Of course you can, as long as you’re using the appropriate unit of measurement, which in this case would be cents. What you mean is that it’s meaningless to describe the variation in pitch in terms of frequency (Hz or kHz).
Right. Or you can talk in terms of factors. To get from one note to the next, you can’t add a constant to the frequency, but you can multiply by a constant. If I remember correctly, the number is log[sub]2[/sub]12. So the frequency of any given C times log[sub]2[/sub]12 is the frequency of the nearest C#.
According to my wife, who has a masters in music, there’s a considerable amount of wiggle-room in the definitions of tremolo and vibrato, such that sometimes the terms can be used interchangeably, and sometimes not.
In general: Vibrato is a rapid, cyclic variation in pitch, usually no greater than a semitone from the tone about which the pitch is oscillating. Tremolo is more widely understood as a fluctuation in the volume (amplitude) of the note being played, but it can be used (though infrequently) to indicate what amounts to little more than a huge vibrato, or “a vibrato you could drive a truck through”, as she likes to put it. I think the reason for this is the vibrato starts to reach the semi-tone range, where you’re technically rapidly repeating two or more individual notes on the chromatic scale. At any rate, it’s hard to call somebody wrong for using one or the other term in any given situation.
Again, thanks to all who have posted, even to those who have been…um…well, wrong. 
For the record, there are different scales one can use, especially if one doesn’t have to be able to play in tune in any key. But, as Bach proved, it is possible to temper the scale so that one can play in any key. It’s an imperfect scale in terms of the pure consonance of intervals (in other words, it’s slightly out of tune!), but it’s such a good compromise that we’ve been using the equal(ly)-tempered scale ever since.
Bach just wasn’t aware, as far as I know, of the mathematics of that scale. Each semitone differs from the next lowest by a factor of the twelfth root of two (2 to the 1/12 power), or a factor of about 1.05946 times the frequency of the previous semitone.
Likewise, a cent (one one-hundreth of a semitone) is a logarithmic interval as well, being equal to the 1200th root of two (2^(1/1200), or about a factor of 1.000578. Not very dang much. For the mathematically-impaired, that’s a 0.0578 percent difference.
Any interval can be conveniently calulated on a spreadsheet such as (shameless plug) OpenOffice, or even Excel, by raising two to a fractional power depending on the number of semitones involved; e.g., a minor third is (2^(3/12)), since there are three semitones in a minor 3rd.
(I offer the above as somewhat of a repayment to the group for all their help. Obviously, I have put in a bit of study on the subject.)
It’s also true that context is as important in the issue of how small an interval that one can hear as it is to the issue of how small a difference in loudness one can hear. Theoretically, the dB is defined as the smallest difference in loudness that a human can percieve, but that’s only in the best of circumstances. Sometimes you won’t notice less than a difference of 2 or 3 dB.
(A dB, of course, is a measurement of difference in intensity, and could be applied to anything, not just sound. There’s no such thing as a sound level of such-and-such dB, unless you reference it to something absolute, usually the smallest humaly-perceivable (sp?) sound pressure level, and go from there.)
And a dB is defined differently when one is measuring differences in power than when one is measuring difference in, say, the voltage level coming out of a microphone. But that’s another subject.
(Back to topic) 
If a singer is varying only the intensity of her voice, there is no difficulty at all in detecting her pitch. If, however, she is varying her pitch all around the note she is singing (vibrato), this poses a difficulty for any electronic device, unless there is some way to discriminate or calculate the intended pitch. So far, I don’t see any way, unless, as GorillaMan implied (to me), the “real” note might be the arithmetic mean of the highest and lowest points on the frequency continuum. But I didn’t think that everyone “vibratoed” that way, especially singers.
You’re quite right, of course, but it was never my intention to debate the definitions of “vibrato” and “tremolo,” although I likely caused this to start with.
What matters is that amplitude fluctuations are not a problem. Frequency fluctuations are. I was using different words to differentiate these two different phenomena, and for no other purpose.
It’s frequency variations I’m trying to qualify here, if that’s even possible.
I just re-read the entire thread, and realized that this comment, more than anything else, actually addresses the point of my original post!
DWD, what I am attempting to do is determine, in real time, whether a musician is on-pitch, and if not, how much she is off by (in cents.)
The difficulty lies in discriminating between vibrato that varies, say, 50 cents below the fundamental note, and a singer who is flat by 50 cents.
Surely you’ve heard some singers (Ethel Waters comes to mind) whose vibrato certainly approached or even exceeded a full semitone from the note she was singing. It’s not a matter of degree, since I have to know if a singer is off by as little as a single cent (yes, I know this may ultimately prove to be impossible) in real time.
If vibrato is always “dancing around” a single note, with equal distance above and below, then averaging out the intended note would certainly be possible. My question is, is this always the case?
And I don’t see how you can be correct in saying
since once you’ve made a tiny (say, 1 cent) variation thereof, it’s by definition no longer on pitch!
Have I made my dilemma clear, or is it still a mystery to some?
[hopefully minor hijack]
Is there a limit, expressed in cents, or some fraction of cents, to the ability of the human perception of sound to discriminate between two tones/fractions of tones?
[/hopefully minor hijack]
I have no doubt whatsoever that this is a matter of considerable debate, but the Hyperphysics site claims about five cents. But, as has already been pointed out, under certain conditions this will be considerably poorer.
I think the only way this can be answered is by experimentation. Once you’ve built your piece of kit that’ll work out the ‘average pitch’ of a note with vibrato, you’ll need to see if what the machine calculates as the pitch matches with what musicians (and laymen as well, I guess) interpret it as aurally.
I could have sworn it’s not even usually the case. When playing slide guitar, oscillations up to about half a semitone below the intended note sound fine, but if I go even a fraction above the intended note, it sounds dreadful. I had always understood, as you did, that the note we “hear” is the peak value of the oscillations. I’m pretty bad at stopping myself overshooting, and it’s the main reason I’m so rubbish at slide. Well, that and the clattery noises I make, and the general lack of talent… 
I think some experimentation is called for, too. If you can set up two oscillators, one producing a vibrato note and one producing a flat tune, you can ask volunteers to tune the second oscillator until they think it’s producing the same note. Should answer the question, at least from the human perception angle.
In college I was a music major with the saxophone being my major instrument.
I was taught that for the sax, and for wind instruments in general, you only go below the pitch.
Trumpet player checking in.
I might have to experiment tonight when I get home, because what I’m about to say I cannot be absolutely sure about, because it is so unconsciously done (and I’m a hack amateur, etc., etc.).
I believe that my vibrato is only in the downward direction.
As an aside, when playing individually the vibrato makes the sound very pleasant, but in an ensemble it’s better to eliminate the vibrato because with everyone “doing their thing” it can sound out of tune even if it isn’t. Obviously, with a trumpet it is easy to use or not use a vibrato, so it’s no big deal.
Though vibrato analysis is not really my area, there are people who study exactly the sort of things you’re asking about. I would consider a trip to your local university library if you’re really interested in digging into the latest research on the subject. The Journal of the Acoustical Society of America (JASA) would be a good place to start.
One analysis I know of (for violin) is:
“The time-frequency characteristics of violin vibrate: Modal distribution analysis and synthesis”, by Mellody and Wakefield, JASA 107 (1): 598-611 JAN 2000.
From the abstract,
“The analysis indicates that the frequency modulation induced by the motion of the stopped finger on the string is accompanied by a significant amplitude variation in each partial of that note… The frequency modulation is well-represented by a single sinusoid with a mean rate of 5.9 Hz and a mean excursion of +/-15.2 cents.”
So, on a cursory reading of the abstract, it appears that the frequency fluctuations in this study were on both sides of the fundamental note and had a mean deviation of ~15 cents.
For information specific to singing voices, you may also want to look at “Modal distribution analysis, synthesis, and perception of a soprano’s sung vowels”, by Mellody, Herseth, and Wakefield, Journal of Voice 15 (4): 469-482 DEC 2001.
The abstract in this article indicates that the vibrato rate and excursion (dictance from the fundamental note) is actually very important in the perception of the identity of the singer. There isn’t any more detail relevant to your questions in the abstract…you’ll have to dig into the article for more.
These are just a few that I knew off the top of my head, and this is not really my field. I’m sure there is much more if you wanted to put the time into learning. I’m sorry I can’t post a more general link to the articles here, but I only have access through a subscription service at my university. Contact me directly if you want me to send you the full abstracts (I may be able to get reprints of the articles if you want them).
GorillaMan, you’ll notice that both articles I mention above use listening experiments with real honest-to-goodness musical instruments (not “sine wave generators”), often performed by genuine musicians, to determine what is perceptually significant to the average listener (I honestly don’t know if any of the listening samples were performed in a sufficiently musical “context”).
Yes, the brain clearly changes the way it operates based on prior expectations (“context”) sometimes, but there is also a natural limit to the basic abilities of the acoustic hardware we have been given. Much of what we have learned about how we understand music comes from scientists who study things like physics, psychophysics, sensory neuroscience, etc.
So, please don’t assume that someone trained in the sciences is ignorant of the issues relevant to legitimate musicians. You would be surprised how many of us have formal training in both a science and music just so we can understand the issues that are important at that boundary. Yes, some of the people lurking around you at the music school are actually scientists in disguise. SHHH! Don’t blow our cover!
GorillaMan must have an interesting tone when he plays, if he truly goes above and below the note, with vibrato.
I sat and thought about this a few minutes, while listening to some unaccompanied violin music. I think Zebra and Algernon are on the money when they said vibrato usually only goes below the note. I know that as a violinist, it’s easier to move your hand in that direction (away from the body) rather than towards. However, I may have to experiment a little the next time I practice.
Randomly pulled out a Menuhin recording as a sample…it reassured me that I’m in good company
panache45 gave a good simple explanation of how the left hand motion produces a pitch moving both above and below the pitch. It shouldn’t be any ‘easier’ to move your hand in one direction than the other - if it is, I’m afraid that’s caused by a problem with your technique.
With wind instruments, as described above, I’m wondering if the timbral changes if a pitch is bent upwards make vibrato less pleasant-sounding if done in that way?
With any of the reed instruments it would. You have a steady pressure with your jaw on the reed/mouthpiece. To lower the note you loosed the pressure but to raise it you need to tighten it. This would cause the note to become harsher, more strident. Generally you need to have a very open and free air passage from your throat through the instrument.
My father played the violin, and my memory of his hand motion to produce the vibrato seems to indicate that it went as much in one direction as the other. In other words, for a stringed instrument I think I’d agree with GorillaMan that the vibrato varies both below and above the desired pitch.
Below the fundamental pitch. Always below.
Violist here. I was taught that vibrato must always occur to the south of true intonation. That acoustically, the human ear hears the “top” of the pitch as the note being played.
I’m not a doctor or a scientist, but this makes sense intuitively to me. Here’s my unsophisticated analogy:
You wave your hand up and down. Then ask an observer “how high is my hand going?” It seems to me the answer you’d likely get is at the top of your motion.
Plus, to my ears, it just plain sounds better. Listen to Frank Sinatra. He goes below. Listen to great 'cellists. They go below. I’ve heard hints from my violin friends that they like to play on the high side of pitch. 
Maybe it’s a violinist thing. But as a violist (read: deeper, mellower), I much prefer to sit a little low in the saddle, intonation-wise, and I think the great majority of string players would agree.
I don’t see that the analogy necessarily carries over to the psychology of sound.
I’ve just picked up a Rostropovich CD to test this - he goes ‘both sides’.
I had to pick up my violin and try this out. Do you use a wrist vibrato, by any chance? I found it hard to force a ‘below’ vibrato at all with arm vibrato, but could do it from the wrist no problem. It did feel that I was forcing myself to not use half the muscles in my hand, though.
…actually, listening further to Rostropovich…the ‘below’ portion of the vibrato does at times sound more prominent. I suspect it’s because of a timbral shift caused by the softer flesh of the finger being toward the sounding part of the string, giving the mellower sound you suggest.