Counterpoint - half and doubled durations or other?

[Dave2002]

In some types of music themes are set against themselves - the most obvious being a round or canon. Sometimes one of the lines is doubled in speed, or alternatively halved. This kind of technique can work well enough if the time signature is an obvious multiple of 2, such as 2/2, 4/4 etc.

What about multiples of three? Have any composers that we are aware of used multiples of 3 for this kind of effect? That could work obviously with 3/4 or even 6/8 or 6/4.

My guess is that with multiples of 5 or 7 or 11 things would just get too complicated for many composers - and also listeners, though why not?

Also note that time signatures such as 6/8 allow both doubling/halving and also multiples/divisors of 3 for this kind of effect.

Some composers could ignore the "straight jacket" of a time signature, in which case themes would flow in odd patterns across bars - maybe someone has done that (successfully?) to reasonable effect.

Basically though, I want to know whether much attention has been paid to these effects, and whether we know of any music which steps outside simple factors of 2.

[Serial_Apologist]

In some types of music themes are set against themselves - the most obvious being a round or canon. Sometimes one of the lines is doubled in speed, or alternatively halved. This kind of technique can work well enough if the time signature is an obvious multiple of 2, such as 2/2, 4/4 etc.

What about multiples of three? Have any composers that we are aware of used multiples of 3 for this kind of effect? That could work obviously with 3/4 or even 6/8 or 6/4.

My guess is that with multiples of 5 or 7 or 11 things would just get too complicated for many composers - and also listeners, though why not?

Also note that time signatures such as 6/8 allow both doubling/halving and also multiples/divisors of 3 for this kind of effect.

Some composers could ignore the "straight jacket" of a time signature, in which case themes would flow in odd patterns across bars - maybe someone has done that (successfully?) to reasonable effect.

Basically though, I want to know whether much attention has been paid to these effects, and whether we know of any music which steps outside simple factors of 2.

Not sure if this is helpful , but metrical diminution and/or diminution by differently subdividing a stated time signature by means of eg dotted values is a common means of adding interest, creating underlying tensions in juxtaposition with leading note enharmonic slips which don't coincide with what is going on or has gone on in other layers of a contrapuntal structure, whether we're talking about sacred Mediaeval and Renaissance masses or for example contemporary jazz arrangements, and I wouldn't have thought whether or not the overarching form operant at any given moment is strict canon or canon elaborated into fugue had any larger principle bearing.

Or perhaps I'm missing the point!

Some of my favourite examples are to be found in Schoenberg's Kammersymphonie No 1, in which such rhythmic superpositions combine unevenly over a texture with mutually conflicting harmonic modulations (from the accepted theoretical standpoint of 1906!) to break up the sense of "where one is", helping prepare the way, alongside various other devices, for atonality and eventually serial pitch organisation. But as with all music in the diatonic tradition the ebb and flow between the elements, continuity or discontinuity of flow act with harmony and rhythm or metre in harness.

[Joseph K]

In some types of music themes are set against themselves - the most obvious being a round or canon. Sometimes one of the lines is doubled in speed, or alternatively halved. This kind of technique can work well enough if the time signature is an obvious multiple of 2, such as 2/2, 4/4 etc.

What about multiples of three? Have any composers that we are aware of used multiples of 3 for this kind of effect? That could work obviously with 3/4 or even 6/8 or 6/4.

Yes, Bach wrote fugues and canons in time signatures like 3/2 and 9/16.

My guess is that with multiples of 5 or 7 or 11 things would just get too complicated for many composers - and also listeners, though why not?

I don't think there is anything intrinsically complicated about odd time signatures. On the other hand, you enjoy the Turangalila Symphony - that features musical ideas set against themselves in the form of heterophony, different rhythmical textures & techniques like rhythmic canons, non-retrogradable rhythms, things like isorhythm... and yet the time signature might just be 2 or 4.

Basically though, I want to know whether much attention has been paid to these effects, and whether we know of any music which steps outside simple factors of 2.

There's loads of 20th century Western music that features odd time signatures... but I am not sure if you are asking just that, or whether you want specifically contrapuntal music in odd times... for some reason.

[crb11]

This might not be quite what you're looking for, but Conlon Nancarrow's studies for piano roll (numbers 13-19) include various unusual canon ratios, including 3/4, 3/5 and some wilder options. They are worth experiencing at least once!

[Richard Barrett]

I don't think there is anything intrinsically complicated about odd time signatures

No indeed - it's fairly typical for folk music in the Balkan region to use 5- and 7-based metre. Irregular metres find their way into 20th century composition through the music of composers like Bartók (who of course first encountered it in the aforementioned folk musics) and Stravinsky, but as for counterpoint in odd time signatures these are predated by the 36 Fugues by Antonin Reicha - no.20 is in 5/8 time while no. 30 is polyrhythmic, combining 4/2 and 3/4.

[Bryn]

This might not be quite what you're looking for, but Conlon Nancarrow's studies for piano roll (numbers 13-19) include various unusual canon ratios, including 3/4, 3/5 and some wilder options. They are worth experiencing at least once!

And if the timbres of the instruments on the Other Minds or Wergo recordings don't suit, try the Ampico/Bösendorfer recordings on MDG. http://www.arkivmusic.com/classical/...lbum_id=148581

[ardcarp]

Nothing new under the sun. Mensuration canons?

https://en.wikipedia.org/wiki/Prolation_canon

eg Ockeghem Missa Prolationum https://www.youtube.com/watch?v=ZWLsLAujZzI

[Dave2002]

There's loads of 20th century Western music that features odd time signatures... but I am not sure if you are asking just that, or whether you want specifically contrapuntal music in odd times... for some reason.

It's not the time signatures which I'm interested in - but rather the notion of something like the mensuration canons mentioned by ardcarp in post 7. I can imagine that someone like Stockhausen might have written a three part canon (or more parts) with ratios of 1:127:131 which would not repeat before 16637 iterations. However that would probably be tending to the extreme side.

Regarding the time signature, if you actually wanted to write such a canonic piece and you wanted to have the musical phrases fit neatly and exactly into bars you'd need to have a time signature of something like 16637/4 or 16637/8 or even 49911/32. Of course there is no need for notes to line up with bar lines at all.

I was indeed thinking that this kind of device has been used since at least the 14th Century, but may have been adopted by many composers since.

Like SA I am enamoured of Schoenberg's Chamber Symphony 1, but I didn't know that it exploited this kind of effect. I don't know whether these effects are always obvious and audible (which they probably are, sometimes), or sometimes so obscure that they can only be spotted either by looking at a score, or if the composer lets us in on the secrets.

I was also interested to know if any composers had used relatively simple factors, such as 3, 5, 7 or 11 before heading off to the "ludicrous" extents hinted at above. Small powers of 2 are probably the most common.

[Richard Barrett]

Ockeghem Missa Prolationum

That is of course an amazing piece of work. I think that even if one were to listen to it not knowing how intricately structured it is, there'd be a feeling that something strange is going on. As has been mentioned, Nancarrow reached the apex of complexity in rhythmical relations between canonic parts in some of his studies, using ratios such as 14:15:16, √2:2, and in a most put-that-in-your-pipe-and-smoke-it kind of way, 1/(3√pi):1/(3√(13/16)), only the first of which IMO is really perceptible as a metrical relationship as opposed to sounding just like simultaneous unrelated tempi.

[Dave2002]

That is of course an amazing piece of work. I think that even if one were to listen to it not knowing how intricately structured it is, there'd be a feeling that something strange is going on. As has been mentioned, Nancarrow reached the apex of complexity in rhythmical relations between canonic parts in some of his studies, using ratios such as 14:15:16, √2:2, and in a most put-that-in-your-pipe-and-smoke-it kind of way, 1/(3√pi):1/(3√(13/16)), only the first of which IMO is really perceptible as a metrical relationship as opposed to sounding just like simultaneous unrelated tempi.

Oh dear. I wasn't going to mention irrational numbers. Those of course are only going to be at most approximations if used in key signatures.

One of my friends and former colleagues used to like filling in forms asking for opinions (say on a scale of numbers up to 10) in the form pi/e or e/pi^2. I don't think the admin people were amused, or even understood what he was trying to do to them.

14:15:16 is, however, feasible.

[Richard Barrett]

Oh dear. I wasn't going to mention irrational numbers. Those of course are only going to be at most approximations if used in key signatures

How would you use a number of any kind in a key signature?

As I pointed out, since Nancarrow didn't have to concern himself with what could be notated, or with what human performers could perceive and perform, he could compose with whatever metres he felt like.

[Serial_Apologist]

How would you use a number of any kind in a key signature?

As I pointed out, since Nancarrow didn't have to concern himself with what could be notated, or with what human performers could perceive and perform, he could compose with whatever metres he felt like.

Alongside the kind of "free pulse" drummers such as Billy Higgins and Ed Blackwell introduced to jazz along with Ornette Coleman as a "response" to the convention that one had to be careful about taking jazz music beyond the 4/4 and 3/4 time signatures still the normal practice in the jazz of the 1950s, use of more complex and especially varying time signatures provided a great way to jerk the front-line improviser out of habitual phrasing and into having to come up with fresh ways of resolving tensions thereby created.

[ardcarp]

As I pointed out, since Nancarrow didn't have to concern himself with what could be notated, or with what human performers could perceive and perform, he could compose with whatever metres he felt like.

Indeed. You would think the modern computer would be able to do the same sort of thing, but avoiding measuring holes on a piece of paper!
(Go on, tell me it can somebody.....)

[Dave2002]

How would you use a number of any kind in a key signature?

As I pointed out, since Nancarrow didn't have to concern himself with what could be notated, or with what human performers could perceive and perform, he could compose with whatever metres he felt like.

We do it all the time with integers! 6/8 etc.

[Richard Barrett]

Indeed. You would think the modern computer would be able to do the same sort of thing, but avoiding measuring holes on a piece of paper!

Of course! Although it also makes possible letting go of regular metre altogether...