I'd like to be able to track my progress in keeping tempo during my practice, not only in working towards increasing my speed, but also the precision of my timing as I work on being able to play faster.

(How) Do people assess the precision of their tempo of their practice performance over time?

I started using librosa beat tracking to plot tempo of rolling windows of my practice performance, as well as reference material. The plots help me quickly identify which phrases that I rush or lag. I can also track the standard error of my tempo day-to-day to see if I'm improving.

librosa's onset detection helps me see if I'm ahead or behind the beat. Is there a quantitative way to summarize and track precision of practice performance so that I can see how/if my practice is improving?

Are there other/better tools or approaches to measure tempo and rhythm? I've seen that Yousician tells you if you're late or early as you play along, but I didn't see a summary of errors or performance. (And I want to practice with my own music.)

I realize that rhythm is only part of being able to play faster or more precisely, but I don't yet know how to measure or summarize tone quality, dynamics, or other aspects.... There's also the prospect of getting lost in the weeds dancing about architecture.... @Athenasius has a great summary of studies about perception of tempo changes providing some limits of necessary precision.

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    Also be aware tha a lot of music - excepting dance music, benefits from being slightly out of tempo at certain points - be it speeding up, or slowing down. That's part of the art of the art. – Tim Jul 24 '20 at 7:05

Based on your description of Librosa, a reasonable quantitative measure would be to track how frequently you get ahead or behind and how long you stay there before getting back on track. Improvement would be a reduction in the number and/or duration of incidents.

Another excellent way to practice rhythm is with a metronome. But rather than set in on every beat, set it for every other beat, or every third beat, or on the first sixteenth of every beat, etc. That will ensure that not only is your pulse accurate with each beat onset but also with each subdivision of the beat.

As a general rule, slow practice is better, even if your goal is speed. Speed comes on its own, with little effort on your part, when your rhythm is accurate and, especially, when your movements are efficient. Relaxation is much more the key to speed than any other factor.

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