What is the definition of rubato? Is there a scientific definition? Why do some performer's attempts at rubato make no sense to the listener, while others do?
I think Bruce is teasing us. There is so much about music that can't be defined scientifically, or even logically. A friend recently came up with the idea of programming a computer to read and play music from old manuscript scores. It doesn't take much thought to realise that this is quite impossible on account of all the interpretative factors that rely on ineffably "human" powers of perception, experience and insight.
Personally I like it when the beat remains discernible during rubato passages. In other words the listener ought to experience the rubato as a deviation from the beat (with certain effects on the emotion of the performance) and not as the absence of any beat.
The best explanation of rubato that I have seen is that the individual beat may be slightly ignored, but the overall pace maintained.
I think it's a very good question as posed. Are there "acceleration and deceleration curves" (borrowing Scott's apt description) that tend to be more pleasing than others in particular situations? Would it be possible, say, to take a passage that a violinist has played completely metronomically, and introduce artificial rubato by electronic means, varying those "curves" while asking some audience to judge which they like best? I don't see why a "scientific" approach should be dismissed so readily. For many among us, it's our go-to line of thinking about any meaningful question.
In my tango days, I met two sorts of rubato:
Martha Argerich tends to use a lot of rubato. As my violin teacher said about vibrato
"The metaphor I use with my students is how you would expect a good driver to pull up to a stop sign."
Rather than just speeding up and slowing down, the Italian "rubato" actually means "stolen" and I think it's widely accepted that the time stolen from one beat is given to another in the same bar. Does the shorter beat occur before or after the longer? Either I guess, according to how you feel about the music. It does puzzle me that pianists' rubato is often hugely exaggerated while string players apply it much more sparingly.
https://en.wikipedia.org/wiki/Tempo_rubato : gives a broader definition.
@Steve Jones "programming a computer to read and play music from old manuscript scores".
I tend to be fairly conservative with tempo, probably because I've heard a lot of unpleasantly bad attempts at rubato. My car analogy, like on that Living Pianos video, is one of slowing down while climbing a hill, then speeding up on the way back down the other side. Done properly, it feels very natural.
The simple, mathematical explanation is that an understandable rubato is a geometric progression. I have found that students immediately understand this.
Ah Bruce, I thought you might have an answer! By "rubato" do you mean accelerando and ritardando in any context? You're saying that the tempo should increase or decrease by the same percentage in each time interval of say half a second in order to sound correct. Maybe you're right but that's a hypothesis that could be scientifically tested, not yet an accepted definition.
It's the fancy name we give to being out of tempo.
I have heard Steve's definition in various formulations but I don't like it. It is too mathematical for me. I do not think that you need to return to the beat of the same metronome that you deviated from. It is enough to return to the same beat wether it is shifted a little in time or not. And the unit may be larger than one measure (e.g. speeding up or down a whole phrase before resuming the initial tempo).
So what are the main points of context for this issue? I'll take a stab:
Spot on, Andres! What we love about music is its predictability (in terms of melody, harmony, pulse, rhythm, texture...) and to be taken slightly by surprise. Just how far it is possible to introduce the surprise factor by stretching those elements without leaving the listener completely confused has been the question that has occupied composers for centuries.
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