Metre and Rhythm in Music

Second years only: Guide to the metres and rhythms on your CD
Important issues in Metre and Rhythm in relation to dance
Revision pages: - not finished yet, but look if you want...
1st and 2nd Years - the Tchaikovsky's Snowflake Page

Introduction
Defining Terms

Pulse and tempo
Metre
Rhythm
"Dance Rhythms"
The relationship between metre and time signature
Time signature
Polymetre
Mixed metre
Additive or asymmetrical metre
Polyrhythm

Some examples of compound metres/hypermetre in the calendar
Random thoughts about rhythm and metre

Circadian Rhythms
Thoughts about notation
How fast is a week?

Footnotes
References

Introduction

Any discussion of metre, rhythm and music in the context of dance, particularly of ballet, must approach the subject from two angles - the prescription of metre (how music is written, e.g. time signature) and the perception of metre (how music "goes", or is perceived to go). Dancers, when they dance, rely on their perception of metre and rhythm rather than following a musical score. Furthermore, choreography imposes another level of metrical and phrasal organisation on what is heard which may be related to, but is not the same as, the metrical and phrasal organisation of the music.

Even when dance notation refers to a musical score during recording or reconstruction, it is only as a framework on which to build the choreographer's scheme of counts and phrases. Since choreographers usually work by listening to music rather than reading it, these counts and phrases will be based on their perception of music - how it goes - not how it is written. {top}

Having listened to the music in this way, choreographers do not always then set out to mirror what they hear in movement. For example, Balanchine's choreography to metrically unambiguous music sometimes imposes a metrical or phrasal scheme which runs counter to that of the music (e.g., the female variation in Tchaikovsky Pas de Deux). Conversely, on music which is - at least on paper - metrically very ambiguous, he sometimes offers a scheme which is simpler and less ambiguous (e.g. the opening of Symphony in Three Movements).

At the interface between music and dance, then, metre is both a perceptual process and a decision-making one, not necessarily an absolute or objective quality inherent in the music itself. Paradoxically, this both begs the question "so why teach it at all?" while at the same time giving a very good reason to teach it. For if metre is not perceived in the same way by individuals in a group, and is not necessarily explicit in the music, then part of teaching ballet - which relies greatly on coherence between music and movement in groups - must be to "manage" the perception of metre[2].

This is not necessary in a social group where agreement about the perception of metre is mediated through social dance forms or singing, or in the case of Western, classically trained musicians, through time signature. Similarly, metrical groupings which come easily or "naturally" to some people through enculturation are perceived as unusual or difficult to others; for example, additive metre is common in the folk music of south-eastern Europe, but not in western Europe, and compound duple metre is common in English children's songs but not in Japanese ones [reference]. {top}

The teaching of ballet, then, poses two challenges with regard to metre. One is that nineteenth century ballet has left us with a multiplicity of social and national dance forms such as the mazurka, polonaise, polka, cachucha and tarantella whose metrical/rhythmic patterns and other characteristics such as tempo or closure, can no longer be assumed to be transmitted socially or culturally. The other is that the metrical information needed to "decode" Western art music in any more depth than as background-music is (at least in England) only infrequently transmitted through education.

This is not to suggest that we should all start dancing the polka again (although it might help if we did it enough), or to advocate the sort of music-appreciation classes which idolized the music of Mozart, Beethoven and Brahms while deeming anything popular or non-European unworthy. On the contrary, it is to propose a form of musical education for ballet students which teaches what it needs to teach without implying to the student that metre is absolute or objectively measurable . In practice, this means teaching the basic components of metre - pulse, tactus, duple, triple, compound and additive metres - and using these terms as analytical tools to aid both the perception of metre and the learning of steps or movements which have to be performed in particular metres.{top}

Defining terms

To those who find the whole subject of metre and rhythm difficult to grasp, it will be of little comfort - and perhaps come as no surprise - that the Harvard Dictionary of Music has this to say about rhythm:

"It would be a hopeless task to search for a definition of rhythm which would prove acceptable even to a small minority of musicians and writers on music." [Apel, 1945]

That being the case, a course such as this which aims to enable students to apply rhythmic or metrical principles to the practice or analysis of dance has to agree some terms of reference at the outset, even when the use of those terms may be disputed. The same problem faced Grosvenor Cooper and Leonard Meyer when they wrote The Rhythmic Structure of Music [Cooper & Meyer, 1960], and this led them to set out their interpretation of terms such as tempo, rhythm and metre in the first chapter, "Definitions and Principles."

Adopting these principles, without necessarily swallowing them whole, is a good enough place to start - one can then, at least, be clear about whose terms are being used. Moreover, The Rhythmic Structure of Music is a much-cited text in musical analysis, and familiarity with its concepts will be helpful when reading work by other authors. Above all, Cooper and Meyer's exposition of metrical organization (of which later) is far friendlier to dance than those catechistic primers on time signature which so often form the central part of books on music for dancers. {top}

Pulse and tempo

Cooper and Meyer (1960) define a pulse as "a series of regularly recurring, precisely equivalent stimuli". This might be a dripping tap or the ticking of a watch, or indeed a heartbeat. The average resting pulse-rate of a human being is between 60 - 80 beats per minute (bpm). The "pulse-rate" in music is called tempo. That pulses are regular and equivalent (that is, equivalent in pitch or volume) is essential in distinguishing pulse from metre.{top}

Metre

Metre is the term used to describe patterns of pulses defined by the prominence of some pulses (by means of accent, for example) over others. A ruler with 300 millimetre tick marks but without longer marks every 10 millimetres to denote centimetres and every five to denote half-centimetres would be very difficult to use. Similarly, the accentuation of certain pulses along the continuum helps us to group pulses.

In music or any other "metric context" as Cooper & Meyer put it, pulses are referred to as "beats"; accented ones are called strong beats and unaccented ones weak beats. Research indicates that our brains seek out patterns even when they are not present; in this sense, metre is not just something inherent in the music, it is a means of processing incoming stimuli.

Moreover, not only is metre "not just something" inherent in the music, it is arguably not an objective, measurable quality of the music at all. Vijay Iyer (1998) argues:

"[...] meter is not necessarily inherent in any audio signal. It is a perceptual and cognitive construction, derived from some perceived periodic patterning of perceived accents (including, paradoxically, accents imposed by the imagined meter itself) but also from some set of assumptions about meter.{top}

Rhythm

"Metre" and "rhythm" are easily confused terms in music. Cooper and Meyer's definition of rhythm relies on a flexible interpretation of the term "accent". Rhythm, they say, is "the way in which one or more unaccented beats are grouped in relation to an accented one" (1960, p. 6) and give the following list of terms from Greek prosody:

iamb - */
anapaest **/
trochee /*
dactyl /**
amphibrach */*

I have followed Kirby-Smith's method of using an asterisk (*) to denote the unaccented syllable, and an oblique (/) for the accented one, since the traditional symbols are difficult to reproduce on a webpage). Kirby-Smith (University of North Carolina) has made table of poetic metres which is a very handy reference to metrical terminology, and also helps to understand the relationship of 'line' to 'phrase'.

But their definition of accent is based on the polarity between arsis and thesis, terms which are interpreted in different ways, depending on what you are talking about (Willett, 1998). The principal interpretations are as follows:

Arsis*	*Thesis or Basis**	"Accent" Type**
Upbeat	Downbeat	Stress
Unstressed	Stressed	Stress
Unaccented	Accented	Stress
Weak	Strong	Stress
Lift	Placement	Stress
Short	Long	Durational
High tone	Low tone	Tonic
Anacrusis	Crusis	Stress

* Confusingly, arsis and thesis are now defined the other way round. I have retained the original Greek definitions. If you use these terms, it is a good idea to define them first!

**"accent type" defined by Cooper and Meyer in footnote 9, p.7

It seems that if one thing definitely distinguishes rhythm from metre, it is that rhythmic patterns usually exhibit differences in duration - i.e. the way beats are spaced. These durational differences can be achieved by pauses in the sequence of beats, or by shortening or lengthening of individual "beats".

Rhythm operates within the framework of metre. Metre without rhythm is a bit like going to book shop and saying "Can I have a book that's seven centimetres thick please?" Even though we often talk about "rhythmic accuracy" or "keeping rhythm", rhythm is usually everything that happens apart from the main beats of the metre. Thus, a really expressive jazz singer will be usually ahead of the beat (i.e. earlier - which is why it's sometimes called backphrasing), using subtle microtimings that give an illusion of being totally free of constraint, while remaining within a metric framework.

If you've ever sat and watched a small boat moored to a quayside, drifting out, turning, lifting, coming back in, drifiting out again but never going out to sea because it's moorings are secure, that is the relationship which rhythm holds to metre in musical performance.

A good example of rhythm with no apparent metre is experienced when listening to morse code (for fun, visit the virtual morse code generator - type in words and hear them played back in Morse). The code is based on the difference between long and short syllables (dits and dahs), which is one of the reasons operators talk about such things as "iambic paddles".

Dance Rhythms

Care should be taken not to confuse the rhythm of a dance step with the rhythm of the music that goes with it. For example, you will often hear people say "a polka goes di-dit dah dah, di-dit dah dah". Now however true that may be of the step that you do when dancing a polka, it neednt' be true of the music. It is only a few polkas that imitate the rhythm of the step - a sort of musical onomatopoeia. Likewise, it would be a very boring waltz that only mirrored the action of the feet - unless you want to satirize the waltz rhythm as in "oom pah pah oom pah pah that's how it goes" from Oliver.

You will realize by now that what we often call a "dance rhythm" is in fact the metre, tempo and rhythmic characteristics of the music which generally accompanies a dance style/form/genre.

Many composers have taken popular dance rhythms (lets call them that for short) and made dance pieces out of them. These can sometimes be quite unsuitable for real dancing, since they are designed to be listened to and judged on their musical merits (and thus provide immortality and adulation for the musical performer and the composer). They are sometimes called things like "Concert Waltz" or "Polka de Concert" or "Grande Polonaise". One of the things that concert music is supposed to do is to develop thematic material , have different colours - dynamics or scoring, and delay enjoyment to fill up a large chunk of a concert programme. Most of these things are inimical to social dancing.

The relationship between time signature and metre

Since we are about to examine the time signature in relation to metre, it is useful to see how Iyer develops his argument about metre to discuss time signature:

Indeed, indicated meters appear in most Western sheet-music scores because of the composer's desire to exploit these assumptions, to allow the implications of a time signature to shape the perfomer's understanding and subsequent rendering of the piece. [...] we tend to speak of the meter of a piece objectively, because it is simply the time signature written to the left of the first bar of written music.

Iyer, 1998 (Chapter 5, "On the Perception of Meter")

Iyer may be right that the composer wishes to exploit assumptions about metre by using a time signature, but those assumptions are in turn subject to other conventions of period, place and common practice. Kirnberger's advice to composers in The Art of Strict Musical Composition (1776) is of little help to composers who do not regularly go out on a saturday night to dance the loure, or sarabande.

"[...] Further more, he [the composer] must have acquired a correct feeling for the natural tempo of every meter, or for what is called tempo giusto. This is attained by diligent study of all kinds of dance pieces. Every dance piece has its definite tempo, determined by the meter and the note values that are employed in it. Regarding meter, those having larger values like alla breve, 3/2 and 6/4 meter, have a heavier and slower tempo than those of smaller values, like 2/4, 3/4 and 6/8 meter and these in turn are less lively than 3/8 or 6/16 meter. Thus, for example, a loure in 3/2 meter has a slower tempo than a minuet in 3/4 meter, and the latter is in turn slower than a passepied in 3/8. Regarding note values, dance pieces involving sixteenth and thirty-second notes have a slower tempo than those that tolerate only eigth and at most sixteenth notes as the fastest note values in the same meter. Thus for example, a sarabande in 3/4 meter has a slower tempo than a minuet, even though both are written in the same meter."
(Strunk, p. 767)

Although it is probably still safe to say that 3/2 and 6/4 tend conventionally to connote a slower, statelier tempo than 3/4 or 6/4, time signatures on the whole do not imply tempo, only metre. Even in Kirnberger's day, a 3/4 was faster or slower depending on the dance you were writing. It is more than a little ironic to be quoting a teacher of musical composition who advised his pupils to study "all kinds of dance pieces" in a course for dancers about metre.{top}

Just for fun, an extract from Georg Muffat in 1695:

"[...] 3/2 requires a very slow movement, 3/4 a gayer one, yet uniformly somewhat slow in the sarabandes and airs; then more lively in the rondeaux, and finally the most lively but without haste in menuets, courantes and many other dances, as also in the fugues and overtures. The remaining pieces, such as are called gigues and canaries, need to be played the fastest of all, no matter how the measure is marked"

Preface to the First Florilegium (1695), Georg Muffat (Strunk 1998, p. 647)

I imagine a dancing master at this point saying "I don't care how the measure is marked, it's fast!"

Time signature

Time signature is an indication given at the beginning of a piece of Western music to tell the musician how the music is metrically organized. They are by no means indispensible: both Stravinsky and Satie wrote piano music without time signatures for example. In improvisatory sections, composers such as Ligeti and Part have used timings in seconds rather than time signatures and barlines.

Johann Kirnberger confident assertion in 1776 that "the measure [bar] consists of two, three, or four equal beats; besides these, there is no other natural type of measure [bar]" is a useful working assumption for non-additive time signatures. That is to say, whatever numbers may be present in the time signature, they all denote metres which are basically duple, triple or quadruple,

The time signatures most commonly in use, and the metres they imply are shown in the table below. The terms "simple" and "compound" are misleading. Just as one has to learn that, contrary to all expectations, "mano" is feminine in Italian even though it ends in "o", there is nothing in the words simple or compound in this context which could remind us that:

Simple means that the beats are subdivided into 2, 4, and 8 (i.e. binarily: 1+2+)
Compound means that the beats are subdivided into 3, 6 and 12 (i.e. ternarily: ) {top}

Simple duple	Compound duple	Simple triple	Compound triple	Simple quadruple	Compound quadruple
2/4	6/8	3/4	9/8	4/4	12/8
1+2+	1++ 2++	1+2+3+	1++2++3++	1+2+3+4+	1++2++3++4++

:Cooper and Meyer, talking of metre say this:

"We are inclined to think of there being only one metric organization, the one designated in the time signature and measured by the bar lines. This is because tonal harmony and homophony, with their emphasis on vertical coincidence, and dance music, with its basic motor patterns, have for the past two hundred years made for the dominance of what we have called the "primary metric level"." [3]

(Cooper & Meyer, 1960, p. 5)

In practice, this means that whatever the time signature may say, there may well be other intended or perceived metrical patterns in music which subvert or contradict the time signature. Add to this a choreographer's metrical or phrasal patterns, and time signature becomes - to the dancer - almost immaterial.

If you are really keen to guess the time signature from listening to music, the following warnings should be issued:

What is written as 16 bars of 3/4 time may be perceived as a compound duple metre (6/8)
What is written as 8 bars of compound duple metre (6/8) may be perceived as 2 bars of compound quadruple metre (12/8)
In jazz music what is written as simple quadruple metre (4/4) may be both performed and perceived as compound quadruple metre (12/8).
What is written in simple triple metre 3/4 may be perceived as duple metre - composers like doing it. [2]
What is perceived initially as the first beat ("1") is frequently anacrustic in notational terms. This is often the case with tarantellas.
The perceived metre may change from one bar to the next - see [2]

Hypermetre

"Hypermetre" is used to describe to higher-order levels of metrical organisation than may be evident from merely looking at the time-signature in a piece of music. Dancer's counts are frequently hypermetric. {top}

Hemiola

Hemiola is a metrical device or phenomenon where a group of 6 beats is:

divided simultaneously into two subgroups, one consisting of (2X3) beats, the other of (3X2) beats ("Panorama" from The Sleeping Beauty),

where the subgrouping alternates between 2x3 and 3x2, as in Bernstein's song "America" from West Side Story.

Hemiola can be effected at more than one architectonic level. Tchaikovsky's "Waltz of the Snowflakes" from The Nutcracker, for example, exhibits almost perpetual hemiola at a higher architectonic level than the time signature (i.e. the melody is effectively in 3/2, whereas the time signature is 3/4).

Because the fun of hemiola is to keep shifting the listener's perception of metre within an established metre, it is not usually "written out" - for example, Tchaikovsky was fond of introducing hemiola at the end of phrases in waltzes (Act I waltz from Sleeping Beauty, Waltz of the Flowers from Nutcracker, male "Bluebird" variation from Sleeping Beauty, final waltz from The Nutcracker) but he never changes the time signature from 3/4 to 3/2 to reflect this.

One reason not to "write out" hemiola is that since hemiola needs both 3/4 and 3/2 or 3/4 and 6/8 to be happening together, someone in the orchestra would lose out, by their part being written in the "wrong" time signature. The other is that if the orchestra began to think in 3/2 rather than 3/4, the chances are that the music would simply sound slower. With the continuation of the 3/4 pulse underneath the 3/2, one tends to perceive the music as faster. However, hemiola is effectively polymetre, q.v.

The best example of hemiola is the triple run. In a triple run, the "down-up-up down-up-up" provides the 2 X 3 metre (6/8) while the the binary opposition of your feet (left-right left-right left-right) as you do it organises that compound duple metre simultaneously into simple triple metre (3 x 2 = 3/4).

Polymetre

The term polymetre is used to describe music where different metres coexist. It is either explicit in a musical score - different time signatures on different staves - or implicit by the grouping and accenting of notes in different parts. {top}

Mixed metre

Very common since Stravinsky - rather than a piece beginning and ending in a single metre or time signature, the metre changes from one bar to the next. Don't confuse mixed metre with additive metre - because unlike additive metre, mixed metre is not cyclical. {top}

Additive or asymmetrical metre

Asymmetrical or additive metres (also known as irrational time signatures) consist of stacked groupings of dissimilar metrical groups. Whereas 9/8 (3+3+3) is a fairly common metre in Western music, a common Turkish metre, for example is 2+2+2+3, but this would be notated as 2+2+2+3, not 9/8, because Western musical notation assumes that each beat of the bar has equal value. 5/4 or 5/8 and 7/4 or 7/8 are relatively common assymetric time signatures which imply additive rhythm (3+2, 2+3 and 4+3, 3+4 respectively) while not explicitly stating the subdivision - this is usually done by means of phrasing or other boundary markers. {top}

Polyrhythm (cross-rhythm)

Playing two rhythms together. Much Latin American and African music is polyrhythmic. Western music of the 19th century (and beyond) privileges melody above nearly every other aspect of musical composition, but is not interested in complex rhythmic or metric devices. Other musics do not make the distinction so clearly between melody and accompaniment - rhythmic complexity and diversity is an important feature of musical expression. and competence.{top}

Gershwin's song I Got rhythm is an example of polyrhythm (or is it polymetre - this is one of the grey areas of rhythm and metre). The metric organisation is simple quadruple metre, but the rhythm of the tune itself is based on an additive metre:

Tune: 12312312312312|1212312312312312
Accompaniment: 1+++2+++3+++4+++|1+++2+++3+++4+++

To show the relationship of the additive metre to the principle

http://www.robotwisdom.com/jorn/jazz.html

The pattern of beats in I Got Rhythm is interesting in another respect.

Some examples of compound metres/hypermetre in the calendar

1. The Year. The year divided into seasons and months is an example of a compound quadruple metre.

Year
Spring			Summer			Autumn			Winter
Jan	Feb	Mar	April	May	June	July	Aug	Sept	Oct	Nov	Dec

2. A season (or quarter year) is an example of both a simple triple and compound duple metre. Three months divided into 3 lots of four weeks gives you a simple triple metre (i.e. subdivisions are made binarily).
Three months divided into fortnights gives you blocks of 3 fortnights (e.g. the weeks before and after half term). This is a classic example of hemiola.{top}

January				February				March				1
Wk 1/2		Wk 3/4		Wk 5/6		Wk 7/8		Wk 9/10		Wk 11/12		Si
1	2	3	4	1	2	3	4	1	2	3	4	Sii

3. A fortnight
A week has 7 days - is that a group of 2+2+3, or is it 7 equally spaced days (i.e. no accented days like Friday night)? What about weekends - do you feel a push towards saturday. Is Sunday slower than saturday? If Monday is the beginning of the week, how come we drift into it slowly and unenthusiastically? Is the metronomic society getting you down?{top}

Week 1

Week 2

Random thoughts about rhythm and metre

Circadian rhythms
The organisation of time is not just a musical issue. Without alarm clocks, windows, electric lights and other external signals, the human body's natural circadian rhythms - when you wake up, when you go to bed - would lead you to get up later and later every day, since our natural day-length is nearer to 25 hours than 24. (see Basics of Sleep Behaviour at UCLA medical school). Not only that, calendars have to do the opposite - which is why we have months of unequal length, and add one day every four years (called intercalation).{top}

Notation
If you have ever laughed at the old joke "dear Auntie Jane, I'm writing this slowly because I know you cannot read very fast" , you will have grasped an important point about musical notation: writing and performing music are different processes, often separated both chronologically and geographically (i.e. a Japanese student in the 21st century plays a piece by Mozart, written in Austria in the 18th century).{top}

How fast is a week?
Our perception of the passing of time is affected both by our response to external events and our mood. Ten minutes spent waiting for a bus in the rain in February seems longer than ten minutes walking along a beach with someone you love on a summer evening. {top}

If you're really interested...
A representation of musical rhythm
A Look at Some Phenomena Underlying Timing and Rhythm
Calendars

Footnotes

1. "Metre management" occurs particularly with new scores for choreographic works. Unable to rely on socially mediated perceptions of metre or phrasing such as social dance forms (it is often these very conventions which composers in the Western art music tradition seek to subvert), dancers have to be acculturated to the composer's metrical scheme. The most obvious example of enforced "metrical acculturation" is the choreographing of Stravinsky's Rite of Spring [quotation and reference].

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More interesting is the case of Prokofiev's Romeo & Juliet (1939) which, according to Ulanova [reference], caused dancers in the first production considerable problems. Not only could they not hear the music adequately (leading Prokofiev to say "they don't need music, they need a drum" [reference]) they also felt that Prokofiev's music did not portray characters in the way they expected or wanted it to [quotation, reference].

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2. There are countless examples in Western art music where what is perceived as a duple metre may be constructed (written, composed, notated) in triple metre. A typical example of this is the last movement of Schumann's Piano Concerto, where a theme sounding rather like a light march is in fact notated in 3/4 time.

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Likewise, "Snowflakes" in Tchaikovsky's Nutcracker is written in 3/4 but the first main theme is so disassociated metrically from the time signature that it is not only difficult to determine the metre, but the perceived metre is also slower than the metre implied by the time signature. This abates when the choir enter - their melody coincides with the waltz time in the time signature - but not not for long.

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3. Though dance music may be responsible for the perceived coincidence between metre and time signature, it is ballet, particularly since the beginning of the 20th century but also even in the 19th century, which challenges this in practice.

References

Apel, W. (1945) Harvard Dictionary of Music. Cambridge, Mass: Harvard University Press, p.639

Cooper, G., Meyer, L. B. (1960) The Rhythmic Structure of Music. Chicago: University of Chicago Press, pp. 1 - 11

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Iyer, V. (1998) Microstructures of Feel, Macrostructures of Sound:Embodied Cognition in West African and African-American Musics. Dissertation, Univerisity of California, Berkeley, found at http://cnmat.cnmat.berkeley.edu/People/Vijay/%20THESIS.html [last accessed March 4th 2001]

Strunk, O. (1998) ed. Source Readings in Musical History, revised edition. New York: W.W. Norton & Company

Willett, S. (1998) "Ancient Rhythmicians and Modern Prosodists: Searching for the Location of Meter" in Versification: an interdisciplinary journal of literary prosody. Vol 2. http://sizcol1.u-shizuoka-ken.ac.jp/versif/97MLA_Willett.html [last accessed 17th March 2001]