Tuning terminology

November 20, 2019, 2:21 PM · I wonder if someone could answer regarding the terminology used when we tune a violin. One common way to tune is using stackable perfect fifths. My question is, when listening to the fifths, what are we listening to in order to decide whether they are in or out of tune? Do we term it a beat? A Tartini Tone? Overtones? Harmonics? Consonance (as opposed to dissonance) and if so what actually causes the consonance or dissonance? Or is it something else?


Replies (40)

November 20, 2019, 2:39 PM · The pleasant, smooth sound of well tuned open strings comes from the coincidence of a maximum of their overtones.
Edited: November 20, 2019, 3:10 PM · What Adrian wrote is certainly correct.

However, it reminds me of an experience I had in Junior high school. I had quit violin lessons about a year earlier (after 7 years) and my family had moved from New York City to some Maryland "farmland" where I was lucky to have a Jr. High system that actually had music classes for non-musicians (my NYC school had nothing of the kind). I remember sitting in music class when we were are asked to identify certain chords and intervals including octaves and fifths. I remember being embarrassed when I mixed up a fifth and an octave. I mean --- I had been tuning my violin(s) myself for at least 5 years. Yes! A fifth sounded as perfect to me as an octave.

I resumed violin playing immediately afterward ---and never stopped. That was 72 years ago. I don't make that mistake these days.

By the way, I've never been sensitive to hearing beats.

November 20, 2019, 3:18 PM · Usually we listen for (lack of) beats when tuning 5ths. It takes some practice to get it right - when the tuning is close, the beats can be very slow.
November 20, 2019, 3:45 PM · Tuning is tricky. If you tune pure fifths (3:2 ratio), then your third between C and E in a string quartet context will be too wide. If you are playing with piano tuned to Equal Temperament, your pure fifths will also be too wide compared to the tempered fifths in ET.

Tuning is always a compromise, you decide which intervals you want pure and which to sacrifice.

Regarding the question of consonance and dissonance, I don't know if you are familiarl with the basic physics of pitch? If you cut a string in a half, you get an octave (2:1), if you have two thirds of a string, you have a pure fifth (3:2), and so on (4:3, pure fourth; 5:4 pure major third; 6:5 pure minor third). Of course it's not always possible to use just pure intervals all the time. Different intervals are also dissonant or consonant. Major thirds and minor thirds were considered dissonant in a lot of medieval music. All depends on context.

Edited: November 20, 2019, 4:13 PM · I apologize in advance if I'm messing up something - but in my world (or should I say instruments/ear/physics combination) the beats get faster = more narrow the closer two strings are tuned to perfect fifths. When the beats cannot be distinguished anymore from each other, then I know I'm almost there.
November 20, 2019, 4:15 PM · So I will start a sentence with a "so" in the hope that Paul will jump in with his knowledge about nature laws and clear this up.
Edited: November 20, 2019, 8:10 PM · "but in my world (or should I say instruments/ear/physics combination) the beats get faster = more narrow the closer two strings are tuned to perfect fifths."

I'm not sure about the math for fifths because I don't know the physical origin of the 'beats' when fifths on the violin are played. I suspect that an overtone of one string is in unison with a different overtone of the other string. (imperfect unison if one hears beats)

If that is the case, then the 'beats' should become less frequent as the two pitches are brought into closer agreement. For unisons, you can prove this to yourself using Excel to graph the sum of two sine waves of very slightly different frequency and make the difference smaller and smaller.

Want to try it? Open MS Excel.

Enter the headings "time," "sum," "I(1)," and "I(2)," and "Delta" in cells A1, B1, C1, D1, and E1 (here and henceforth, without the quotation marks). In the A column, in cells A2 through A1001, enter the numbers 1 through 1000. These will represent time in arbitrary units. In the B column, enter "=C2+D2". In the C column, in Cell C2, enter "=SIN(0.2*A2)", and copy that down to cell C1001. In Cell D2, enter "=sin((0.2+E$2)*A2)", and in Cell E2, enter 0.04. Graph the B column (the signal sum) vs. the A column (time). Your chart should show six beats over 1000 arbitrary time units. Change the Delta value in cell E2 to 0.02 (moving frequencies closer together). The chart will now show three beats over the same 1000 arbitrary time units. The difference between the two frequencies has gone by half, and the so has the frequency of the beats. QED. In fact, for unisons, the frequency of the beats is the difference of the two pitch frequencies. So you should hear beats at 1 Hz if you strike a 440 Hz and a 441 Hz tuning fork at the same time.

I tried modifying my scheme for fifths by changing the formula in cell D2 to "=sin((0.3+E$2)*A2)" (and copying that down to cell D1001) so that the two frequencies would be in close to a 3:2 ratio. You see the same thing as before except the "beats" are much less pronounced. For easier visualization I recommend starting with a Delta (E2) value of 0.01 and then changing it to 0.005. After doing this "experiment" I can see why fifths might be intrinsically harder to tune than unisons.

Want my Excel sheet? Just send me an email (pdeck at vt.edu) and I'll respond with the file.

November 20, 2019, 11:58 PM · Paul, thanks for your efforts. I think what we here is an interference pattern of the two frequencies we're playing. And about the direction it goes - I'll check this in the evening. (Memory often 0lays tricks to us...)
November 21, 2019, 10:53 AM · When we tune perfect fifths , frequency ratio 3/2, slightly out of tune, the beats that we hear are the interference pattern , the difference between the second overtone of the lower string with the first overtone of the upper string. When that beat pattern gets fast enough, more than 16 Hz, we start to hear it as a difference tone, or Tartini tone. I usually hear it as an annoying buzz. This is easily demonstrated by playing a high double-stop third on the A and E strings. The difference between a perfect fifth and a piano fifth is only 2 cents, (2/100 of a half-step), not enough to worry about in practice. Sometimes Cellos and Violists will want to tune the C- string up 6 cents to match the piano C.
November 21, 2019, 11:35 AM · People might be conflating different psychoacoustics phenomena that are all caused by the same process: two tones sounding at the same time.

The perception of "beating" is strongest when the frequency of the two tones are very close to each other. The ear hears a SINGLE tone with a fundamental frequency as the average of the two, which is almost a perfect unison.

But the amplitude of this perceived pitch is dramatically and periodically varying at a rate equal to the difference between the two frequencies. When the difference is around 20hz or more, a distinct beating of the base tone is detected.

When tuning notes that are far apart, such as perfect fifths, this amplitude modulating a perceived frequency effect is not present. What you can perceive with some practice, is two separate tones, i.e., the Tartini Tones. One has a frequency that is the average of the two frequencies. The other is the difference.

When tuning perfect fifths, the tone that is the difference of the two frequencies has a much larger amplitude than the one which is the sum. So what you perceive is a distinct lower frequency tone playing louder than a higher frequency tone, not the "beating" of a single tone.

If you have perfect pitch, you might detect this distinct lower tone as being an octave lower than the low note you are trying to tune. For example, if you are tuning the D string to the A string, you might perceive a pitch an octave lower than the D as you get close to the A.

For most of use mere mortals, we actually use the ability of the ear to detect amplitude variations in time. For perfect fifths that are not exactly 3/2 in frequency apart, the amplitude variation in time appears rather chaotic to the ear over short time intervals. This is "dissonance".

As the ratio of two frequencies approaches integer multiples, 3/2 for perfect fifths, the amplitude variation over time becomes much smoother and more regular with fewer shifting peaks and valleys. This is "consonance".

It is possible to tune a perfect fifth by using one's sense of consonance, but still hear a distinct low frequency tone. Thinking this is "beating", the player becomes confused and tries retuning to eliminate the tone.

One advantage of tuning the violin by playing softly is that it becomes more difficult to pick up the Tartini tones and confuse them for beating due to mistuning.

Edited: November 21, 2019, 12:08 PM · A concrete example - if you had a string at 200Hz and another at 300HZ, you'd listen for the beating of the 600Hz overtones of each.
November 21, 2019, 1:40 PM · Thanks, Carmen, for that elaboration. jq
November 21, 2019, 2:26 PM · Thank you for all your enlightening replies.

Are we saying that beats and Tartini tones are the same thing? Or different but based on the same issue? I ask because I have heard the “beats” for years when tuning but only just recently perceived what I think are Tartini tones. The TTs I heard while playing a scale to a drone, the TT literally popped up in my perception and I could say I “felt” as well as heard the tone. It felt almost like my skull or jaw was vibrating whereas the beats I hear but don’t feel.


Edited: November 21, 2019, 2:41 PM · Two pitches a perfect fifth apart share at least two coincident partials. The first, in order from low to high, is the
3rd partial of the lower pitch and the 2nd partial of the higher pitch. The next is the 6th partial of the lower pitch and the 4th partial of the higher pitch.

For example, take A4 and E5: The first partial in common between those two strings will be the pitch E6, which is the 3rd partial of A4 and the 2nd partial of E5.
When we tune those two strings, we are listening for beating of the E6 partial.

The next highest coincident partial of A4 and E5 would be E7, the 6th partial of A4 and the 4th partial of E5.

There is another partial pair, but I doubt we use it for tuning. I assume we are listening only to the 3:2 and 6:4 partials (unless the string is false, in which case we may not be able to match both partial pairs at the same time).

Gordon's math works out for the lower partial pair. We don't necessarily need to listen to anything more specific than beating--just get whatever beating you hear as slow as you can.

November 21, 2019, 9:52 PM · The effect that gives rise to the two Tartini tones also gives rise to the "beating" one hears when two tones are close to the same frequency.

Gordon and Scott's examples use the "beating" method by listening for the overtones shared by the two strings one is trying to tune.

So the A and E strings would exhibit beating an octave higher than the open E: 1320hz (E6).

The D and A strings would exhibit beating an octave higher than the open
A: 880hz (A5).

The G and D strings would exhibit beating an octave higher than the open D: 587hz (~D5).

Depending on the violin, it may be difficult to hear this effect. But the transition from a dissonant sound to a consonant as one approaches a perfect 5th is very noticeable.

November 22, 2019, 9:21 AM · While we are in a scientific mood, I should like to point out a very common error: Tartini tones are not accelerated beats!

If we play A=440Hz against A=441Hz we get a combined tone of 440.5Hz modulated at 0.5Hz But we hear a pulsing beat of 1Hz because the ear responds to pressure gradients more than pressure amplitudes.

Tartini (Combination) tones:
These are a form of distortion in the ear itself due to non its non-linear response: they are not in the surrounding room! They also occur in microphones, amplifiers and speakers, more so in poor quality equipment.

November 23, 2019, 12:02 PM · Adrian, are you sure? There are so many phenomenon like additive tones, subtractive tones, and various other combinations which can result in many volume pulses in between (which can be perceived as tones), that I am questioning that.
November 23, 2019, 5:03 PM · David, of course in double stops, beats and difference-tones will occur simultaneously,but the beats will disappear when the difference-tones are in tune!

In Wikepedia, the article on "beats" confuses the two; but not the article on "difference-tones", where the maths are somewhat more complicated...

Edited: November 24, 2019, 11:15 AM · Tuning is easy.... right... at least that is what I thought! And how is the base resonnance frequency of the instrument body (which is somewhat different for every instrument) affecting the perception of being in tune? We keep talking about two different strings resonnance and how they interact, but the body has resonnance of its own (not to mention the other 2 strings). Are some instruments easier to tune than others? Is an easier to tune instrument better or worse?
November 24, 2019, 5:13 PM · Okay so here's a question for y'all wave-mechanics physicists. If I want to bring up my viola intervals by 1 Hz, how many beats per second should I hear considering I'm not tuning a unison?
November 24, 2019, 6:25 PM · Paul,
One beat per second equals is one herz. So if your A string is at A=440 and you raise it so you hear a 1 beat per second difference with your 440 source, you've raised the pitch one hz. You would be tuning a unison to a 440 source, wouldn't you?
November 25, 2019, 4:14 AM · Piano tuners have to learn what different numbers of beats sound like when they are learning to tune. When I was in band instrument repair school I was right next to a piano tuning/rebuilding class and all of us students became friends and I was fascinated to learn that they use a special metronome to help them learn the sound of the different numbers of beats for different intervals.

Regarding Paul's question, one way to accomplish that would be to have one tuner playing the reference A440 pitch and have another digital tuner on the viola to show you the actual frequency. Tune so that's showing A441 and you'll learn exactly what 1Hz beats sound like.

Edited: November 25, 2019, 7:28 AM · Oops, I wasn't clear. I was talking about tuning fifths not unisons on my viola (or violin for that matter). Previously in this thread, Joel Quivey wrote, "When we tune perfect fifths, ..., the beats that we hear are the difference between the second overtone of the lower string with the first overtone of the upper string." Those are the overtones at the octave point for the upper string. Shouldn't they "beat" twice as fast as the respective principal, because I'm actually hearing the "beating" of two frequencies close to 880 Hz (e.g., for the A and D string)? So if I'm listening for a 1-Hz increase in the lower string (to tighten the interval slightly), shouldn't I be listening for a 2-Hz beat? That's what I'm trying to figure out. Scott, since you do this professionally (yes?), maybe you can tell me by how many Hz a fifth in the middle of the piano keyboard should be tightened (say, Middle C and the G above it), according to strict mathematical equal temperament (I guess I can calculate that easily enough), and how many "beats" you would listen for to tune that interval.
November 25, 2019, 9:29 AM · Paul, beats at 880 Hz are probably too fast to be perceived as "beats", but more as tones.

For anyone who has a decent sound recording and editing program, you can generate a combination of any two tones, record them, and then expand the waveform such that peaks and valleys in sound volume are easily viewable, and you can even count how many there are per second.
Depending on the frequency, these volume spikes and valleys can be perceived either as "beats", or as "tones".

This may be a more "caveman" approach than used by the latest mathemagicians, but direct viewing of sound pressure levels may also be more valuable than theory.

Theory sometimes is modified to conform better to observations, and observations are also sometimes modified to take better advantage of theory.

Edited: November 25, 2019, 12:16 PM · No no no David. Okay ... let's go back to unisons. Let's say we have two tuning forks, one at 440 Hz and one at 441 Hz. We will hear "beats" at 1 Hz. But suppose we could magically listen to just the first overtone instead of the principals? Then we'd be hearing 880 Hz and 882 Hz. That would be a 2 Hz beat. When we listen to fifths, the principals are too far apart to hear actual beating, but as Joel says, we can hear the first overtone of one beating against the second overtone of the second. So I'm envisioning that if one wants the interval to be 1 Hz wide, one needs to hear a 2 Hz beat.

"For anyone who has a decent sound recording and editing program, you can generate a combination of any two tones, record them, and then expand the waveform such that peaks and valleys in sound volume are easily viewable, and you can even count how many there are per second."

Yes this can be done in MS Excel but I was hoping someone already knew the answer. Like Scott.

Edited: November 26, 2019, 2:32 AM · Yup, lots of different ways to choose intonation, whether from an Excel program, or via listening.

I am prepared to mess with either one (and more), if anyone claims that there is only one right way. ;-)

November 25, 2019, 6:51 PM · I am Not an expert, but, I do remember an electronics experiment in school. Two sine wave signals, at slightly different frequencies, sent into an oscilloscope showed a combined sine wave that varied much slower in amplitutde, there was an overall "envelope" and time between the amplitude peaks would be the difference, heard as "beats"
November 25, 2019, 8:20 PM · Joel, yeah that's how it works and it's easily simulated in Excel.
November 26, 2019, 2:31 AM · "When we listen to fifths, the principals are too far apart to hear actual beating, but as Joel says, we can hear the first overtone of one beating against the second overtone of the second. So I'm envisioning that if one wants the interval to be 1 Hz wide, one needs to hear a 2 Hz beat."

That would be my speculation as well.

Edited: November 26, 2019, 5:41 AM · 880Hz with 882Hz give a tone of 881hz: (880 plus 882) divided by 2
and an amplitude modulation 1Hz: (881 minus 880) divided by 2
So why do we hear a beat of 2Hz?
My physics teacher (50 years ago..) explained that the pulsing we perceive is not the peak of the modulation but the gradients each side of the peak: i.e. change of pressure rather than actual pressure peak.

Tartini (combination) tones are a completely different mechanism.

Many books and articles a mistaken about this.

November 26, 2019, 6:31 AM · Roger, the natural modes of vibration of the violin will vibrate at the frequencies of the notes being played.

The further the frequency of the note is from the natural frequency of the mode, then the less power that mode contributes to the overall sound.

When first learning about natural modes of vibration, people tend to think that each mode is stuck vibrating at its natural frequency. But that is not what happens.

November 26, 2019, 6:49 AM · When two pure tones are sounded together, call them f1 and f2, it *always* creates an acoustic effect that is the same as one tone sounding with a frequency of (f1+f2)/2, and another tone sounding with a frequency of (f1-f2)/2.

This has nothing to do with how close or far apart the frequencies of the tones are, or if they are fifths, octaves or unisons.

What does change is how the ear perceives the effects. When f1 and f2 are very close, the ear cannot perceive the difference tone, (f1-f2)/2, because the frequency is too small. However, the way the tones combine causes the sum tone, (f1+f2)/2, to have its amplitude pulse high then low with a frequency equal to f1-f2.

So one hears a single tone, (f1+f2)/2, that appears to pulse or "beat" at a frequency of f1-f2.

When the frequencies are some distance apart, the pulsing phenomenon no longer occurs. Your ear does not perceive beating of the amplitude of a single tone. The combined wave forms appear chaotic and thus the perception of dissonance, until their frequencies get closer to integer multiples of each other. Then the combined waves appear very regular again and sound consonant.

For violin strings, we do not have pure tones, but rather a combination of overtones that each are close to pure (i.e., perfect sine waves).

For perfect fifths, the octave of the higher note and the 2nd overtone of the lower note are both close to the same frequency, but with greatly diminished amplitude. So perception of the "beating" effect can be challenging.

November 26, 2019, 10:36 AM · ... wow ...
Edited: November 26, 2019, 2:59 PM · Carmen wrote:
"When the frequencies are some distance apart, the pulsing phenomenon no longer occurs."

I believe it does, from having looked at combined waveforms of many various frequency combinations. While the human hearing system may not consciously "hear" these in the sense that they can be described from hearing alone, I believe such things can highly contribute to overall impression of sound. Many things which can contribute to an impression cannot be consciously nailed-down or defined by a listener.

November 26, 2019, 4:28 PM · There I agree with David: reality is always richer than pure maths!
Edited: November 26, 2019, 6:52 PM · This takes me back to my youth, when I studied both music and radio. The music part was the cornet. Perhaps the beating phenomenon is easier to detect with wind instruments, but I had no trouble hearing the beat frequency when tuning to another instrument. I could easily hear a warbling that would slow down to a fraction of a cycle per second; when it stopped completely I was perfectly tuned.

Now that I'm playing violin, I first use this technique to tune my A string to a tuning fork. The beating is more subtle, but it's there. Once I've tuned the A string, I tune an adjacent string to a perfect fifth. As others have pointed out, the second harmonic of the higher string beats against the third harmonic of the lower one. The effect is even more subtle, but noticeable if you listen carefully.

Yes, you could get a beat at twice the frequency if you compared the fourth harmonic of the higher string with the sixth harmonic of the lower one. But the higher the partial, the less the amplitude; the 6/4 beat will likely be drowned out by the 3/2 beat (not to mention the fundamentals).

Now, getting back to the radio theory I mentioned... One of the basics is that when you mix two signals of different frequencies, you get two additional signals: one is at the sum of the original frequencies, and one is at the difference. This process is known as heterodyning, and is used in many places - including just about any radio you might have - to change one frequency to another. Remember those old-time radio sounds with squeals going up and down? That's various radio frequencies heterodyning up and down as you turn the dial. If you're careful you can adjust the dial so a squeal (created as the difference of two frequencies) goes down into the sub-audible range, and you might hear a beating that slows to a stop as you match the frequencies exactly. It's exactly the same concept that we use when tuning a musical instrument - the difference in the frequencies of the two instruments goes to zero when you're in tune.

November 26, 2019, 7:44 PM · "There I agree with David: reality is always richer than pure maths!"

And feelings generally are more important to people than facts. Sure, I get it. But I read somewhere that violas and cellos in orchestras should squeeze their fifths by a certain amount, and if I'm targeting that amount, then I want to know what I should listen for.

November 27, 2019, 3:33 AM · Paul, I think you'd need to ask someone in the particular ensemble whose tuning you wish to match.

By the way, the interference phenomena I mentioned are real, since they are actual air pressure deviations which can be picked up with a microphone, displayed graphically in a sound file, and reproduced through a playback system. It's just that the human ear/brain system isn't as overtly aware of some as it is of others, so some might be described as more of a feeling or an impression.

November 27, 2019, 6:14 AM · Squeezing the fifths is what happens in Equal Temperament!
In the middle of the keyboard, they will beat at around 1Hz which is acceptable. And the thirds & sixths will be a little less acidic.

Squeezing them even more happens in various meantone tunings to get a few really sweet, near-pure thirds.

If we want C & E pure, we must squeeze the four included fifths rather a lot. Or maybe just one or two of them....

November 27, 2019, 8:35 AM · We might be moving from the challenge of tuning tones in Just Intonation to perceptions of polyphonic sound and confusing different psychoacoustic effects.

If one stares at time traces of multiple notes being played at the same time and compresses the time scale, various wave-like amplitude variations can be seen. They all affect the perception of the sound, but are not all caused by the same phenomenon nor are they even perceived the same way although they look similar.

Here is an easy experiment one can perform on a violin to get a sense of dissonance versus Tartini tones versus pulsing. It involves playing a sixth interval by using the 1 finger on the A string to play a B while also bowing the open D string.

Start with the 1 finger close to the nut so the sixth will be too narrow for a Just Intonation sixth. The chord will sound harsh but no pulsing can be detected. It sounds more like annoying buzzing. This is classic dissonance.

Move the finger up the A string. As you approach a Just Sixth, the dissonance will melt into two very distinct tones, one slightly quieter than the other. These are the Tartini Tones.

As you slowly stretch the sixth, you MIGHT be able to hear a low frequency pulsing effect. The variation in the amplitude of one of the Tartini tones is subtle can be a challenge to hear, but is distinct from the sound of too narrow a sixth.

I find it easier to approach a Just Sixth from a wide position because the transition to the "pure" Tartini tones is more rapid than approaching from too narrow an interval. Both the pulsing phenomenon and the dissonance phenomenon disappear more rapidly from a wide approach.

As I continue to stretch the sixth to approach an octave, the enveloping variation becomes more pronounced and eventually the distinct, low frequency pulse or beating becomes very evident to the ear, especially as one gets close to a perfect octave.

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