I thought I'd share a few thoughts about a general concept/technique in Arabic music that I think of as "rubato on the beat" or "out-of-time but in-time." I've never heard anyone use a specific term for this in Arabic, if anyone knows an Arabic musical term for this, please let me know in the comments.
I hear it a lot from vocalists and violinists and it's really useful and beautiful in taqasim.
The idea is that you play in a way that your phrasing is not metrically related to the beat, except for certain key points that will line up with the beat (usually the beginning and ending of the phrase at least, and sometimes key points in the middle).
The best way I can think of to explain this except that it's sort of like you feel a much bigger beat—you feel a whole measure—or even two measures—as just one big beat. Within that "big beat" you can subdivide any way you like, or even play completely freely.
A couple of my students a little while back were asking about how to achieve this effect but stay connected to the underlying pulse. I've noticed many oud players seem to either play in time (especially if they have a background in other styles of music) or they can play a traditional rubato taqasim that's not connected to the beat, but this in-between way of playing can be an elusive skill. It's worth developing though! It allows one to bridge the gap between free, expressive rhythm and energetic, precise rhythm. While one can certainly just wing it and keep trying to get it, there are a few exercises that I feel help achieve this more methodically and also give one's imagination a bigger repertoire of sounds to choose from while improvising.
The first thing to try is to just experiment with playing freely (out of time) but making sure the first and last notes you play in a phrase land on the beat. If you are unsure about landing the last note, try playing a tremolo and listening for the beat.
For a more systematic way of deepening one's ability to feel two meters simultaneously, below are some practice ideas for developing more advanced facility and vocabulary. Fair warning - it involves some math and complex rhythmic relationships.
Personally, I've practiced both ways and each has its benefits, so the approach described above is a perfectly good way to practice — especially to start. Since everyone is a different individual, I find having multiple approaches helpful; what makes something 'click' will be vary from one person to the next.
This may be overkill, but for the true nerds, here's a deeper dive on working on a polymetric approach as a basis for this kind of freer-sounding playing. I'm not at all saying that this is the foundation for what one hears in Arabic music, just that it is another path that leads to some similar sounds and can help solidify one's sense of time and ability to play creatively with rhythmic relationships.
A more thorough and methodical approach to this is to work on superimposing a different meter—it gives a similar feeling of being not "in-time" or at least "in a different time" but you retain awareness and control in relation to the underlying pulse.
If you take a measure of 4, for example, and subdivide the whole measure as 3, 5 or 7 then the downbeats will line up but nothing else will (see ex.1 below, note that I am including the basic 4 beat structure for reference throughout).
For an additional challenge, you can also displace this by a beat or a half measure so that the points where it lines up are in the middle of measures and not on the downbeat. This gives it even less of a feeling of being on the grid (ex. 2).
Superimposing 5 or 7 is often a bit too challenging at first, so as a first step towards feeling this kind of freedom to play in a different time than the prevailing meter (while still staying connected to the underlying beat) I suggest simply practicing improvising with quarter note triplets (3:2), giving you 6 beats in a bar of 4/4. Once you feel comfortable with playing 6 notes per bar, try feeling each pair of notes as a larger beat, so that you are feeling 3 beats over 4. In this scenario, the 'real' beat 3 feels like the middle of your 'new' beat 2 (ex. 3):
In the other direction, you can "fill in" the quarter note triplets with 8th notes to create rhythmic variety and solidify the feeling of the "new" time (ex. 4). The next step would be to do the same thing, but start the triplets on beat 2 (ex. 5).
From there, you could repeat the whole process but trying to feel a measure of 5 in one bar of 4 (5:4 quintuplet).
Another great alternative is playing a 4:3 rhythm. Playing as if one is in 3/4 or 6/8 while the underlying beat/phrase is in 4/4 is a very common technique in both Arabic composition and improvisation. In a regular free-meter taqasim, because there is no stated 4/4 pulse to contrast against, the end result is closer to just alternating between phrasing in 3 and 4. However, the principle is the same (as is much of the vocabulary used). Ex. 6 shows a common phrase of this type (shown in C 'ajam, but could be in any maqam). Brackets show the 3/4 phrasing (could be felt as 6/8 as well, this is just a matter of phrasing and emphasis).
(note: A phrase like this would usually not be played with the 4 full iterations of the rhythm shown here - it might end early, or start on a different beat so that it ends after 3 cycles, for example.)
Playing a 4:3 rhythm is just extending this concept to the next level. Since 4 bars of 3 has the same number of beats (12) as 3 bars of 4, the downbeats of the resulting rhythmic phrases will only line up every three bars. If you don't start the cross-rhythm on 1, it will happen sooner, as you can see from example 7 below:
This rhythm can be felt as a polyrhythm, but it can also be understood as a simple dotted rhythm, as in example 8:
Examples 7 and 8 show the same rhythm, but two ways of conceptualizing the sound.
You might notice that this phrasing is nearly identical to the common iqa' Bamb Masri (also simply called Bambi), shown in ex. 9.
These kind of exercises can continue on in various combinations. For example, you can alternate between 3:2 and 4:3 to get something that very close to 7:5, 11:8, or 17:12 (example 10).
While it is possible to play these in rhythmically precise ways that clearly differentiate between the rhythms written below and the complex polyrhythms like 11:8, the main goal here is developing the ability to maintain awareness of the underlying pulse while playing and feeling a contrasting pulse.
After a while, these rhythmic superimpositions just become a natural extension of your vocabulary.
Note that I've notated both the triplets and the quadruplets in two ways to help understand the subdivided relationship to the underlying beat.
To me, this feeling of simultaneously being in time and out of time is closely related to the feeling while playing tremolo of being at once connected to the beat but also detached from it. Tremolo can also be used in conjunction with this technique to help smooth the transition of playing rubato to playing in time.
I hope this is of some interest and use to somebody! Please comment below if you found it useful or if you have thoughts or questions.