This is the aspect of music that cerebral nerds fixate on and screw around with to produce "weird" time signatures -- prog rock, math rock, etc.
But it turns out that plain old folk music can produce unusual time signatures without a cerebral orientation -- indeed, from a purely corporeal approach, which simplifies the process. Alternating the number of steps within a pass in a dance -- e.g., a 2-step followed by a 3-step, and repeat -- actually provides motivation for an unusual time signature -- rather than pulling it out of your ass to showcase your brainpower.
The fundamental meters have either 2 or 3 beats per measure, with main stress on the first beat and weak beats trailing after: "ONE two, ONE two" or "ONE two three, ONE two three".
The standard is 4 beats per measure, but you actually hear it and move your body to it as though it were two pairs of two beats apiece, each pair having stress on the first beat and a weak beat after, but with the first pair's stress being even stronger than that of the second pair. This yields three degrees of stress -- strong on 1, medium on 3, weak on 2 and 4. "ONE two, THREE four, ONE two, THREE four".
If you don't subdivide the measure into those two pairs, and try to preserve only one degree of stress, that would yield 1 stressed beat and 3 weak beats. For example, "ONE two three four". It's not impossible, but those three consecutive weak beats trail off a little too long. The natural tendency is to break up three weak beats in a row, and give one of them stress -- albeit to a lesser degree than the main stress. Hence grouping them into two pairs, each with its own stress (strong, weak, medium, weak).
Before getting to the unusual meters, let's ground the discussion of 4 beats in a kinesthetic activity. Normally I'd say marching in place to the rhythm, but that won't work if we're using 2 or 3 weak beats in a row. For walking or marching, one foot marks the strong beat, the other marks the weak beat. Two or three weak beats in a row would require putting the same foot forward for 2 or 3 steps -- doesn't work. So instead, try hopping -- hop on one foot for a strong beat, hop on the other foot for a weak beat. There's nothing impossible about hopping on the weak-beat foot for 2 or 3 times in a row.
Now try hopping to a count of "ONE two three four" -- strong foot once, then weak foot three times in a row, and repeat it over again. There's too many weak hops in a row, and only one strong hop to relieve the monotony! It feels like you're going to keel over in the weak-foot direction, or buckle under the pressure, or otherwise shut down your activity level.
But alternate back and forth, so that the 3rd beat gets the strong foot, and it feels totally natural and sustaining of your activity level. If your strong foot hits equally hard on 1 and 3, that's really just a 2-beat meter. So try to give the 3rd beat only medium strength -- more than the weak foot does, but not as forceful as the main 1st beat. Now you've got a 4-beat measure that's subdivided into two pairs, with primary and secondary stress.
We've solved the mystery of why 4-beat measures in music almost always get subdivided into two pairs -- because 3 weak beats in a row, with only 1 strong beat to relieve them, is conceptually imaginable, but corporeally destructive. It shuts down your dancing activity. Solution: subdivide 4 beats into 2 pairs with strong and medium stress, and suddenly you can alternate your feet and thereby preserve your energy level and balance. Musical choices are constrained by the requirements of dancing.
So why allow 3-beat measures, with their 2 weak beats in a row? Don't think, just give it a try -- hop on the strong foot once, then hop on the weak foot twice, and repeat. It's not as awkward as three weak beats in a row. It does throw you slightly off-balance, but not so much that you can't recover. And it does drag your energy down slightly, but not as much as when there are three straight beats worth of low-energy motions.
If we want any departure at all from the simple "ONE two, ONE two" meter, and its "RIGHT left, RIGHT left" motions, let's only depart as minimally as possible, just to keep it grounded. And in fact, you're getting a lot more leeway than you'd think. With "ONE two three..." the number of strong beats is the same, but now you've got twice as many weak beats -- a 100% increase. That's plenty.
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This brings us naturally to a meter with 5 beats per measure. We're fine with a 2-beat pair, we're fine with a 3-beat triplet, and we're fine with combining 2 pairs into a 4-beat measure. So why not try another minimal combination? -- two groupings as before, but instead of 2 + 2, it's 2 + 3, which gives us 5 beats total. There's still stress on the first beat of each grouping, and still primary stress on the first grouping, and secondary stress on the second grouping. Then it's, "ONE two THREE four five, ONE two THREE four five".
To dance to this rhythm, hop on your feet: strong weak, strong weak weak. Again, on the 3rd beat, your strong foot should only get medium-level force, not as much as on the 1st beat, but clearly more than on the weak beats. And if that's too difficult at first, just give your strong foot the same force on both 1 and 3. Having medium force on the secondary stress is more important when there's 4 beats per measure, where using the same force on 1 and 3 would make it sound like only 2 beats per measure.
When there's 5 beats, subdivided into 2 + 3, there's no way to confuse the first and second grouping because each has a different number of beats. So distinguishing the first from the second grouping does not depend so much on varying the force delivered on the stressed beat of each one, and differing primary from secondary stress is not necessary. (But it does help emphasize the different groupings.)
In the comments later, I'll add some thoughts on why 2 + 3 is more natural than 3 + 2, the other possible way to combine a pair and a triplet together to make 5 beats. It's not relevant now, though. But it does derive from the mechanics of dancing.
For a more interesting dance, rather than just hopping in place, try moving sideways in a line. Move in the direction of your strong foot, so that you land on a strong beat. For the weak beat, just hop in place on your weak foot. So it goes: "STEP hop, STEP hop hop" and repeat. Maybe it's more of a jump than a step to the side -- something more forceful than the hop.
And that's it -- "unusual" time signatures with 5 beats per measure are not so strange if the music is motivated by dancing, like Mother Nature intended. We could do 7 beats per measure, but it would break down according to the same principles -- e.g., combine a pair, a pair, and a triplet. The sideways dance would go, "STEP hop, STEP hop, STEP hop hop" and repeat.
These asymmetrical meters allow for some pleasant variety in the rhythm, while also keeping us from falling all over ourselves. It mixes things up while still keeping them in a simple, orderly structure.
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Now for some actual music with 5 beats per measure, driven entirely by dances that involve a 2-step and then a 3-step movement. The ground zero for their popularity seems to be Thrace, which is today split between Bulgaria, Greece, and Turkey, in Southeastern Europe. From my cursory survey, 5-beat meter seems to be a fairly reliable marker of Indo-European lineage, but that's a topic for another post.
National folk music saw a resurgence after the end of the Communist era in Eastern Europe, and in Bulgaria this was mixed with pop music to produce "pop-folk". It also went under the name "chalga," but that label has rapidly changed in meaning over the past 10-15 years to mean generic international genres that are mixed together (techno, rap, etc.), without a strong Bulgarian folk influence. The songs below are from before that transition, and the instrumental backing band hails from Thrace (Yambol).
The singer in the first one was a Turkish-speaking Gypsy of Bulgarian nationality -- talk about being at the crossroads between Europe and Asia. It may be hard to hear the 2 + 3 rhythm at first, but wait until the clarinet solo when there's just a spare bass rhythm otherwise. The primary stress is on 1, then a rest or silence on 2. There's a little offbeat before 3, the secondary stress on 3 itself, which builds through weak beats 4 and 5, then crashes down again on 1.
Try tapping your fingers in order to the rhythm. If you use only your four fingers, and not your thumb, you'll notice that the main stress falls on a different finger from one measure to the next. Namely, the next one in order from whatever it was previously. If it were a 4-beat measure, it should fall on the same finger each time. Since it's moving over one finger each measure, that means there's 1 more beat than 4 -- 5 beats per measure.
If you start with your pinky, that gets the heavy stress, the ring gets a rest, the middle gets the secondary stress, and the fore and thumb get two weak building-up beats, before cycling back to the heavy stress on the pinky again. Slow it down in YouTube if you need to. That also goes for getting used to dancing to it.
The second song is a little harder to hear, but it's the only other example I could quickly find while browsing a country's music that I'm pretty unfamiliar with. It's also a faster tempo than the first example, so try this one out after you've got the first one down.
The first music video brings back memories of the good ol' days here, where the band was ugly guys who could compose and play instruments well, a cute songbird on the microphone, and random babes moving their bodies to fully integrate dance with music and singing, as Mother Nature intended.
"Kaynana" by Reyhan and Orchestra Kristal (2002):
"Limon" by Toni Dacheva and Orchestra Kristal (1993):