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

 Circadian Rhythms
 Thoughts about notation
 How fast is a week?

Introduction

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

"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]

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}
 

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Pulse and tempo

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Metre

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}

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The relationship between time signature and metre

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 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)

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)

Time signature

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}

: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]

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

Polymetre

Mixed metre

Additive or asymmetrical metre

Polyrhythm (cross-rhythm)

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. 
 
 

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}
 

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}
 

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}

Recommended reading:
http://www.stetson.edu/departments/human/handbook/PDF/Basic%20Elements.pdf
http://www.stetson.edu/departments/human/handbook/handbook_text.html
Glossary of french terms used in 17th century french music 

Footnotes

References

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

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