Polymetres and polyrhythms

In a recent blog I referred to symmetry in rhythm. The meaning was clear enough within the context but I decided to think a little more about the notation of rhythm, a subject most of us take for granted due to the conditioning we receive from an early age.

Talking about music is always tricky, and sometimes irrelevant. As Duke Ellington once remarked: ‘Too much talk stinks up the place’.  But if we’re here to exchange information and advice via a blog, which is essentially verbal in nature, then we can’t really avoid it. The point is this: different classifications often overlap one another.

The following extract from the book is a good place to start:

The most significant difference between the work of the graphic artist and the musician is that music is a temporal art form. Musical ideas are linked to the elusive notion of time. Our perception of time relates to the coordination of the sequence of events in our lives in a way that can be ordered and measured. If we promise to be at a certain place at an agreed time we are able to ensure, by reference to the metrics involved, that we arrive when expected.

That which we perceive to be the flow of time is, in everyday experience, a system of classifying the process of cause and effect. This is certainly true of the planning and arrangement of the temporal configuration of musical materials.

No one understands exactly what time is but, as musicians, we can only accept the idea as ‘given’. The Earth’s orbit is responsible for the years and seasons, its axial rotation the days and nights, which are further subdivided into hours, minutes and seconds. Further subdivisions are necessitated by specialist requirements of one kind or another where fractions of a second might become significant.

Why is time such an important concept to a composer? Here’s another book extract:

When an artist has been concentrating on an area of fine detail in his picture he can take a few steps back to obtain an overall view of the work. The more he moves away, the more he perceives it to be a single, unified whole. A composer faces a different set of problems. His composition may occupy a considerable period of time and he needs to develop specialized skills to achieve the same end. The controlled arrangement of the components of music to produce a balanced and unified whole as they extend through time is what we refer to as form.

Form and rhythm

Form and rhythm are inseparable. A composer can achieve a consistency of style, not only regarding the rhythms within the bars, but also the arrangement of bars, phrases, sentences, harmonic changes and thematic groups. It is exactly the same process but over a longer period. Rhythm is the unifying factor.

The beat is the underlying pulse, or scheme of regularity, representing our notion of time as it affects our daily routine. There is no doubt in my mind that the patterns we superimpose on to this continuum, constituting the rich variety of rhythms in the various styles of music, are the result of interference patterns that are a common feature of physical systems.

Time signatures

When the need to commit music to paper arose, enabling composers to share their music with people who hadn’t heard it before, musicians needed to represent the temporal features of music in a way that would be widely understood. They developed an ordered approach that has links with the structure of verbal language in poetry. The numerator in the time signatures they invented usually expresses the number of beats to the bar and the denominator shows the temporal value of each beat. Popular music written in 4/4 was often conceived as being in 8/8, since the quaver was generally the smallest unitary building block from which the rhythms were constructed. Cut common, though written with four beats to the bar, is conducted in two due to the difficulty of conducting in four at faster tempos.

The 6/8 and 12/8 series are hybrids – compound time signatures but with two and four beats in the bar. 6/8 is a compound duple, 9/8 a compound triple and 12/8 a compound quadruple.

9/8 is a true compound. Introducing duples into this metre leaves a remainder (1), but 2+2+2+3 (or its permutations) could still be used, just as 12/8 could use 3+3+2+2+2 and its permutations, or 4+3+2+3. There are other possibilities but if you want your music to ‘swing’ best give them a miss!

It will occasionally be necessary to make a judgement regarding which time signature to select. Common and cut common time signatures overlap at slower tempos so that either can sometimes apply. The compound quadruple (12/8) signature should only be used where triplets predominate in both the melody and underlying rhythms. We can write 12/8 feeling above the part or use a dual time signature, 4/4 (12/8). The more general use of multiple signatures can also obviate the need to write a profusion of time signatures throughout a composition featuring regular changes of metre.

My computer notation program can’t cope with this but the bracketed signature could be typed in as a text object. This would still necessitate turning off the automatic spacing option (if possible), where the computer’s understanding of the set metre is contradicted.

Similarly, a waltz may have an underlying 9/8 feeling although we may write in 3/4, with the triplet being implied in the performer’s interpretation.

Music in compound time signatures often groups itself into numbers of bars that are also divisible by three. As a result of habit and conditioning, there’s a tendency to add bars to produce duple or quadruple phrase construction. The symmetry of our own bodies may cause us to assimilate duple and quadruple forms more naturally. We should also consider using phrases and sentences that are divisible by three, where applicable.

At faster tempos, a waltz in 3/4 may be counted or conducted in one, i.e. one beat to the bar. The jazz waltz is often conceived and performed in this way.

2/2, 3/2 and other metres encountered in straight music are used to avoid the incongruous effect of eighth notes at slow tempos. Dividing the beat into quarter notes instead of having a quarter note divided into eighths is more in keeping with the longer duration of the notes.

2/2 is read the same as cut common and differs with regard to the characteristic degree of activity of the rhythms.

Other time signatures, such as 5/8, 7/8, 11/8 etc., have geographic associations and occurred in primitive societies.

Continuity and simultaneity

There are two methods composers can use to combine different musical forms: in continuity and in simultaneity. Differing rhythms and time signatures in continuity are fairly common but using them in simultaneity is more difficult to handle. Interestingly, the simultaneous use of different keys is easier for the listener to assimilate than the use of simultaneous time signatures. Various reasons have been offered to explain this. My own favourite is the importance of momentum and inertia in our daily experiences, which is disturbed by irregularity.

Polymetre in this blog refers to the simultaneous use of different time signatures. Here again, there are two methods. We can maintain the vertical alignment of bar lines, or not. If we combine, for example, 4/4 and 5/4 signatures, whilst keeping bar lines in alignment, we can no longer express the rhythms in the two signatures by means of a common, symmetric* system of units, for the obvious reason that a fifth of the temporal (clock time) duration of a bar cannot be equal to a quarter. This is why musicians have to work so hard to play such music and the less the two (or more) time signatures have in common, the stranger it gets.

*To revert back to the incentive for this blog, the word symmetric will hereby refer to a regularity of structure. Some writers on this subject use the word to refer to other characteristics, e.g. reversibility, etc. 

On the other hand, we can allow the longer metre to overlap the bar line delineated by the shorter signature(s). (We can use more than two simultaneous metres.) There was nothing new about this when modern composers began to experiment in these areas. Mozart’s Dissonance Quartet had already featured 5/4 figures over a 3/4 metre, although they really resulted from an interesting treatment of delayed resolutions, rather than from a conscious attempt to use polymetres.

It’s a matter of choice whether the shorter or the longer metre forms the basis of the piece.

The effectiveness of this latter technique can be enhanced by using harmonic, orchestral and other characteristics to delineate the subdivisions, otherwise they can become buried. Anyone who paid attention at school will realize that the different metres will ‘come out even’ when the divisors reach a common product, or denominator. In the case of 4/4 and 5/4, 20 (= 20 quarter notes) will end each cycle. That means 4 bars in 5/4 and 5 bars in 4/4.

Some composers, Charles Ives, for example, used complex simultaneous combinations of metre. The main difficulty, apart from the mental gymnastics involved, is in laying out the score. Each line of orchestration will not end conveniently at the edge of the page and blank areas will be left here and there because the conventional score layout doesn’t use a ‘piano roll’ method of notating music on paper. Similarly, the vertical alignment of bar lines can only be approximated because of the lack of symmetry. In a funny kind of way, music evolved to the point that it required written forms of expression but now finds itself hemmed in by them.


The definitions we use are often chosen out of expediency, so it’s worth pointing out that the terms polymetric and polyrhythmic are also, occasionally, interchangeable. (The book has a number of references to overlapping definitions such as that existing between accented passing notes/suspensions/auxiliary notes etc.)

The main distinguishing factor that defines polyrhythms is the use of symmetric (see above) structures where rhythms can be expressed by the use of common units. If different, coexisting rhythms can be written on graph paper, using orthogonal lines without cheating, they’re symmetric polyrhythms.

Obviously, 2/4 against 4/4, or 3/4 against 6/4 (for example) will possess the above characteristics, leading to another instance of an overlap of identification (as is 2/4 against 3/4, which would come out even after 2 bars). Often, these forms will result from the distribution of accents and stresses, not from an awareness of metre. A decision must then be made regarding whether or not to use more than one metre. Which makes more sense? Which will seem more natural, consistent or easy to read?

A slight complication occurs in the case of 3 against 2, and vice versa (the hemiola). These effects are sometimes called ‘cross metre’ or ‘cross beat’. In the case of a quarter note triplet in 6/8 and 2 quarter note triplets in 12/8 (the double hemiola) the rhythms employed possess a regularity of structure (in these examples, the eighth note). The unitary ‘building blocks’ are rearranged to form both rhythms without leaving a remainder. This is why these rhythms are called hybrids.

2, 3, 4 and 6 are all factors of twelve, and 2 and 3 are factors of 6.

Quarter note triplets against a 4/4 metre cannot be explained in this way without reference to the 12/8 rhythm, which is alien to the metre. The clock time duration of the bar is divided by 12 units in 12/8 but by only 8 in 4/4. The resulting polymetre effect might be another instance of overlapping terminology but, in any case, the original stimulant for the introduction of ‘threes’ was the desire to insert compound ‘fluidity’ into the more rigorous duple metres. Theoretical justification comes later.

Tuplets, where a group of notes foreign to the underlying metre occur, are regarded as embellishments, rather than as rhythmic entities in their own right. The forms are: duplets, triplets, quadruplets, quintuplets, sextuplets, septuplets, octuplets, nonuplets etc. (Look them up. Your notation program will have a sub-menu.)

Primitive cultures created complex and highly arresting polyrhythms that were produced intuitively and yet are difficult for even well trained musicians to assimilate. Obviously, cultural traditions and day-to-day conditioning are an important factor. Complex polyrhythms extending over more than one bar are found in various parts of the world.

The best way to appreciate these forms is to look them up on the www. Actions speak louder than words, something else the Duke would have agreed with.



3 thoughts on “Polymetres and polyrhythms

  1. There is so much inaccuracy even in relatively simple scores, like boogie woogie written in 4/4 time with dotted quarters and 16th notes with the textual annotation of “swing time” or “lazy rhythm”. 12/8 with quarters and 8ths are precise in this instance.

    You also made me think of Chopin who largely dismissed annotating musical instruction, concluding that a competent musician doesn’t need them and an idiot will not get it right no matter how precise you are.

    Frank Zappa found that he had to dumb down his music, limiting his compositions to triplets because the most competent players available to him could not perform his larger tuplet figures. To that I say, god bless the computer which does not have these human failings.

    Brahms wrote an exercise book demanding freedom between hands on the piano. It scores increasingly larger tuples against each other. I just laughed as I saw 16 against 17 as I paged through. I laughed then, but sometimes while playing if I am not focused sometimes my hands drift apart and become out of sync while each keeping their own tempo so maybe this is not as far fetched as I thought although doing so consciously seems beyond me. Unfortunately “being of two minds” is more of a conflict than a freedom, such is my limitation.

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