March 4 Information

Can anyone think of a psychological experiment that could determine if the perception of tempo is logarithmic? Is anyone here aware of anything like that having been done already, and, if yes, what was the methodology used? If not, would it be enough to ask people to compare tempos and ask them if they perceive the increase in tempo from 20 to 30 BPM to be roughly equivalent to the increase from 40 to 60 PBM, etc. and then do a statistical analysis of the results, over many tests, and many people, to see if there is a consistent pattern, and voila? Basemetal 11:26, 4 March 2018 (UTC) reply

Some context: It is well known that the perception of pitch is logarithmic, that is we perceive the distance between 100 Hz and 200 Hz as equivalent to that between 1000 Hz and 2000 Hz. Now the logarithmicity of the perception of pitch (is this a word?) is explained (as far as I know) by the way a structure in the ear that looks like a logarithmic spiral (and whose name I forget) works to analyze pitch. The perception of tempo is all in the brain probably. At least I don't see where else it could be. Still it is obvious that the perception of tempo is not linear: we don't perceive the change in tempo from 240 BPM to 241 BPM the same as the change from 30 BPM to 31 BPM. That already would be enough of a reason to graduate metronomes logarithmically rather than linearly, as even if we're not able to determine exactly if it is logarithmic, it is certainly closer to logarithmic than to linear. In that case we could use the same unit of measurement for pitch and tempo. A tempo of 60 BPM would be 1 Hz, 120 BPM would be 2 Hz. Basemetal 11:26, 4 March 2018 (UTC) reply

PS: Does this belong to this desk? Is it more likely to be answered at the Science Desk? Thanks. Basemetal 11:28, 4 March 2018 (UTC) reply

It is worth noting in this context that the notches on a standard metronome are not a constant distance apart, and indeed get farther apart as the tempo increases. A typical range of BPM markings would be 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 126, 132, 138, 144, 152, 160, 168, 176, 184, 192, 200, 208. Double sharp ( talk) 11:46, 4 March 2018 (UTC) reply

Ok. (That should have been in small because it refers to an aside and not to the main question). Looks logarithmic more or less. That still doesn't answer the question except possibly you're saying some metronome manufacturers believe the answer to my question to be "yes". That's not what I was asking. Not all of them, incidentally, because as far as I can remember my old electronic Korg MA-30 was linear although I can't check since I haven't bought new batteries for it in ages, as I mostly use the Google metronome these days. However note the measurement unit, the BPM is not logarithmic. You're right, you're right, the Hz wouldn't be either. My mistake. (You see, I've spared you an edit). If we wanted to really measure tempos logarithmically we'd have to measure tempos with "notes" of 12-TET (for example) instead. If for example we had C4 = 15360 BPM = 256 Hz we'd have 60 BPM = 1 Hz, 120 BPM = 2 Hz, 240 BPM = 4 Hz all be the "note" C. The difference ("interval") between a tempo of 60 BPM and 120 BPM would be an "octave", etc. Basemetal 12:47, 4 March 2018 (UTC) reply

So noted and corrected – now it is in small print! ^_^ Double sharp ( talk) 13:29, 5 March 2018 (UTC) reply

This book (Guide to Computing for Expressive Music Performance) states, admittedly without sources, that "Human perception of tempo changes logarithmically not linearly." Double sharp ( talk) 13:31, 5 March 2018 (UTC) reply

I guess that's the reasonable guess you'd expect, but it would be interesting to see a real test conducted and examine its methodology. I've been also thinking of the interference between purely auditory memory and muscular memory. If one actually plays an instrument (which you do) ones "muscular" idea of tempo, e.g. how the difference of playing at 40 vs 80 BPM compares with that between playing at 150 BPM and at 300 BPM. I would guess the results may be different for musicians, or even between singers and instrumentalists, between players of low instruments (cello, bassoon) and those of high instruments (violin, flute). It'd be interesting to see how a real rigorous test would come out. Of course muscular tests of tempo and how they interfere with the estimation of tempo when no muscular effort is involved can be conducted in a lab with non-musicians too. You ask people to estimate the relation between four tempos t0, t1, t2 and t3 of a click track and ask them if the relation between t0 and t1 is about that between t2 and t3. Then you examine how the results differed (if they did) when they were asked to push a button at the same time at the tempos of the click track. Unfortunately there doesn't seem to be much out there, judging from what I and the editors here have found. Either that, or it's the wrong RD. Basemetal 15:40, 5 March 2018 (UTC) reply

I think a possible issue with focusing on tempo relations is that to me, far-apart tempi do not really feel related unless they come in small-integer ratios, so that they are, in a sense, metronomically equivalent. This is either a problem or an answer, depending on how you look at it. ^_-☆ If we were to focus on close tempi, then it would seem to boil down to looking for the just-noticeable differences, and seeing if they indeed get farther apart for faster tempi. If you think about it, the idea that these should be logarithmic gets more and more convincing. Classical tempo hierarchies tend to proceed according to the sequences 1, 2, 4, 8, ..., and the same piece can accelerate from two to four to eight beats per bar without really changing the speed (e.g. the first movement of Beethoven's Fourth Piano Concerto and the slow movement of Mozart's G minor String Quintet). Now if one accomplishes a ritardando or accelerando, then the shorter beats are obviously stretched or squeezed faster than the longer beats in direct proportion with their value; nothing else would preserve the 1-2-4-8 power-of-two sequence. But I'm not sure if this has been proved through science, rather than just suggestive arguments like this one. Double sharp ( talk) 07:38, 6 March 2018 (UTC) reply

I was hoping there had been experimental studies. If not, too bad. I'll wait until this goes to archive and if no one else has answered I'll post the same question to the Science RD. Incidentally, even experimental studies would have their limitations, I'm fully aware of that, and I'm not so naive to think such experiments will tell the whole story, since musical perception of tempo is something more complex than the elementary task of estimating the speed of a click track. It involves a hierarchical perception which may be influenced by other features of a piece that the speed of the notes. For example a piece can be largo and yet be full of 32nd notes. The tempo hasn't changed and even the least musically educated person will feel that. People also will be rather good, I'd guess, at picking the beat of the piece that all the other temporal values organize around even though the length of each note individually varies. But how do you test for that and still keep control of all the parameters in the experiment. If you test with real musical pieces there's the danger you no longer know what had really been operating. You're reduced to the level of those idiotic "experiments" that showed that listening to "Mozart" had milking cows produce more milk. What piece by Mozart? How loud was it played? Why Mozart and not Vivaldi or Bach? Have they compared the effect of that "Mozart" with the effect of (say) the sound of flowing water? On the other hand if the experiment is too simple, just a click track, a series of pitches, etc then it may fail. So there is something in real music that may be missing. Then you'd have to build up structure little by little in many steps while controlling all the parameters and see when the feeling of musical tempo appears. One can imagine how expensive such experiments would turn out to be. Basemetal 10:42, 7 March 2018 (UTC) reply