What happens when two instruments that are out of pitch play the same note?

Answer 1

This depends on the instruments and the difference between their pitches.

In its simplest form, two simple tones sounded together are heard in the ear as their individual pitches, plus the sum and difference tones, which are the mathematical result of adding and differencing their respective frequencies.

This may not sound terribly simple. Frequency (measured in Hz, which stands for "cycles per second") is the characteristic of periodic sounds that we interpret as pitch. When we tune instruments in an orchestra, for instance, we use a standard pitch, A = 440Hz. When the Oboe player produces this note for the orchestra to tune to, she plays the A on her oboe, and that sound primarily is made up of a sine wave (which is the simplest of motions) which repeats 440 times a second. The Concertmaster tunes her A string against this tone from the oboe, until they match exactly.

To tune the note, the Concert Master changes the tension on her string until the "beats" go away. The beats are, in fact, the sum and difference tones I mentioned up there in that stuffy definition. If, for instance, the Concert Master's A string is flat, say 435Hz, her ear perceives both the 435Hz and 440Hz tones, but also she hears the sum and difference tones, which are both 5Hz. This, she hears as a volume change in an overall 440Hz-ish pitch. As she tightens her string and draws it up to 440Hz, to match the Oboe note, the speed of the beats decreases, until the sound perceived by the violinist smooths to an un-wavering A.

There is a range in which pitch difference is perceived only by the ear as beats. Outside of this range, the difference tones begin to be perceived as actual notes, themselves. Within the range, there is a smaller range where the beats are perceived as vibrato, wavering, "wet sound" (as opposed to the dry sound of a single pitch) and many other names. Generally, this is perceived by non-musicians (and many untrained musicians) as adding "warmth" or interest to the sound. In the range between warmth and separate pitches, the beat is perceived as "uncomfortable", "noise" or even an inexplicable harshness.

Now, a side-bar, if I may. No 'natural' instrument (which is a term applied to all instruments made by human hands, operated by fingers, lips and breath, fingers bows and strings, etc) produces a simple sine wave. In fact, most of them produce many sine waves, which are added by the ear to make the particularly recognizable "timbre" which identifies, for the listener, the difference between Oboe and Violin. (There are other mechanisms, including how the attack is formed, how the sound decays, etc. ) When two instruments of the same kind play together, out-of-tune, all of these sines will interfere to produce beats.

Having said this, let's go back to the difference between the instruments playing out-of-pitch. If the difference is small, the result is a beat, perceived as a steady warble, usually slowly varying. If they are a bit more out of tune, the warble becomes uncomfortable. If they are very far apart, the difference and sum tones can even become audible notes, and this is where things get interesting. If, for instance, the difference and sum tones are exact integer multiples of the difference between the frequencies of the original tones, they are perceived as being related to the two individual pitches of the instruments, and if they are also integer multiples (or divisions) of the lower tone, they are perceived as chord-tones!

For this reason, when a barbarshop quartet warms up, they will often sing chord tones, starting with the bass and lead, who sing in octaves. Octaves are in a ration of 2:1, the higher pitch to the lower, and the difference tone is therefore the same as the lower note. (i.e., lower tone = A220, upper is A440, the difference is A220). Then, the baritone will insert a note between them at the fifth. A properly tuned fifth has a frequency which is in a ratio of 3/2 to the note below it, in our case, it will be E330. (220 *3 = 660, /2 = 330.) The difference tone between the bass and baritone (330Hz - 220Hz) is 110Hz, and can, if they are all in tune, be heard as the A an octave below the bass: A110. (220/2=110. It is an A because, as 440 is twice 220, 110 is one-half 220: octaves are relations of x2 or 1/2.) Between the baritone and lead, the difference tone is also 110Hz, so that lower A110 is reinforced.

Now the fun begins. The sum tones between the bass and lead (which we neglected before) is 660hz, which is twice 330hz. The first overtone of the baritone is also 660hz, so it is easier for the baritone to tune: he listens to the note an octave higher than the lead, and tunes until the beats are gone from both the A110 and the E660. What happens now? The sub-bass note A110 interacts with the baritone note. The difference tone is 660-110 = 550, and the sum is 770. First, 550.

To understand 550, we have to consider the harmonic structure: each harmonic (or overtone) is an integer multiple of the fundamental tone (the first harmonic) where the integer names the harmonic (and, because the "fundamental" isn't really counted in overtones, the #-1 overtone.) Our fundamental pitch is A110, and all the other pitches we're finding are integer multiples of that frequency. So far, we have identified A110, A220, E330, A440, E660 and we could pretty much expect to see A880 and E1320.

550 comes between A440 and E660, our fundamental pitch (up two octaves) and the fifth above it (up an octave) and it is another harmonic pitch. In fact, we can identify the harmonics as musical pitches:

A110 1st harmonic fundamental

A220 2nd harmonic octave above fundamental

A330 3rd Octave and fifth

A440 4th second octave

A550 5th two octaves and a Major third

A660 6th two octaves and a fifth

A770 7th two octaves and a very special 7th

A880 8th third octave.

The sequence goes on, and eventually reaches the point where the pitches can't be heard by the human ear. (The harmonics also get quieter as the pitch goes up, part of the timbre of the sound.)

So A550 is the Major third, placed just above the lead...and this is where the tenor sings to complete the chord! Once the tenor has been added, the sum and difference tones between his note and the other notes being sung _and_ being produced in the ear, are all multiples of 110Hz, so they are all harmonic sounds: this is why a barbarshop quartet which is in tune sounds like more than just four guys!

So. If two instruments are slightly out-of-pitch, you will hear beats. If they are more-than-slightly out-of-pitch, the beating will be annoying, and if they are far enough apart (even a semitone!) you may perceive additional notes as well as the two instruments' notes!