How accurately can a good musician judge an interval to be out of tune? [closed]

Question

This is for a music/physics crossover experiment. Almost everyone will be familiar with the Doppler Effect, even if they don't know the name. Imagine that a vehicle is blaring its a siren or otherwise making a noise with a distinct pitch. As the vehicle approaches you and then passes you, the note will appear to drop in pitch.

The apparent note will be higher in pitch than the "true" note when the vehicle approaches, and the apparent note will be lower in pitch than the "true" note when the vehicles recedes. (By "true" note, I mean the original frequency emitted, prior to any Doppler shift.) There will be a glissando as the car passes. The closer the vehicle passes you by (e.g., the closer you are standing to the side of the road as the vehicle passes), the more rapidly the glissando will occur.

It occurred to me that it should be possible to calculate the speed of the vehicle by the interval between the approaching and receding notes. So, I have performed some calculations and here are a couple of examples: if the perceived pitch drops by a (well tempered) minor third, then the vehicle was travelling at 106.8km/h (66.4mph). To produce a perceived drop equal to a major third, the vehicle must have been travelling at 142.1km/h (88.3mph).

Personally, with my skills, even an accuracy of 10km/h would be a challenge. Increasing the speed by only 10km/h would produce something I might recognize as "a slightly sharp minor third," but it might be difficult for me to discern precisely how sharp the interval is.

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Here is my question: how close could musicians come to determining how sharp/flat an interval is, in cents? For example, would musicians be able to detect (by ear) that an interval is 10 cents above a minor third, or will their unaided ear be limited to detecting that the interval is, e.g., 100 cents above a minor third? I can distinguish how sharp/flat an interval is to a semitone, but I expect that good musicians could do rather better.

So, does anyone know how good/precise musicians are at judging interval intonation in this fashion?

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NOTES

• Perfect pitch is not required, just the ability to judge the interval.
• The temperature will have an effect on how much Doppler shift occurs in the perceived frequency. The values above assume 20C / 68F. Oddly, the air pressure does not have an effect. Humidity does in principle but it is very small.

CLARIFICATIONS

• I am not expecting a musician to answer with a speed, but rather with a musical interval and, ideally, with a judgement (in cents) of how far off an interval is (e.g., "this is a minor third that is sharp by 100 cents").
• I am not expecting to make radar guns obsolete. The purpose is for fun and for education. Forget my proposed use for the moment.
• Whether the musician is a professional or a good amateur is not important. There may be some amateurs who are better at this than some professionals. I am also interested in how well piano tuners would be.

Answer 1

I can't answer for "some" musicians, but I can answer for myself and my answer might give you some insight on why you might have a hard time getting a definite answer to this question.

Background: I am a jazz musician, a sometime professional. I do a lot of ear training. I rarely deal with non-Western tuning systems.

I've tested my threshold for identifying the (in)accuracy of intonation and it varies considerably. For perfect intervals of all kinds, I'm pretty accurate; I can generally tell within a few cents if a perfect interval is off. It gets progressively more difficult when you get into thirds and sixths, although I tend to be pretty good at identifying whether sevenths are in tune or not. Minor seconds and tritones are extremely difficult, particularly melodically, and my accuracy could be as much as ten times lower in those situations.

In many respects, that's just the tip of the iceberg. Melodic is much more difficult than harmonic. Higher pitch is correlated with greater accuracy. Accuracy is generally high when the two pitches are close to forming an inveral but generally low when the two pitches are at the midpoint of two western intervals. I might be able to tell you, for example, that "these two notes are a perfect fifth sharp about ten cents" but it is unlikely that I'd be able to tell you that a perfect fifth sharp 40 cents is anything more than somewhere vaguely between a very sharp perfect fifth and a very flat minor sixth.

I have never tried this with glissando but I would imagine that this would make things significantly more difficult.

In conclusion, I think that my ability in this regard varies so significantly (and it is my somewhat unscientific belief that this is typical among good musicians, based on my experience) that the only real answer to the question is "it depends".