet), and each containing subdivisions of three eighth notes (quavers). It is felt as
8: one two three four five six ... (or, if counting dotted-quarter beats, one and a two and a)
The table below shows the characteristics of the most frequently used time signatures.
|Simple time signatures|
|Time signature||Common uses||Simple drum pattern||Video representation|
|Common time: Widely used in classical music and most forms of popular music. Most common time signature in rock, blues, country, funk, and pop.|
|Alla breve, cut time: Used for marches and fast orchestral music.|
|Used for polkas, galops, marches, and many styles of Latin music (including bolero, cumbia, and merengue).|
|Used for waltzes, minuets, scherzi, polonaises, mazurkas, country & western ballads, R&B, and some pop|
|Also used for the above but usually suggests higher tempo or shorter hypermeter. Sometimes preferred for certain folk dances such as cachucha|
|Compound time signatures|
|Time signature||Common uses||Simple drum pattern||Video representation|
|Double jigs, polkas, sega, salegy, tarantella, marches, barcarolles, loures, and some rock music|
|Compound triple time: Used in slip jigs, zeibekiko, and lullabies, otherwise occurring rarely ("The Ride of the Valkyries", Tchaikovsky's Fourth Symphony, and the final movement of J.S. Bach's Violin Concerto in A minor (BWV 1041) are familiar examples. Debussy's "Clair de lune" and the opening bars of Prélude à l'après-midi d'un faune are also in 9
|Also common in slower blues (where it is called a shuffle) and doo-wop; also used more recently in rock music. Can also be heard in some jigs like "The Irish Washerwoman". This is also the time signature of the second movement of Beethoven's Pastoral Symphony.|
While changing the bottom number and keeping the top number fixed only formally changes notation, without changing meaning – 3
2, and 3
1 are all three beats to a meter, just noted with eighth notes, quarter notes, half notes, or whole notes – these conventionally imply different performance and different tempi. Conventionally, larger numbers in the bottom correspond to faster tempi and smaller numbers correspond to slower tempi. This convention is known as tempo giusto, and means that the tempo of each note remains in a narrower, "normal" range. For illustration, a quarter note might correspond to 60–120 bpm, a half note to 30–60 bpm, a whole note to 15–30 bpm, and an eighth note to 120–240 bpm; these are not strict, but show an example of "normal" ranges.
This convention dates to the Baroque era, when tempo changes were indicated by changing time signature during the piece, rather than by using a single time signature and changing tempo marking. For example, while 3
2, and 3
1 have the same beat pattern, they would conventionally be used for increasingly slow music. A 20th century example is "O Fortuna" (1935–1936) by Carl Orff, which begins slowly in 3
1, and then speeds up and changes to 3
Complex time signatures
Signatures that do not fit the usual simple or compound categories are called complex, asymmetric, irregular, unusual, or odd—though these are broad terms, and usually a more specific description is appropriate. The term odd meter, however, sometimes describes time signatures in which the upper number is simply odd rather than even, including 3
4 and 9
Irregular meters are common in some non-Western music, and in ancient Greek music such as the Delphic Hymns to Apollo, but the corresponding time signatures rarely appeared in formal written Western music until the 19th century. Early anomalous examples appeared in Spain between 1516 and 1520, plus a small section in Handel's opera Orlando (1733).
The third movement of Frédéric Chopin's Piano Sonata No. 1 (1828) is an early, but by no means the earliest, example of 5
4 time in solo piano music. Anton Reicha's Fugue No. 20 from his Thirty-six Fugues, published in 1803, is also for piano and is in 5
8. The waltz-like second movement of Tchaikovsky's Pathétique Symphony (shown below), often described as a "limping waltz", is a notable example of 5
4 time in orchestral music.
Examples from 20th-century classical music include:
- Gustav Holst's "Mars, the Bringer of War" and "Neptune, the Mystic" from The Planets (both in 5
- Paul Hindemith's "Fuga secunda" in G from Ludus Tonalis (5
- the ending of Stravinsky's The Firebird (7
- the fugue from Heitor Villa-Lobos's Bachianas Brasileiras No. 9 (11
- the themes for the Mission: Impossible television series by Lalo Schifrin (in 5
4) and for Room 222 by Jerry Goldsmith (in 7
In the Western popular music tradition, unusual time signatures occur as well, with progressive rock in particular making frequent use of them. The use of shifting meters in The Beatles' "Strawberry Fields Forever" and the use of quintuple meter in their "Within You, Without You" are well-known examples, as is Radiohead's "Paranoid Android" (includes 7
Paul Desmond's jazz composition "Take Five", in 5
4 time, was one of a number of irregular-meter compositions that The Dave Brubeck Quartet played. They played other compositions in 11
4 ("Eleven Four"), 7
4 ("Unsquare Dance"), and 9
8 ("Blue Rondo à la Turk"), expressed as 2+2+2+3
8. This last is an example of a work in a signature that, despite appearing merely compound triple, is actually more complex. Brubeck's title refers to the characteristic aksak meter of the Turkish karşılama dance.
However, such time signatures are only unusual in most Western music. Traditional music of the Balkans uses such meters extensively. Bulgarian dances, for example, include forms with 5, 7, 9, 11, 13, 15, 22, 25 and other numbers of beats per measure. These rhythms are notated as additive rhythms based on simple units, usually 2, 3 and 4 beats, though the notation fails to describe the metric "time bending" taking place, or compound meters. See Additive meters below.
Some video samples are shown below.
While time signatures usually express a regular pattern of beat stresses continuing through a piece (or at least a section), sometimes composers place a different time signature at the beginning of each bar, resulting in music with an extremely irregular rhythmic feel. In this case, the time signatures are an aid to the performers and not necessarily an indication of meter. The Promenade from Modest Mussorgsky's Pictures at an Exhibition (1874) is a good example. The opening measures are shown below:
Igor Stravinsky's The Rite of Spring (1913) is famous for its "savage" rhythms. Five measures from "Sacrificial Dance" are shown below:
In such cases, a convention that some composers follow (e.g., Olivier Messiaen, in his La Nativité du Seigneur and Quatuor pour la fin du temps) is to simply omit the time signature. Charles Ives's Concord Sonata has measure bars for select passages, but the majority of the work is unbarred.
Some pieces have no time signature, as there is no discernible meter. This is sometimes known as free time. Sometimes one is provided (usually 4
4) so that the performer finds the piece easier to read, and simply has "free time" written as a direction. Sometimes the word FREE is written downwards on the staff to indicate the piece is in free time. Erik Satie wrote many compositions that are ostensibly in free time but actually follow an unstated and unchanging simple time signature. Later composers used this device more effectively, writing music almost devoid of a discernibly regular pulse.
If two time signatures alternate repeatedly, sometimes the two signatures are placed together at the beginning of the piece or section, as shown below:
To indicate more complex patterns of stresses, such as additive rhythms, more complex time signatures can be used. Additive meters have a pattern of beats that subdivide into smaller, irregular groups. Such meters are sometimes called imperfect, in contrast to perfect meters, in which the bar is first divided into equal units.
For example, the time signature 3+2+3
8 means that there are 8 quaver beats in the bar, divided as the first of a group of three eighth notes (quavers) that are stressed, then the first of a group of two, then first of a group of three again. The stress pattern is usually counted as
8: one two three one two one two three ...
This kind of time signature is commonly used to notate folk and non-Western types of music. In classical music, Béla Bartók and Olivier Messiaen have used such time signatures in their works. The first movement of Maurice Ravel's Piano Trio in A Minor is written in 8
8, in which the beats are likewise subdivided into 3+2+3 to reflect Basque dance rhythms.
Romanian musicologist Constantin Brăiloiu had a special interest in compound time signatures, developed while studying the traditional music of certain regions in his country. While investigating the origins of such unusual meters, he learned that they were even more characteristic of the traditional music of neighboring peoples (e.g., the Bulgarians). He suggested that such timings can be regarded as compounds of simple two-beat and three-beat meters, where an accent falls on every first beat, even though, for example in Bulgarian music, beat lengths of 1, 2, 3, 4 are used in the metric description. In addition, when focused only on stressed beats, simple time signatures can count as beats in a slower, compound time. However, there are two different-length beats in this resulting compound time, a one half-again longer than the short beat (or conversely, the short beat is 2⁄3 the value of the long). This type of meter is called aksak (the Turkish word for "limping"), impeded, jolting, or shaking, and is described as an irregular bichronic rhythm. A certain amount of confusion for Western musicians is inevitable, since a measure they would likely regard as 7
16, for example, is a three-beat measure in aksak, with one long and two short beats (with subdivisions of 2+2+3, 2+3+2, or 3+2+2).
Folk music may make use of metric time bends, so that the proportions of the performed metric beat time lengths differ from the exact proportions indicated by the metric. Depending on playing style of the same meter, the time bend can vary from non-existent to considerable; in the latter case, some musicologists may want to assign a different meter. For example, the Bulgarian tune "Eleno Mome" is written in one of three forms: (1) 7 = 2+2+1+2, (2) 13 = 4+4+2+3, or (3) 12 = 3+4+2+3, but an actual performance (e.g., "Eleno Mome"[original research?]) may be closer to 4+4+2+3.[clarification needed] The Macedonian 3+2+2+3+2 meter is even more complicated, with heavier time bends, and use of quadruples on the threes. The metric beat time proportions may vary with the speed that the tune is played. The Swedish Boda Polska (Polska from the parish Boda) has a typical elongated second beat.
In Western classical music, metric time bend is used in the performance of the Viennese waltz. Most Western music uses metric ratios of 2:1, 3:1, or 4:1 (two-, three- or four-beat time signatures)—in other words, integer ratios that make all beats equal in time length. So, relative to that, 3:2 and 4:3 ratios correspond to very distinctive metric rhythm profiles. Complex accentuation occurs in Western music, but as syncopation rather than as part of the metric accentuation.
Brăiloiu borrowed a term from Turkish medieval music theory: aksak. Such compound time signatures fall under the "aksak rhythm" category that he introduced along with a couple more that should describe the rhythm figures in traditional music. The term Brăiloiu revived had moderate success worldwide, but in Eastern Europe it is still frequently used. However, aksak rhythm figures occur not only in a few European countries, but on all continents, featuring various combinations of the two and three sequences. The longest are in Bulgaria. The shortest aksak rhythm figures follow the five-beat timing, comprising a two and a three (or three and two).
Some video samples are shown below.
A method to create meters of lengths of any length has been published in the Journal of Anaphoria Music Theory and Xenharmonikon 16 using both those based on the Horograms of Erv Wilson and Viggo Brun's algorithm written by Kraig Grady.
Irrational time signatures (rarely, "non-dyadic time signatures") are used for so-called irrational bar lengths, that have a denominator that is not a power of two (1, 2, 4, 8, 16, 32, etc.). These are based on beats expressed in terms of fractions of full beats in the prevailing tempo—for example 3
10 or 5
24. For example, where 4
4 implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), 4
3 implies a bar construction of four third-parts of it. These signatures are of utility only when juxtaposed with other signatures with varying denominators; a piece written entirely in 4
3, say, could be more legibly written out in 4
According to Brian Ferneyhough, metric modulation is "a somewhat distant analogy" to his own use of "irrational time signatures" as a sort of rhythmic dissonance. It is disputed whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-irrational signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in John Adams' opera Nixon in China (1987), where the sole use of irrational signatures would quickly produce massive numerators and denominators.
Historically, this device has been prefigured wherever composers wrote tuplets. For example, a 2
4 bar of 3 triplet quarter notes could be written as a bar of 3
6. Henry Cowell's piano piece Fabric (1920) employs separate divisions of the bar (1 to 9) for the three contrapuntal parts, using a scheme of shaped noteheads to visually clarify the differences, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he finds that "such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base tempo". Thomas Adès has also used them extensively—for example in Traced Overhead (1996), the second movement of which contains, among more conventional meters, bars in such signatures as 2
14 and 5
A gradual process of diffusion into less rarefied musical circles seems underway. For example, John Pickard's Eden, commissioned for the 2005 finals of the National Brass Band Championships of Great Britain, contains bars of 3
10 and 7
Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in 4
5 is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only 4⁄5 of a reference whole note, and a beat 1⁄5 of one (or 4⁄5 of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.
Some video samples are shown below.
These video samples show two time signatures combined to make a polymeter, since 4
3, say, in isolation, is identical to 4
Some composers have used fractional beats: for example, the time signature 2+1⁄2
4 appears in Carlos Chávez's Piano Sonata No. 3 (1928) IV, m. 1. Both 2+1⁄2
4 and 1+1⁄2
4 appear in the fifth movement of Percy Grainger's Lincolnshire Posy.
Music educator Carl Orff proposed replacing the lower number of the time signature with an actual note image, as shown at right. This system eliminates the need for compound time signatures, which are confusing to beginners. While this notation has not been adopted by music publishers generally (except in Orff's own compositions), it is used extensively in music education textbooks. Similarly, American composers George Crumb and Joseph Schwantner, among others, have used this system in many of their works. Émile Jaques-Dalcroze proposed this in his 1920 collection, Le Rythme, la musique et l'éducation.
Another possibility is to extend the barline where a time change is to take place above the top instrument's line in a score and to write the time signature there, and there only, saving the ink and effort that would have been spent writing it in each instrument's staff. Henryk Górecki's Beatus Vir is an example of this. Alternatively, music in a large score sometimes has time signatures written as very long, thin numbers covering the whole height of the score rather than replicating it on each staff; this is an aid to the conductor, who can see signature changes more easily.
Early music usage
Mensural time signatures
In the mensural notation of the 14th, 15th and 16th centuries there are no bar lines, and the four basic mensuration signs indicate the normal ratio of duration between different note values. Unlike modern notation, the subdivisions could be either 2:1 or 3:1. The relation between the breve and the semibreve was called tempus, and could be perfect (triple 3:1 indicated by circle) or imperfect (duple 2:1, with broken circle), while the relation between the semibreve and the minim was called prolatio and could be major (3:1 or compound, indicated by dot) or minor (2:1 or simple meter).
Modern transcriptions often reduce note values 4:1, such that
N.B.: In mensural notation actual note values depend not only on the prevailing mensuration, but on rules for imperfection and alteration, with ambiguous cases using a dot of separation, similar in appearance but not always in effect to the modern dot of augmentation.
|Proportion||Notated values||equivalent to||Notated values|
Besides showing the organization of beats with musical meter, the mensuration signs discussed above have a second function, which is showing tempo relationships between one section to another, which modern notation can only specify with tuplets or metric modulations. This is a fraught subject, because the usage has varied with both time and place: Charles Hamm was even able to establish a rough chronology of works based on three distinct usages of mensural signs over the career of Guillaume Dufay (1397(?) – 1474). By the end of the sixteenth century Thomas Morley was able to satirize the confusion in an imagined dialogue:
it was a world to hear them wrangle, every one defending his own for the best. "What? You keep not time in your proportions." "You sing them false. What proportion is this?" "Sesquipaltry." "Nay, you sing you know not what; it would seem you came lately from a barber's shop where you had 'Gregory Walker' or a Curranta played in the new Proportions by them lately found out, called 'Sesquiblinda' and 'Sesquihearkenafter'."
- Plaine and Easie Introduction to Practicall Musicke (1597)
In general though, a slash or the numeral 2 shows a doubling of tempo, and paired numbers (either side by side or one atop another) show ratios instead of beats per measure over note value: in early music contexts 4
3 for example is unrelated to 'third-notes'.
A few common signs are shown:
- tempus imperfectum diminutum, 1:2 proportion (twice as fast);
- tempus perfectum diminutum, 1:2 proportion (twice as fast);
- or proportio tripla, 1:3 proportion (three times as fast, similar to triplets).
In particular, when the sign was encountered, the tactus (beat) changed from the usual whole note (semibreve) to the double whole note (breve), a circumstance called alla breve. This term has been sustained to the present day, and though now it means the beat is a half note (minim), in contradiction to the literal meaning of the phrase, it still indicates that the tactus has changed from a short to a doubled value.
Certain composers delighted in creating mensuration canons, "puzzle" compositions that were intentionally difficult to decipher.
Irregular bars are a change in time signature normally for only one bar. Such a bar is most often a bar of 3/4, 5/4 or 2/4 in a 4/4 composition, or a bar of 4/4 in a 3/4 composition, or a bar of 5/8 in a 6/8 composition.
If a song is entirely in 4/4 a change to 3/4 will make the song feel like it has skipped a beat, the opposite is true for 5/4 where it feels like the song adds a beat. If a song changes to 2/4 is will make it feel like that bar is half as long as all the others
Some popular examples include "Golden Brown" by The Stranglers (4/4 in a 3/4 composition), "I Love Rock 'n' Roll" originally by the Arrows (3/4 in a 4/4 composition), "Hey Ya!" by Outkast (2/4 in a 4/4 composition), and "Wuthering Heights" by Kate Bush (different kinds of irregular bars in a 4/4 composition).
- Schaffel, a kind of swing in rock and techno music
- Tala, meter in Indian music
- Colotomy, a coinage by Jap Kunst to describe the metric structure of gamelan music.
- ^ Alexander R. Brinkman, Pascal Programming for Music Research (Chicago: University of Chicago Press, 1990): 443, 450–463, 757, 759, 767. ISBN 0226075079; Mary Elizabeth Clark and David Carr Glover, Piano Theory: Primer Level (Miami: Belwin Mills, 1967): 12; Steven M. Demorest, Building Choral Excellence: Teaching Sight-Singing in the Choral Rehearsal (Oxford and New York: Oxford University Press, 2003): 66. ISBN 0195165500; William Duckworth, A Creative Approach to Music Fundamentals, eleventh edition (Boston, MA: Schirmer Cengage Learning, 2013): 54, 59, 379. ISBN 0840029993; Edwin Gordon, Tonal and Rhythm Patterns: An Objective Analysis: A Taxonomy of Tonal Patterns and Rhythm Patterns and Seminal Experimental Evidence of Their Difficulty and Growth Rate (Albany: SUNY Press, 1976): 36–37, 54–55, 57. ISBN 0873953541; Demar Irvine, Reinhard G. Pauly, Mark A. Radice, Irvine’s Writing about Music, third edition (Portland, Oregon: Amadeus Press, 1999): 209–210. ISBN 1574670492.
- ^ Henry Cowell and David Nicholls, New Musical Resources, third edition (Cambridge and New York: Cambridge University Press, 1996): 63. ISBN 0521496519 (cloth); ISBN 0521499747 (pbk); Cynthia M. Gessele, "Thiéme, Frédéric [Thieme, Friedrich]", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001); James L. Zychowicz, Mahler's Fourth Symphony (Oxford and New York: Oxford University Press, 2005): 82–83, 107. ISBN 0195181654.
- ^ Edwin Gordon, Rhythm: Contrasting the Implications of Audiation and Notation (Chicago: GIA Publications, 2000): 111. ISBN 1579990983.
- ^ Scott Schroedl, Play Drums Today! A Complete Guide to the Basics: Level One (Milwaukee: Hal Leonard Corporation, 2001), p. 42. ISBN 0-634-02185-0.
- ^ See File:Bach BVW 1041 Allegro Assai.png for an excerpt from the violin part of the final movement.
- ^ Dokter, Julia (2021). Tempo and Tactus in the German Baroque. University of Rochester Press. p. 144. ISBN 9781648250187.
- ^ "Odd Time Signatures: A Complete Guide | Hello Music Theory". hellomusictheory.com/. 6 March 2020. Retrieved 2022-02-20.
- ^ a b Tim Emmons, Odd Meter Bass: Playing Odd Time Signatures Made Easy (Van Nuys: Alfred Publishing, 2008): 4. ISBN 978-0-7390-4081-2. "What is an 'odd meter'?...A complete definition would begin with the idea of music organized in repeating rhythmic groups of three, five, seven, nine, eleven, thirteen, fifteen, etc."
- ^ "Tchaikovsky's Symphony # 6 (Pathetique), Classical Classics, Peter Gutmann". Classical Notes. Retrieved 2012-04-20.
- ^ Edward Macan, Rocking the Classics: English Progressive Rock and the Counterculture (New York: Oxford University Press, 1997): 48. ISBN 978-0-19-509888-4.
- ^ Radiohead (musical group). OK Computer, vocal score with guitar accompaniment and tablature (Essex, England: IMP International Music Publications; Miami, Florida: Warner Bros. Publications; Van Nuys, California: Alfred Music, 1997):[page needed]. ISBN 0-7579-9166-1.
- ^ Manuel, Peter (1988). Popular Musics of the Non-Western World: An Introductory Survey (rev. ed.). Oxford University Press. p. 131. ISBN 9780195063349.
- ^ Gardner Read, Music Notation: A Manual of Modern Practice (Boston: Allyn and Bacon, Inc., 1964):[page needed]
- ^ Constantin Brăiloiu, Le rythme Aksak, Revue de Musicologie 33, nos. 99 and 100 (December 1951): 71–108. Citation on pp. 75–76.
- ^ Audio: "Eleno Mome" from The Dances of the World's Peoples, Vol. 1: Dances of the Balkans and Near East, Smithsonian Folk Ways
- ^ Gheorghe Oprea, Folclorul muzical românesc (Bucharest: Ed. Muzicala, 2002),[page needed] ISBN 973-42-0304-5
- ^ "The Journal of Anaphorian Music Theory".
- ^ "Frog Peak Artist: John Chalmers".
- ^ a b c d "Brian Ferneyhough", The Ensemble Sospeso
- ^ John Pickard: Eden, full score, Kirklees Music, 2005.
- ^ Jaques-Dalcroze, Émile (1967). Rhythm, Music and Education. Translated by Harold F. Rubenstein. London: Dalcroze Society. p. 84, and Appendix, example 2. – Page 210 in the French original
- ^ Apel 1953, p. 150.
- ^ C. Hamm: A Chronology of the Works of Guillaume Dufay based on a Study of Mensural Practice (Princeton, New Jersey, 1964)
- ^ Thomas Morley: Plaine and Easie Introduction to Practicall Musicke (1597), quoted in the unsigned "Proportions" in New Grove XV p. 307 (1980)
- ^ Apel 1953, p. 189.
- ^ Apel 1953, p. 148.
- ^ Ernst Friedrich Richter, A Treatise on Canon and Fugue: Including the Study of Imitation, translated from third German edition by Arthur W. Foote (Boston: Oliver Ditson, 1888): 38. [ISBN unspecified].
- ^ Songs That Skip a Beat, retrieved 2022-09-22
- ^ Songs That Add a Beat, retrieved 2022-09-22
- Apel, Willi (1953). The Notation of Polyphonic Music 900–1600. The Medieval Academy of America Publication No. 38 (fifth, revised and with commentary ed.). Cambridge, Massachusetts: Medieval Academy of America. ISBN 9780910956154.