(or BPPR for short).
This is my name for the process of combining common polyrhythms, like 4 against 3 or 3 against 2, and blurring them in ways that allow unexpected polyrhythms, like 7 against 5, to work as morphable grooves. [Note: When I say "3 against 2", I mean "3 in the space of 2" aka crotchet triplets].
I've written 4 songs that feature Blurred PolyPolyRhythms:
Giant Gippsland Earthworm (5 against 7), Eunice (9 against 13), Peak (8 against 5 (and 3, 13 and 21!), Make Stuff Up (8 against 11) and Osculate (12 against 17).
The BPPR page of possibilities
When I was little, I used to enjoy singing the 5-to-the-bar classic Everything's Alright (from Jesus Christ Superstar) with the first 4 notes of the title sung as dotted quavers (4 against 3). Then when I was about 13, I realised that if I put crotchet triplets over the last 2 beats of the bar (3 against 2) it made it all pretty close to 7 against 5. This kept me quietly amused for the next 30 years, but I didn't do anything with it. Then my friend Carl Pannuzzo wrote a song, for his acapella trio, the Acapelicans, called Seven. I immediately heard it as being in 5 with 7 against it. It turns out that Carl hadn't intended it. The bass singer, Simon, felt the song in 5 while Carl and Sarah felt it in 7, yet they all happily grooved together.
In 2008, I helped record Penelope Swales' version of Baterz's song Twee, which is in 17 (grouped 6-6-5), but for the final singalong, Penelope squashes a 9 across it and it works! Yeah, that got me thinking! Then in 2014, I incorporated some 7 against 5 into my score for Little Wing Puppets' show, "Spike".
Weirdly, my next foray into BPPR was 9 against 13. I was playing a 13 groove with Linsey Pollak* and found myself spontaneously putting 9 against it (as 4 against 6 then 2 against 3 then 3 against 4). The "wonky waltz" nature of it got me inspired. In working out the ratios, I realised that saying"4 over 3" is misleading**, as it means dividing 3 beats into 4, so therefore the fraction is actually 3/4 (thus 0.75). So rather than "over" (or even "across"), I tend to say "4 against 3" for dotted quavers. Crotchet triplets (3 against 2) are thus 2/3 or 0.66... , which means dotted quavers (0.75) are that little bit slower that crotchet triplets (0.66...). Therefore, to even out the blurred polypolyrhythm, you need the rush the 4 against a little and drag the 3 against 2s a bit. Or play with which beats are going to flam. Eventually, my experiments with 9 against 13 became my song Eunice, which starts in a straight 13, then puts a 9 disco over a two bar cycle. Then for a only a few bars it goes into a 9 waltz before making it an 18 grouped in 3s (shuffle) with the one of the groove displaced 3 beats (subdivisions)*** later... such fun! It then displaces back to the 9 disco before doing the straight 13 to the end (except for the 4 bars where all the above grooves are stacked on eachother).
I soon drew up my BPPR page of possibilities (see below). 8 against 11 is particularly satisfying, due to its symmetricality and having an 8 involved and that eventually became my song, Make Stuff Up , with a rather Bossa Nova vibe for the 11 and a 16thy groove (with the one displaced an 1/8 later) for the 8.
Even though 5 against 3 (thus 0.6) isn't a particularly common polyrhythm, I was drawn to the Fibonacci / Golden Ratio vibe of 5 against 8 and 8 against 13. Here's my 13 against 8 version of the Mission Impossible theme: http://malwebb.com/AimHigh.mp3 . I call this "Aim High", as the morse code for AH is dot dash dot dot dot dot, which is what my 13 rhythm is, and Lalo Schifrin based the 5/4 rhythm of Mission Impossible on the morse code for MI, which is dash dash dot dot. I've also subtly ripped off Baterz's song, Darling, which is a lovely 4 against 3 thang in itself. He would've appreciated the odd timey-ness!
My song Peak goes full Fibonacci, being 8 against 5, but eventually also puts 13 and 21 across the bar in a few spots. It actually begins in 5, with 3 and 4 against it (yes, I know 4 isn't a Fibonacci number, but it sounds good!). It soon settles to a 5 against 8 mbira riff which underpins (well, overpins) the rest of the song. The middle part of the song has a fairly straight 8 beatbox groove against it (with its one displaced a beat later) which fits remarkably well with the 5 beatbox groove.
Then I finally wrote a 5 against 7 song, in Giant Gippsland Earthworm . The choruses are a straight up 7, but the rest of the song is various combinations of the 5 and 7 grooves. For the recording, I actually put the "pivot point" of bar (the 5th beat of the 7 and the 4th beat of the 5) at 41/70s of the bar (halfway between 40/70s and 42/70s), so both grooves were equally compromised (but, to me, imperceptably so). I'm thinking of doing a similar "pivot point of the bar" comprimise when I make a studio recording of "Peak", putting it somewhere between 0.6 (3/5) and 0.666... (2/3) perhaps at 0.6180339... (the inverse of the golden ratio, Phi) which would be appropriate, but only if it sounds good! For more fun with Phi, check out my Irrational number delays.
Next came Osculate, in 12 against 17 (see on the diagram below). 12 is already rich with polyrhythmic possibilities, so blurring 17 into it is amazing. While the 2-3-2-3-2 grouping of 12 is nicely symmetrical, I've gone with 2-3-2-2-3 like the classic west african 12 beat bell pattern (2-2-1-2-2-2-1), so the 17 is 3-4-3-3-4. If you put 8 against the 12 and do it over 2 bars, the pulse can shift from 16 to 17! Here's a rough of the workings of it: http://malwebb.com/12against17thang.mp3 . I've made the guitar riff more 12-ish than 17-ish, to help all the 12 and 24 polyrhythms sit better. I've made those 5 accents G E C A D, which is C pentatonic, but I'm then adding B and F# in the long beats to make it G major scale, hence G E B C A D F#. Then I've put the remain five notes in the remaining five gaps to make it a chromatic tone row over the 12: G Eb E Bb B C Ab A Db D F F# . . Harmonically and rhythmically, it's all very 5+7=12, but also 5+12=17... hmmm.
And with this, as with all the songs mentioned above, I'm keen to do it in a way that an unwitting listener can still enjoy it as an inviting piece of music, without tripping over the maths: That's the dream!
I recently noticed that "Electric Sunrise", by Australian guitarist Plini, is in 13, grouped 3,3,3,4 with duplets over the 3s, and Dave Brubeck's "Blue Rondo à la Turk" is in 9 (mostly grouped 2,2,2,3), so I made a BPPR mashup of them: https://youtu.be/kKUUL4nOvP4 . I'm not sure if the Plini fans will approve!
Just a note on Polyrhythms in general, the western and Indian approach to polyrhythm is generally quite melodic, in that the polyrhythmic element floats over the groove. Whereas in African music the polyrhythm is intrinsic to the groove. i.e. a 12 beat cycle where the pulse might be 2, 3, 4 or 6 at any moment, depending on the listener and/or how the players and/or dancers are accentuating it. To me this is true polyrhythm. I try to leave all the pulse options open to the listener, and yet still guide the listener through a journey of different accetuations of the polyrhythm, switching between the pulse options and even displacing them (moving which pulse is the "one").
*When Linsey Pollak first heard the riff, he said it was like "Brazilian Kopanitsa" (Kopanitsa being a Bulgarian 2 2 3 2 2 dance/rhythm). He also wrote (and I'm sure he won't might me sharing it): "Interestingly, the Macedonians often thought of these 'bent' rhythms more as short short long short short rather than 11/8 126.96.36.199.2, which could also be interpreted as 18/8 188.8.131.52.3 (still short short loong short short). I have heard the same melody (Baba Djurdja) played as 7/8 2.2.3 and also as 10/8 3.3.4 (both short short long). Originally (a lot of the music being dance music) the musicians would take the rhythm from the steps of the leading dancer".
It was great to hear that, as it kind of sets a cultural precedent for my little BPPR rabbit hole... I was a little worried that I was losing my mind!
**Note: The fact that western time signatures look like fractions adds to the confusion created by a term like "4 over 3" also seeming like a fraction, where it's actually the inverse. [You might've notice I don't use any time signtures in the above description.] The "8" in 7/8 is fairly arbitary: 7/4, 7/8 and 7/16 could really all represent the same groove. So if a groove is 7-in-the-bar, I'll just call it a 7.
That said, if you treat time signatures as fractions, then they represent the fraction of a 4/4 bar (i.e. 7/8 is seven eights of a 4/4 bar). I feel like this is part of why 4/4 is so dominant, to the point where it's known as "common time"! Sure, a 4 groove is very appealing, but maybe it wouldn't be so dominant if our western system of notation didn't default to it so much. And then there's metronomes, sequencers and recording software that always default to 4!
***The word "beat" has too many meanings within rhythm: It can mean the groove (a latin beat), or the pulse (on the beat), or the subdivisions (upbeats, off beats, etc.) And then actors use it to mean a pause! I decided that when writing, I'd try to only use "beat" to mean subdivisions and use "pulse" and "groove" for the other meanings. Soon after that, I got Greg Sheehan's book and noticed that he does the same! I particularly like his use of the term "3 in the pulse", "5 in the pulse", etc. for how many beats (subdivions) the pulses are divided by. Nice!
I feel like this page will continue to evolve over the years... stay tuned!