I'm glad to see some discussion of clicks happening, they certainly have been a key issue in my research in Turkey. Some of you may know that Turkish folkloric and Ottoman art musics have a stunning variety of metrical structures, with musics that are notated with meters ranging from 2/4 to 120/2 (and nearly every combination in between).
Yusuf Cemal Keskin is widely regarded as the best living performer of the Trabzon-Maçka regional style of kemençe playing, and is very much in demand as a stage, wedding, and private event performer. He's recorded a number of albums, but is better regarded as a live performer. Here's an example of his playing/singing of a song that might accompany a slower horon dance:
If this were to be arranged into an ensemble form and recorded, immediately the first "violence" that would be done to the piece would be to give it a 5/8 time signature. The song (and others like it) don't have any inherent "five-ness" to them; the meter of the song is felt as a two, with one shorter beat followed by a longer beat. But how much longer is the longer beat than the short? If we round the durations up and down we arrive at 2+3, but if you analyze the waveform display of the audio file, we see something different:
While the tempo is actually quite consistent throughout, the lengths of the "3s" (long beats) and "2s" (short beats) are not, and moreover the precise ratios change between measures as well. The picture above shows from about 0:01 until 0:07 in the video. In the first measure, the first short beat is slightly "too long," and for several beats the kemençe is "behind" the beat, only catching up in bar 4, with the downbeat to bar 5 coming quite a bit "early" although the next beat (the vertical line to the right of 5.1.00 is "perfectly in time."
What makes this music distinctive is not the 5/8 meter that can be abstracted from this, but rather the ways in which Yusuf Cemal Keskin plays with millisecond-level listener expectations, placing things slightly behind or earlier than what would be expected. Studio recordings – of him or other kemençe players – typically are based around both a click-track and a grid, and lose this entire layer of rhythmic significance. Perhaps tellingly, Keskin is not the best studio kemençeci; Tahsin Terzi and Selim Bölükbaşı are more "sought after" for multitrack recording sessions, in part since they have figured out how to adapt kemençe performance practice so that it lacks these expressive microtimings. From anecdotal evidence, when Keskin has made albums, engineers have typically quantized parts like you see above, as the seductive draw of the DAW's interface informs a user that a musical event is "discrepant" with regards to the unwavering bar/beat grid.
What I've discussed is related to what Charlie Keil terms "participatory discrepancies" or Vijay Iyer terms "expressive microtimings," but I'm not thoroughly comfortable with wholesale adopting either of their theorizations. There is nothing participatory nor discrepant about the timing of Yusuf Cemal Keskin's playing (although certain choices might relate to how audiences participate in terms of dancing along or watching the video). I'm also not convinced by the embodied cognition aspects of Iyer's work – I think it's part of it, but doesn't account for everything happening here, in part since his analysis is largely acultural.