1 George Armitage Miller, ‘The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information’, Psychological Review, 63 (1956), 81–97 (p. 96).

2 The chorus reverts to 4/4.

3 The Appendix contains a chronology of works making significant use of septuple time. In this study, septuple metre includes time signatures such as 7/4, alternating metres such as 3/8+2/4, and multiples of seven such as (3+3+4+4)/8. It includes music written before modern barlines and time signatures, where a seven-beat metre can be inferred. I will also discuss non-Western musics whose concept of metre and rhythm is different, but where rhythms are based on a seven-beat structure. The terms ‘septuple time’, ‘seven time’, ‘septuple metre’, etc. will be used interchangeably.

4 The term ‘meme’ was first coined by Richard Dawkins in The Selfish Gene (Oxford, 1976). A good overview of memetics and its application is given by Susan Blackmore, The Meme Machine (Oxford, 1999). For some musical applications, see Steven B. Jan, The Memetics of Music (Aldershot, 2007). Blackmore (p. 14) identifies the three characteristics of memetic replication that are direct parallels of those outlined by Darwin: variation, or slight imperfection in the copying process, allowing the meme to evolve over time and adapt to different situations and contexts; selection, or the ability to grab the attention and be passed on; and retention, the existence of essential features that persist from one generation to the next.

5 The genetic parallel of a memeplex is an organism (a plant or animal), or even an entire ecosystem, which can be regarded as a vehicle created to maximize the survival chances of its constituent genes.

6 Some works listed in the Appendix to this article were identified by searching for key phrases in concert reviews in the online archive of The Times. To illustrate the potential weaknesses of this approach, I searched for reviews of Berlioz's works between op. 15 and op. 26 inclusive. Three of these 12 received no reviews. L'enfance du Christ, op. 25, had three reviews, only one of which mentioned its use of septuple time. Based on this small sample, the chance of finding a septuple-time work by this method is 25%. The hit-rate can be substantially improved by searching multiple sources, but it is nevertheless likely that I have missed at least as many seven-time works as I have found, probably more among music by less well-known composers.

7 Tiny, that is, compared to the total population of works of Western art music. The total number of published works was perhaps of the order of two million by 1900 (according to estimates given in Music Printing and Publishing, ed. Donald Krummel and Stanley Sadie (Basingstoke, 1990), 129), and with the expansion of music publishing in the twentieth century this figure will have multiplied substantially. Septuple time might thus occur in no more than around 0.01% of Western musical works.

8 Gustave Reese, Music in the Middle Ages (New York, 1940), 51.

9 Martin L. West, Ancient Greek Music (Oxford, 1992), 151.

10 All extant fragments have been transcribed by Egert Pöhlmann and Martin L. West, Documents of Ancient Greek Music (Oxford, 2001). The metre of the Oslo Papyrus fragment is by no means certain, with Pöhlmann and West commenting that it is in ‘iambic trimeters’, and that ‘the resources of rhythmic notation are employed sparingly’ (p. 131). West's earlier transcription (Ancient Greek Music, 313) is notated in 6/8 metre, with the first four quavers of each bar written as quadruplets covering the time of three quavers.

11 It does seem that ancient Egyptian music was strongly rhythmic. Curt Sachs, in Rhythm and Tempo: A Study in Music History (New York, 1953), suggests that an Egyptian inscription from around 2000 BC naming three professional hand-clapping women indicates that ‘the [clappers] must […] have given to melody the intricate patterns of additive rhythm that the Near and Middle East has preserved through thousands of years to the present’ (pp. 94–5). Hans Hickmann, in ‘Rythme, mètre et mesure de la musique instrumentale et vocale des anciens Egyptiens’, Acta musicologica, 32 (1960), 11–22, observes how little we know about the stresses of ancient Egyptian speech, making extrapolation from poetic to musical metre almost impossible. However, he concludes that the Egyptians must have had simple regular metrical rhythms: ‘Toute chanson ou autre forme de musique vocale, accompagnée par le battement des mains si souvent représenté, doit être relativement simple dans sa structure métrique et rythmique’ (p. 14).

12 Julian Ribera, Music in Ancient Arabia and Spain, trans. Eleanor Hague and Marion Leffingwell (London, 1929), esp. Chapter 3 (pp. 3–42).

13 Although Mukund Lath, in A Study of Dattilam: A Treatise on the Sacred Music of Ancient India (New Delhi, 1978), fails to find evidence of septuple metres in the ancient Dattilam text, which he dates as ‘pre-seventh-century’ (p. 444), he remarks that complex rhythmic patterns including sevens ‘were certainly used in non-dramatic popular forms’ at the time (p. 324).

14 Henry G. Farmer, in A History of Arabian Music (London, 1929), for example, describes how the rhythm of ancient Arabian music was intimately related to the poetic metre of the text, whilst Curt Sachs observes that later Arab musicians were the first to use ‘rhythms specifically musical, beyond the simple feet and lines of versification’ (Rhythm and Tempo, 85).

15 Several examples from both Indian traditions are given in Peggy Holroyde, Indian Music: A Vast Ocean of Promise (London, 1972), 30–42.

16 Tolga Bektas, ‘Relationships between Prosodic and Musical Meters in the “Beste” Form of Classical Turkish Music’, Asian Music, 36 (2005), 1–26 (p. 5).

17 Sidney Moore, ‘Thai Songs in 7/4 Meter’, Ethnomusicology, 13 (1969), 309–12.

18 Anna Czekanowska, Polish Folk Music: Slavonic Heritage, Polish Tradition, Contemporary Trends (Cambridge, 1990), 194.

19 Boris A. Kremenliev, Bulgarian-Macedonian Folk Music (Berkeley, CA, 1952), 27.

20 Helen Rees, ‘The Many Musics of a Chinese County Town: A Case-Study of Co-Existence in Lijiang, Yunnan Province’, Asian Music, 27 (1995), 63–102 (pp. 68–70).

21 James Irsay, Dances and Trances (CD liner notes, Arbiter Records #2002, New York, 2000).

22 Rose Brandel, The Music of Central Africa: An Ethnomusicological Study (The Hague, 1961), 225.

23 Helen Myers, Music of Hindu Trinidad: Songs from the India Diaspora (Chicago, IL, 1998), 75–6.

24 Godfried Toussaint, ‘The Euclidean Algorithm Generates Traditional Musical Rhythms’, Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, ed. Reza Sarhangi and Robert V. Moody (Banff, 2005), 7–11.

25 Stephen G. Hatherly, A Treatise on Byzantine Music (London, 1892), 152.

26 Kremenliev, Bulgarian-Macedonian Folk Music, 27, mentions an alternative theory regarding septuple time's arrival in the Balkans: ‘Christov believes that the Ruchenitza might have come to Bulgaria from the Orient through the Turkish-Tartar tribe which came to the Balkans with Isperikh in the seventh century’ (citing Dobri Christov, Tekhnicheskiiat stroezn na bulgarskata narodna muzika (Sofia, 1928), 31–2). This is supported by the observation that Greek theory used a basic time unit (the chronos protos), which is inconsistent with the Balkan practice of uneven beats, whereby a dance in 7/16 metre might be regarded by a Balkan musician as being in triple time, with three beats of length 2, 2 and 3 (Kremenliev, op. cit., 21).

27 It is likely that Alfonso himself was involved in creating some of these songs.

28 Ribera, Music in Ancient Arabia and Spain, 3.

29 According to Anna Maria Busse Berger, in ‘The Evolution of Rhythmic Notation’, The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge, 2002), 628–56, Franco broke away from the tradition of rhythmic ‘modes’, although interpretation of this notation still requires an understanding of perfect (three-beat) and imperfect (two-beat) intervals.

30 The phrases all end with a pattern. The short ‘barlines’ in the facsimile relate to the notation of the dotted crotchet, rather than demarcation of bars. There is scope for interpretation in works such as this: for example, the rhythm of ‘en’ in bar 5 is transcribed here (and in the Dufay Collective's recording) as a ligature, although the literal notation is a quaver longer than this. It is worth noting that scholars disagree on the extent to which the Cantigas would have been performed in a mensural manner (see, for example, Hendrik van der Werf, ‘Accentuation and Duration in the Music of the Cantigas de Santa Maria’, Hispanic Seminary of Medieval Studies (1987), 223–34). No. 254 is perhaps unusual in its metrical regularity as well as its septuple metre.

31 Reese, Music in the Middle Ages, 276, writes that ‘a vertical line […] was not used to rule off measures […] in copies of mensural music intended for practical use until the second half of the 15th century, and even then its employment was restricted’. See also the discussion in Andrew Hughes, ‘Mensuration and Proportion in Early Fifteenth Century English Music’, Acta musicologica, 37 (1965), 48–61.

32 Sylvia Townsend Warner, ‘An Aspect of Tudor Counterpoint’, Music and Letters, 2 (1921), 35–49, gives such an example from a duo by a ‘Mr Gyles’ from the sixteenth-century Baldwin MS, but also comments that ‘the more recondite and far-fetched of the proportions were kept […] for the use of the learner and the learned man’ (p. 40).

33 For example, John Stainer, ‘Morley's Plaine and Easie Introduction to Practicall Musicke’, Musical Times and Singing Class Circular, 43 (1902), 457–60, comments as follows: ‘At the close of the fifteenth and in the early part of the sixteenth century, learned musicians loved to revel in the mixture of notes of all sorts of relative lengths […]. But this silly attempt to combine notes of such different relative lengths was happily confined to the works of theorists, [and] it rarely found its way into compositions intended for use. Old [Thomas] Morley seems to have been laughing up his sleeve while discoursing upon it’ (p. 459).

34 The aria was published in Telemann's serialized collection Der getreue Musikmeister in 1728 and is a survivor from his lost opera Sancio. Earlier examples of this technique are the arias ‘Bel piacere’ from Act 3, scene x of Handel's Agrippina (1709), and ‘Tornami in seno o'core’ from Act 2, scene vii of L'Orione by Cavalli (1653), both of which mix 2/4 and 3/8 time signatures but never achieve a regular seven-time effect.

35 Sven Hansell and Carlida Steffan, ‘Adolfati, Andrea’, Grove Music Online, Oxford Music Online, <http://www.oxfordmusiconline.com> (accessed 26 July 2009).

36 The claim was made in David Kettlewell, ‘7/8 Metres’, Early Music, 2 (1974), 203. Shield's String Trios nos. 1, 3 and 8, published in 1791, all have final movements in 5/4 metre, so he may also have used seven time. However, no self-standing Scherzo is listed among Shield's works, and no septuple or quintuple metres are mentioned in his pedagogical writings.

37 ‘Eusebius’ is an Adagio notated in duple metre, although it is largely in a seven-over-two cross-rhythm. Eusebius was Schumann's name for the dreamy, introspective side of his character, contrasting with the passionate, outgoing Florestan (in Carnaval, a ‘Passionato’ in 3/4 time).

38 Arthur Henry Fox Strangways, The Music of Hindostan (Oxford, 1914).

39 Béla Bartók, Rumanian Folk Music, trans. E. C. Teodorescu, ed. Benjamin Suchoff (The Hague, 1967), 44, identifies 69 of his collection of 914 Rumanian folk tunes as being in so-called ‘Bulgarian’ metres. Of these, 19 are in a septuple metre, most commonly (3+4)/16. In Bartók's Hungarian Folk Music, trans. Michel D. Calvocoressi (Oxford, 1931), only Example 34 (out of 320) is in septuple metre.

40 Several examples are listed in the Appendix.

41 Many works written after the 1930s are still in copyright, which may account for their being less frequently cited in some types of source. Books in copyright are less likely to be available in an electronically searchable form, so some of the sources most easily searched will exclude works written after this time. For the same reason, some full-text online resources such as <http://www.archive.org> or <http://imslp.org> have little material published after around 1930.

42 Much credit must also go to the quartet's drummer, Joe Morello, who died whilst this article was in preparation. His obituary in the Guardian (14 March 2011) describes him as the ‘understated drummer who anchored the Dave Brubeck Quartet’, and observes that ‘his implacably mathematical yet hypnotically dramatic solo over the piano vamp at the close of the band's most famous piece, Take Five, is a jazz landmark, and a tutorial for generations of drummers’.

43 According to Fred Hall, It's About Time: The Dave Brubeck Story (Fayetteville, AR, 1996), 74, there were ‘occasions in Afghanistan, Turkey and India, where Dave immersed himself in the music of the East, that later influenced much of his composing’.

44 The Wikipedia article ‘Septuple Metre’ lists many examples from ‘progressive rock’ and its successors, a subgenre that emerged in the late 1960s, which aimed to extend the technical and compositional boundaries of rock music. Pink Floyd, Jethro Tull and Peter Gabriel are among the few exponents who enjoyed some chart success with septuple time.

45 See, for example, Thomas S. Omond, English Metrists in the Eighteenth and Nineteenth Centuries (Oxford, 1907), 180.

46 Several relevant studies are mentioned in Dalia Cohen and Ruth Katz, ‘Rhythmic Patterns Reflecting Cognitive Constraints and Aesthetic Ideals’, Journal of New Music Research, 37 (2008), 15–35.

47 Richard Roe, The Principles of Rhythm, Both in Speech and Music, Especially as Exhibited in the Mechanism of English Verse (Dublin, 1823), 5. Roe (1764/5–1853) was a stenographer, writer and, between 1816 and 1834, assistant secretary and assistant librarian to the Irish Royal Academy. He was also a ‘popular bass singer’, according to Page Life, ‘Roe, Richard Baillie (1764/5?–1853)’, Oxford Dictionary of National Biography (Oxford, 2004; online edn, January 2008, <http://www.oxforddnb.com/view/article/23942>, accessed 24 January 2012).

48 Johann Philipp Kirnberger, The Art of Strict Musical Composition (1779), trans. David Beach and Jurgen Thym, with introduction and explanatory notes by David Beach (New Haven, CT, and London, 1982), 383.

49 Dirk-Jan Povel, ‘Internal Representation of Simple Temporal Patterns’, Journal of Experimental Psychology: Human Perception and Performance, 7 (1981), 3–18.

50 Some languages achieve stress by changing the pitch rather than (or as well as) lengthening the syllables. Ancient Greek is thought to have been of this type, which perhaps explains the peculiarity, to modern Western ears, of ancient Greek metre. Several oriental languages are also in this category, and it is perhaps no coincidence that non-binary metres are extremely rare in native Chinese music (Sachs, Rhythm and Tempo, 58–9).

51 René Descartes, A Compendium of Musick (London, 1653), 6–7.

52 Miller, ‘The Magical Number Seven’.

53 Philip N. Johnson-Laird, ‘Rhythm and Meter: A Theory at the Computational Level’, Psychomusicology: A Journal of Research in Music Cognition, 10 (1991), 88–106, citing Miller, speculates that seven represents a limit to the number of metrical divisions it is possible to hear.

54 An early exception is In Nomine IX, by John Bull (?1562–1628), which is in (8+3)/4 time throughout (The Fitzwilliam Virginal Book, ed. John A. Fuller Maitland and William Barclay Squire, 2 vols. (New York, 1979), ii, 34–9). The next example I know of is Saint-Saëns's Prière, op. 7 no. 3 (1866). Scriabin's Prelude op. 11 no. 24 (1895) is in (6+5)/8.

55 In books on Indian music, tables of tala, more often than not, skip straight from 10 to 12 beats (see, for example, Holroyde, Indian Music, 281–2). Martin Clayton, in Time in Indian Music: Rhythm, Metre, and Form in North Indian Rag Performance (Oxford, 2008), mentions 11-beat tala only once (p. 131). Tabla-player Krsna Dhenu, in ‘Nine, Eleven, and Thirteen Matra Talas’ (<http://kksongs.org/tabla/chapter30.html>, accessed 18 April 2009), mentions two rare Indian 11-beat tala, one of which is ‘almost extinct’.

56 Mihaly Csikszentmihályi, Flow: The Psychology of Optimal Experience (New York, 1990).

57 Reed Larson, ‘Flow and Writing’, Optimal Experience: Psychological Studies of Flow in Consciousness, ed. Mihaly Csikszentmihályi and Isabella Csikszentmihályi (Cambridge, 1992), 150–71 (p. 171).

58 Eric Clarke, ‘Rhythm and Timing in Music’, The Psychology of Music, ed. Diana Deutsch (New York, 1999), 473–500 (p. 495).

59 For a discussion of entrainment in general and its application to music, see Martin Clayton, Rebecca Sager and Udo Will, ‘In Time with the Music: The Concept of Entrainment and its Significance for Ethnomusicology’, ESEM CounterPoint, 1 (2004), 1–82. Recent research has found links between parts of the brain involved in musical expectation, emotion and movement. See, for example, Marcus T. Pearce et al., ‘Unsupervised Statistical Learning Underpins Computational, Behavioural, and Neural Manifestations of Musical Expectation’, NeuroImage, 50 (2010), 302–13.

60 Jessica Phillips-Silver and Laurel J. Trainor, in ‘Feeling the Beat: Movement Influences Infants’ Rhythm Perception’, Science, 308 (2005), 1430, conclude that physical movement is important for very young children in interpreting two- or three-beat rhythms, and in ‘Hearing What the Body Feels: Auditory Encoding of Rhythmic Movement’, Cognition, 105 (2007), 533–46, they reach the same conclusion for adults. They do not consider rhythms in five or seven.

61 Aniruddh D. Patel, ‘Musical Rhythm, Linguistic Rhythm, and Human Evolution’, Music Perception, 24 (2006), 99–104 (p. 100).

62 Ian Cross, ‘Music and Meaning, Ambiguity and Evolution’, Musical Communication, ed. Dorothy Miell, Raymond MacDonald and David Hargreaves (Oxford, 2004), 27–44 (pp. 29–30).

63 Thaddeus L. Bolton, ‘Rhythm’, American Journal of Psychology, 6 (1894), 145–238 (p. 192). Bolton makes similar comments about another respondent (with ‘some musical talent and training’). Seven-groups are not mentioned for any other of the 30 respondents.

64 Stephen Handel and Gregory R. Lawson, ‘The Contextual Nature of Rhythmic Interpretation’, Perception and Psychophysics, 34 (1983), 103–20. The experiments are also described in Stephen Handel, ‘Rhythm and Meter: Using Polyrhythms to Study Rhythm’, Music Perception, 1 (1984), 465–84, where he adds the caveat that ‘polyrhythms provide a good context for studying rhythm as a foreground melody at the expense of providing a poor context for studying rhythm as a background-organising factor for tonal melody’ (p. 465).

65 Indian classical music, Balkan folk music, and the music of other cultures utilizing septuple time all make use of improvisation, so this observation cannot be assumed to apply beyond Western musicians.

66 Dave Brubeck, liner notes, Countdown: Time in Outer Space: The Dave Brubeck Quartet, CD COL 512894-2 (Columbia/Legacy, 2004), 6.

67 John Ross Frampton, ‘Some Evidence for the Naturalness of the Less Usual Rhythms’, Musical Quarterly, 12 (1926), 400–5 (p. 403). Frampton's case would be stronger if he had cited examples without a Balkan connection.

69 Christopher Francis Hasty, Meter as Rhythm (Oxford, 1997), 146. Hasty argues that an important aspect of metre perception is a process of ‘projection’, where events and their timing are anticipated on the basis of what has gone before.

68 Bruno H. Repp, Justin London and Peter E. Keller, ‘Production and Synchronization of Uneven Rhythms at Fast Tempi’, Music Perception, 23 (2005), 61–78.

70 Mark J. Steedman, ‘The Perception of Musical Rhythm and Metre’, Perception, 6 (1977), 555–69 (p. 577; italics original). Steedman used his principle as the basis of a computer program that successfully identified musical metre from the placing of musical events (in Bach fugues).

71 Whilst this line of reasoning may be valid in the West, it is hard to argue that these characteristics of septuple time cause problems in, say, Balkan music.

72 Erin E. Hannon and Sandra E. Trehub, ‘Metrical Categories in Infancy and Adulthood’, Psychological Science, 16 (2005), 48–55.

73 Erin E. Hannon and Sandra E. Trehub, ‘Metrical Categories in Infancy and Adulthood’, Psychological Science, 16 (2005), 53.

74 Erin E. Hannon and Laurel J. Trainor, ‘Music Acquisition: Effects of Enculturation and Formal Training on Development’, Trends in Cognitive Sciences, 11 (2007), 466–72 (p. 468).

75 It is unclear whether this effect is psychological/neurological or memetic.

76 Joel S. Snyder, Erin E. Hannon, Edward W. Large and Morton H. Christiansen, ‘Synchronization and Continuation Tapping to Complex Meters’, Music Perception, 24 (2006), 135–45 (p. 135).

77 Handel's caveat may also be relevant (see n. 64).

78 Examples of these are Peter Gabriel's ‘Solsbury Hill’ (1977), Holst's ‘Hymn to the Waters’ (1910) and the Andante from Brahms's Piano Trio op. 101 (1886) respectively.

79 Justin London, Hearing in Time (Oxford, 2004), 103. Godfried Toussaint, in ‘The Euclidean Algorithm’, observes that such rhythmic combinations are generated mathematically by a procedure known as the Euclidean Algorithm, and lists the formulae for many examples of asymmetric rhythms.

80 As in Pink Floyd's ‘Money’ (1973), where the anticipated eighth beat of each bar of the repeated bass line (below) is actually the first note of the next:

(Transcription from The Dark Side of the Moon, Capitol SMAS-11163, track 6.)

81 This is explicit in the folksong ‘Cronan Bleoghain’ (1909), and in Alkan's ‘Air à 7 temps’, no. 8 from the Impromptus op. 32 (Paris, 1849; opening illustrated below), where the seventh beat of the bar is always a repetition of the sixth:

This might be a formalization of the practice of leaving a pause or fermata at the end of phrases in hymns and chorales; see n. 157.

82 Martin Clayton, ‘Culture, Cognition and Additive Rhythm: A Comparative Case Study’, Music, Time and Place: Essays in Comparative Musicology (Delhi, 2007), 55–9 (p. 58).

83 London, Hearing in Time, 31–3.

84 A good example is Shostakovich's Prelude no. 14 from his 24 Preludes and Fugues, op. 87 (1950).

85 Even in Balkan music, phrases almost always consist of two, four or eight bars (Béla Bartók, Rumanian Folk Music, i, 45). The broader question of septuple hypermetre is beyond the scope of the present article, but it is a topic on which there appears to be almost no existing literature, other than occasional mentions in the context of individual works.

86 Indian music using seven-beat tala, typically based on a seven-beat repeated melodic pattern, is usually more like the Western ‘regular’ septuple metre than the Balkan version, although fast, Balkan-like septuple rhythms are also found in Indian music (Clayton, ‘Culture, Cognition and Additive Rhythm’, 57).

87 West, Ancient Greek Music, 51.

88 Charles Francis Abdy Williams, The Aristoxenian Theory of Musical Rhythm (Cambridge, 1911), 75 and 58 respectively.

89 Sachs, Rhythm and Tempo, 205.

90 Anne Elizabeth Walters, ‘Review: Six Publications from the Colorado College Music Press by General Editor Albert Seay’, Journal of Music Theory, 26 (1982), 349–56 (p. 353).

91 Sylvestro di Ganassi, Opera intitulata Fontegara (Venice, 1535), cited by Sachs, Rhythm and Tempo, 207.

92 Judson D. Maynard, ‘An Anonymous Scottish Treatise on Music from the Sixteenth Century, British Museum, Additional Manuscript 4911: Edition and Commentary’, 2 vols. (Ph.D. dissertation, Indiana University, 1961), ii, 335. Roughly translated, this says that although such proportions may not be pleasant to sing, they should not be rejected, because they are properly situated within the gamut of proportions.

93 Thomas Morley, A Plaine and Easie Introduction to Practicall Musicke Set Downe in Forme of a Dialogue (London, 1597), 55 (see also n. 33). Andreas Ornithoparcus, Micrologus, or Introduction: Containing the Art of Singing (London, 1609), 60–1.

94 Elway Bevin, A Briefe and Short Instruction of the Art of Musicke to Teach How to Make Discant, of All Proportions that Are in Use (London, 1631), 3.

95 Charles Butler, The Principles of Musik, in Singing and Setting (London, 1636), 29.

96 Descartes, A Compendium of Musick, 7.

97 Lorenzo Penna, Li primi albori musicali per li principianti della musica figurata; distinti in tre’ libri (Bologna, 1672), cited in David Damschroder and David Russell Williams, Music Theory from Zarlino to Schenker: A Bibliography and Guide (New York, 1990), 232.

98 Sébastien de Brossard, A Musical Dictionary; Being a Collection of Terms and Characters, as well Ancient as Modern; Including the Historical, Theoretical, and Practical Parts of Music (London, 1740), 302 and 282.

99 Kirnberger, The Art of Strict Musical Composition, 383.

100 Jean-Jacques Rousseau, The Complete Dictionary of Music. Consisting of a Copious Explanation of All Words Necessary to a True Knowledge and Understanding of Music, trans. William Waring (London, 1779), 225. Rousseau follows this with an unidentified piece in 5/4 time.

101 Charles Burney, A General History of Music, from the Earliest Ages to the Present Period (1789), ed. Frank Mercer, 2 vols. (New York and London, 1935), ii, 778. The comment refers to the last scene of the second act of Handel's Orlando.

102 Augustus Frederic Christopher Kollmann, An Essay on Musical Harmony, According to the Nature of that Science and the Principles of the Greatest Musical Authors (London, 1796), 76. Kollmann, a German who spent most of his adult life in England, goes on to describe sextuple metre as ‘the last sort of extraordinary simple measures’, not even acknowledging septuple time as a possibility (p. 76).

103 Roe, The Principles of Rhythm, 6.

104 Berlioz's Orchestration Treatise: A Translation and Commentary, trans. Hugh Macdonald (Cambridge, 2002), 342.

105 Gottfried Weber, The Theory of Musical Composition, trans. James F. Warner (London, 1851), 98–102. Moritz Hauptmann, The Nature of Harmony and Metre (1853), trans. William Edward Heathcote (London, 1888), 196–9.

106 Ferdinand Hand, Aesthetics of Musical Art: or, The Beautiful in Music, trans. Walter E. Lawson (London, 1880), 42.

107 François-Joseph Fétis, ‘Du développement futur de la musique: Dans le domaine de rhythm’, Revue et gazette musicale de Paris, 19 (1852), 281–476.

108 Mary I. Arlin, ‘Metric Mutation and Modulation: The Nineteenth-Century Speculations of F.-J. Fetis’, Journal of Music Theory, 44 (2000), 261–322 (p. 265).

109 William E. Caplin, ‘Theories of Musical Rhythm in the Eighteenth and Nineteenth Centuries’, The Cambridge History of Western Music Theory, ed. Christensen, 657–94 (p. 679).

110 Frederick J. Crowest, Musical Groundwork: Being a First Manual of Musical Form and History, for Students and Readers (London, 1890), 77.

111 Ebenezer Prout, Musical Form (London, 1893), 149.

112 John Frederick Rowbotham, ‘On the Differences between Ancient and Modern Art’, Proceedings of the Musical Association, 14 (1887), 23–42 (p. 29). Hatherly, A Treatise on Byzantine Music, 152.

113 George Grove and John Alexander Fuller-Maitland, ‘Time’, A Dictionary of Music and Musicians (A.D. 1450–1889), ed. George Grove, 4 vols. (London, 1900), iv, 120.

114 Leo Smith, Musical Rudiments (Boston, MA, 1920), 46.

115 John Ross Frampton, ‘Some Evidence for the Naturalness of the Less Usual Rhythms’, Musical Quarterly, 12 (1926), 400–5 (p. 400).

116 [Unsigned], ‘Indian Music’, Musical Times and Singing Class Circular, 37 (1896), 519–21.

117 Arthur Hutchings, ‘Indian Traditions, Classical and Popular’, Music and Letters, 27 (1946), 79–83.

118 Gardner Read, Modern Rhythmic Notation (Bloomington, IN, 1978), 84.

119 Molly Gustin, ‘A Theory of Roots’, Journal of Music Theory, 6 (1962), 178–98; Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music (Cambridge, MA, 1983); Benjamin O. Miller, ‘Time Perception in Musical Meter Perception’, Psychomusicology: A Journal of Research in Music Cognition, 12 (1993), 124–53. This limitation is also noted in Edward W. Large and John. F. Kolen, ‘Resonance and the Perception of Musical Meter’, Connection Science, 6 (1994), 177–208 (p. 181).

120 Lerdahl and Jackendoff, A Generative Theory of Tonal Music, 279. Limiting consideration to duple and triple metres was arguably a reasonable simplifying assumption by Lerdahl and Jackendoff in developing an innovative, complex and influential theory covering many aspects of Western music.

121 See above, p. 368.

122 London, Hearing in Time.

123 Indeed, little space has been devoted to rhythm and metre in general in books of Western music theory, in comparison, for example, with the extensive discussion of harmony and structure.

124 Leslie A. Osborn, ‘Notation Should Be Metric and Representational’, Journal of Research in Music Education, 14 (1966), 67–83 (p. 82).

125 Whilst we are familiar with complex notation of the twentieth century, music was generally notated more simply in earlier periods. The extra complexity involved with septuple metre would arguably have been relatively much greater than it is today.

126 This is still reflected in standard measures of time and imperial measures of length.

127 Busse Berger, ‘The Evolution of Rhythmic Notation’, 643–4.

128 Sachs, Rhythm and Tempo, 171.

129 Reese, Music in the Middle Ages, 51.

130 Sachs, Rhythm and Tempo, 92.

131 Carl Dahlhaus recognizes similar trade-offs in his Esthetics of Music, trans. William Austin (Cambridge, 1982), 92, where he observes that ‘there seems to be solid evidence of concern to maintain a balance between complication in one direction and simplicity in another; such concern has prevailed in all periods […]. Some simple aspect – unity of meter or limitation of chord-vocabulary – typically formed a support and foil for complications in rhythmic detail or in harmonic-tonal relationships.’

132 The way in which notation supports and reinforces the Western musical aesthetic is evidenced by the difficulty of using Western notation to transcribe other musics. Martin Clayton, ‘A. H. Fox Strangways and the Music of Hindostan: Revisiting Historical Field Recordings’, Journal of the Royal Musical Association, 124 (1999), 86–118 (p. 106), describes a case where a transcription in seven time would have been more appropriate than that produced by Arthur Fox Strangways in his study of the music of Hindustan. Cecil Sharp also commented on the difficulty of identifying the metre of some folksongs (see ‘My Boy Billy’ (1912) in the Appendix). Such difficulties perhaps reflect other cultures’ more flexible approach to metre than we are used to (and, arguably, than is encouraged by our notation) in the West.

133 A similar argument applies, for example, to non-diatonic scales. Tunes in the major and minor scales are easily notated, but other scales are more difficult to notate (and thus to perform), requiring many accidentals or non-standard key signatures. They were largely ignored by Western composers until the late nineteenth century, despite being common elsewhere.

134 Many radical works, such as Stravinsky's Rite of Spring, became exemplars once their innovations were accepted into the memeplex.

135 Beethoven wrote to his publisher expressing concern that Reicha's experiments might overshadow his own reputation for originality. According to Julia Ronge, on the Beethoven-Haus Bonn website, ‘Beethoven rejected Reicha's fugues and was particularly upset by his claim that they had been composed following a “nouvelle Méthode”, which according to Beethoven would only lead to “the fugue no longer being a fugue”. In order to distance himself from this new trend and to highlight his own originality (which he really considered to be “modern”), Beethoven sent the publishers the exact words for the preface which he wanted in the original edition of the variations [opp. 34 & 35].’ Quoted from Ludwig van Beethoven: Brief an Breitkopf & Härtel in Leipzig, Wien, etwa 18. Dezember 1802, Autograph, <http://www.beethoven-haus-bonn.de/sixcms/detail.php?id=&template=dokseite_digitales_archiv_en&_dokid=b295&_seite=1> (accessed 13 February 2009).

136 This chart uses the dates in the chronology of works appended to this article. The gradually increasing average age can be largely attributed to rising life expectancy.

137 According to Peter Eliot Stone, in ‘Reicha, Antoine’, Grove Music Online, Oxford Music Online, <http://www.oxfordmusiconline.com> (accessed 26 July 2009). Reicha is believed to have read Kirnberger, The Art of Strict Musical Composition. However, see also n. 141.

138 This information is from the composers’ entries in Grove. I have searched unsuccessfully for septuple-time works by Franck, Milhaud, Stanford or Gédalge.

139 As well as the French composers on the list, Reicha spent two years in Paris shortly before writing his 36 Fugues; Liszt spent several periods in Paris, and French became his preferred language; Cui's father was a French army officer who remained in Russia after Napoleon's retreat from Moscow; and Tcherepnin moved to Paris in his early twenties.

140 Deirdre Donnellon, ‘French Music Since Berlioz: Issues and Debates’, French Music Since Berlioz, ed. Richard Langham Smith and Caroline Potter (Aldershot, 2006), 1–18 (p. 9).

141 Antoine Reicha, writing in French in the introduction to his Trente six fugues pour le piano-forté composées d'après un nouveau système (Vienna, 1805), acknowledges both an oriental and a European folk influence: ‘Il existe des chants et des danses nationales dans plusieurs pays de l'Asie, qu'on ne peut rendre que par des mesures composées; et il en existe même dans plusieurs contrées de l'Europe.’

142 Bartók, Hungarian Folk Music, 9.

143 Sachs, Rhythm and Tempo, 171.

144 Aniruddh D. Patel and Joseph R. Daniele, ‘An Empirical Comparison of Rhythm in Language and Music’, Cognition, 87 (2003), 35–45. David Huron and Joy Ollen, in ‘Agogic Contrast in French and English Themes: Further Support for Patel and Daniele’, Music Perception, 21 (2003), 267–71, characterize the distinction with the examples of ‘Frère Jacques’, a melody with evenly spaced notes, and ‘English Country Garden’, with notes of contrasting long and short durations. This research is presented as a comparison of a ‘stress-timed’ language (English, where stresses tend to be regularly spaced) with a ‘syllable-timed’ language (French, where syllables tend to be regularly spaced). This distinction, as measured by Patel and Daniele and others, although statistically significant, is subtle. Patel and Daniele, in ‘Stress-Timed vs. Syllable-Timed Music? A Comment on Huron and Ollen’, Music Perception, 21 (2003), 273–6, obtained different results with German speech and music. Antonio Pamies Bertrán, ‘Prosodic Typology: On the Dichotomy between Stress-Timed and Syllable-Timed Languages’, Language Design, 2 (1999), 103–30 (p. 125), argues, on the basis of his own measurements and previous studies, that the distinction has little validity, commenting that ‘on the basis of the data presented, it follows that languages considered stress-timed, and others considered syllable-timed give a rather similar response to the three tests, with results that openly contradict the typological models they are supposed to represent’.

145 The subsequent paper was Aniruddh D. Patel, John R. Iversen and Jason C. Rosenberg, ‘Comparing the Rhythm and Melody of Speech and Music: The Case of British English and French’, Journal of the Acoustical Society of America, 119 (2006), 3034–47, with the chart on p. 3042 being of particular interest in this context. Colin Matthews, in ‘Holst, Gustav’, Grove Music Online, Oxford Music Online, <http://www.oxfordmusiconline.com> (accessed 26 July 2009), describes Holst's style as ‘enigmatic’ and ‘very different from that of other English composers’. He had a particular interest in Hindu literature and philosophy. Raymond Head examines this issue in more detail, and describes how it influenced Holst's style, including, among other factors, his quintuple- and septuple-time settings of the Sanskrit Rig Veda texts: ‘Holst and India’, Tempo, 158 (1986), 2–7; 160 (1987), 27–36; 166 (1988), 35–40.

146 Cecil J. Sharp, English Folk Song: Some Conclusions (London, 1907), 80. The song in question was ‘Riding Down to Portsmouth’ (1906) (see Appendix).

147 Cecil J. Sharp, English Folk Song: Some Conclusions (London, 1907), 80. The song in question was ‘Riding Down to Portsmouth’ (1906) (see Appendix), 81.

148 Cecil J. Sharp, English Folk Song: Some Conclusions (London, 1907), 80. The song in question was ‘Riding Down to Portsmouth’ (1906) (see Appendix), 80.

149 Of the 70 songs in Ralph Vaughan Williams and Albert Lloyd's The Penguin Book of English Folk Songs (London, 1959), just four (‘The Bramble Briar’, ‘The New York Trader’, ‘The Ship in Distress’ and ‘Ye Mar'ners All’) are mainly in quintuple time.

150 Sharp, English Folk Song, 88.

151 Ian Kemp, Tippett: The Composer and his Music (London, 1984), 69.

152 Quoted by Tippett himself in Those Twentieth Century Blues: An Autobiography (London, 1991), 127.

153 Percy A. Scholes, The Listener's History of Music, 2 vols. (Oxford, 1923), i, 8.

154 See Robert Kaplan, Rhythmic Training for Dancers (Champaign, IL, 2002), 37; Max T. Krone, ‘Music in Turkey’, Music Educators Journal, 39 (1952), 28–30 (p. 30); Moore, ‘Thai Songs’, 310; and Rees, ‘The Many Musics of a Chinese County Town’, 68–70.

155 Kaplan, Rhythmic Training, 37.

156 Martin Clayton, ‘Culture, Cognition and Additive Rhythm’, 57, observes that in Balkan dances to such rhythms, the longer beat simply represents a ‘heavier dance step’, analogous to the downbeat of a waltz or minuet. Reicha, in the introduction to his 36 Fugues, quotes a ‘waltz’ from the Alsace in 5/8 time (2+2+1).

157 The only hymn I have found in septuple time is ‘Old Hundredth’, a tune from the second edition of the Genevan Psalter of 1551. It is notated in an implicit (3+2+2)/2 metre:

(First line of transcription from the original in Pierre Pidoux, Le psautier huguenot du XVIe siècle, 2 vols. (Basle, 1962), i, 120.) The minim rest after each phrase corresponds to the fermata often found in chorales, and is often omitted in modern versions of ‘Old Hundredth’. To my ears, the tune sounds perfectly natural performed in this way, suggesting that many hymns may in effect have been routinely sung in septuple time, with a fermata (often indicated by a double barline) becoming an un-notated (and silent) seventh beat. This practice might have influenced some folk tunes and other works (see n. 81).

158 Examples for piano include Alkan's Impromptu op. 32 no. 8 (1849), Rebikov's Echo rustique, op. 8 no. 11 (1897), and Cui's Prelude op. 64 no. 15 (1903).

159 Lath, A Study of Dattilam, 323.