# In a musical note (A for an example) are all the other frequencies harmonic?

In a musical note (A for an example) are all the other frequencies harmonic (integer multiples)? Is there any inharmonic frequencies in A?

Edit: A4 on the piano for example.

Im very grateful for all the answers I get. Thanks so much!

• Which A? From what I've read, timpani A's have non-integer multiples of the loudest frequency in their overtone series. I've also read that the fundamental frequency of a timpani note is not the loudest. Feb 22 '18 at 18:57

are there any inharmonic frequencies in A?

Simplistically speaking, 'A' tells us the fundamental pitch of the note (or at least it would if we knew which A - e.g. A4 is often, though not always, considered to be 440Hz).

However, whether or not there are any inharmonic frequencies completely depends on the timbre of the note. Looked at another way, it's (partly) the frequencies of the harmonics that define the timbre of the sound.

In a the sound of any 'real' instrument, there will almost certainly be frequencies that are not integer multiples of the fundamental. Some of these might be nearly integer multiples (for example, the 'stretched' partials you find on a string); some of them might not be close to integer multiples (for example, the partials in some bell sounds). There will also often be lots of energy in the sound that does not come from an identifiable partial of stable pitch, but is often described as noise. However, the boundary between noise and partials isn't really a clear one, as partials often have unstable frequencies and may be short-lived.

So, when we're talking about 'real' or acoustic instruments, we can almost certainly say NO, not all the other frequencies are harmonic. However, using a computer or synthesizer, it's possible to get close to a sound that has only integer harmonics.

• in an acoustics class once the professor took a note (from a synth if i remember correctly) and slowly removed the harmonics. once certain number of harmonics were removed we could no longer tell what instrument was being played. even removing attack and changing decay didn't have as much effect on being able to identify the instrument.
– b3ko
Feb 22 '18 at 20:08
• @b3ko yes, it's an interesting experiment - to do this kind of experiment yourself, a piece of software like Spear that deals in terms of partials can be useful. Feb 22 '18 at 22:15
• "enharmonic" should be "inharmonic" (I don't have the rep to make trivial edits). Feb 23 '18 at 14:38
• I'm surprised removing the attack made so little difference. Remember 'hybrid' synthesis - notably the Roland D50 - where a sampled attack was grafted onto a synthesised sustain and release? It was remarkably effective. Feb 23 '18 at 15:08
• Laurence Payne it was many years ago but if I remember correctly if he altered the attack we could still guess what instrument it was. Altering the harmonics made it next to impossible to tell what instrument it started as.
– b3ko
Feb 23 '18 at 19:52

Depends what's playing it. The harmonic series - the 2,3,4 etc. frequency ratios - are the overtones of a theoretical 'perfect' instrument. That's a simple string, of negligible mass, vibrating as a whole, as two halves, as three thirds etc.

In real life, instruments are not perfect. They produce overtones that are displaced from those theoretical frequencies, and not of uniform strength. Sometimes, like the flute, they're pretty close. Sometimes they are wildly displaced - the classic example is the church bell which can have a very strong overtone a major 7th above the fundamental.

Then there's the matter of the attack transient, the very beginning of each note when the bow bites into the string, the piano hammer strikes the string, the clapper hits the bell, a brass instrument note is 'tongued'... That's generally completely inharmonic. And it contains much of the characteristic of the note, the information as to WHAT instrument it is.

So yes, in most cases a note will contain plenty of 'imperfect' overtones.

If you select one (of the many) A, it is defined by one single frequency. One example might be the A having frequency 440Hz. If you dial in 440 Hz on a frequency generator, on setting sinus (it should be sine, see comments) wave, it will have only that frequency and sound an A.

But I guess, you are asking about A-s sound from other sources, say from instruments. There are a lot of different instruments which sound quite different. Some of these have mostly integer multiplier frequencys, ie 2x 3x 4x and so of the base frequency. In the example it would be 440Hz, 880Hz, and so on.

The actual relative strengths of these overtones in part define the sound of the instrument. A clarinet will sound different from a flute, partly because the volume of the different overtunes will differ. Here we are talking about the "overtones".

But it never is quite as easy. All instruments I have ever seen in addition creates various noises, one example is wind noise. This is not related to overtones.

• Sinus waves get up my nose...
– Tim
Feb 22 '18 at 19:21
• Tim: Sorry -- English not beeing my main language, can you decode the comment for me. Feb 22 '18 at 19:38
• A sinus is a space in one's head, etc. The word you needed was sine. Just a joke...
– Tim
Feb 22 '18 at 20:29
• Aah. Point taken on the Sinus vs Sine. Feb 23 '18 at 5:44

In addition to the other answers, two further reasons for non-integer multiples:

• A piano often has 3 strings which are hit on each note. They may not be perfectly in tune with each other, in which case it cannot be the case that each of them vibrates at the same basic frequency.
• The low notes of a piano in particular are built from coils of wire, which will provide a slightly imperfect vibration (i.e. even ignoring harmonics you may not see a sinusoidal vibration because the force on the string is not exactly proportionate to its displacement) - for instance the spring itself may vibrate with an additional transverse low frequency vibration changing the tension in the spring over time causing a slight change in pitch at e.g. 10Hz which would be heard as a small amount of vibrato. Inevitably that means you are not going to get a harmonics which are constantly at frequencies integer multiples of a theoretical base note.

Assuming you are asking about the mathematical relationship between pure sine wave tones corresponding to the fundamental frequency of notes on a piano, the relationship between the notes is close to but not exactly an integer multiple.

Starting with an A4 sine wave at 440 hz, we can get to A5 (880 hz) by doubling the frequency. We can get to A3 (220 hz) by cutting the frequency in half. Continuing to double the frequency goes to higher A's, and dividing the frequency goes to lower A's. This works for any note--doubling or halving the frequency takes you to the next octave of that note.

Going back to A4, if we multiply the frequency by 3, we get a note that sounds good when played in a chord with A4. If you tried to find it on a piano, you would get E6 (E5 is the first E above A4, E6 is the next one). Ignoring the extra octave for a moment, the interval between A and E is called a fifth, so multiplying frequencies by 3 is the origin of our concept of fifths.

Continuing, you could multiply THAT frequency by 3 to go from E6 to B7. Multiply by 3 again and you get a really high F#. Then a C#, G#, D#, ... This continues to cycle through the notes on the piano. (And remember that we can always divide by two to get lower octaves of the note, so I am ignoring the fact that we keep going to higher notes as we multiply.)

Now wait a minute. If we keep cycling through the notes on the piano, we are eventually going to get back to A. From the second paragraph above, every A is just 440 hz multiplied by some number of 2's, but somehow we cycled back to A by multiplying 440 hz by some number of 3's. How can `2*2*2*2... = 3*3*3*3...`?

It doesn't. When you get back to A, you've multiplied the frequency by `3^12`, which is 531441. This is pretty close to going up 19 octaves from the original note, which would mean multiplying the frequency by `2^19`, which is 524288. The difference between these two tones is about 1.36%. Given that the frequency difference between half steps on a piano is about 5.95%, this is very badly out of tune.

If the fact that these frequencies are well outside the range of hearing bothers you, imagine instead that you start at a note and go down an octave. Then go up an octave-plus-a-fifth. Keep alternating. Whenever you get more than an octave above your original note, go down an extra octave. Keep doing this until you get back to the original note. You will have gone up an octave-plus-a-fifth 12 times, and down an octave 19 times. If you had actually been multiplying the frequency by 3 going up, you would now be at a frequency `(531441 / 524288)` times your original note's frequency.

Modern instruments for western music typically average out this error smoothly across the scale, which is called equal temperament. This results in a fifth that is `(2.996614 / 2)` times the original frequency rather than `(3 / 2)` times. Many different temperaments exist that balance getting a fifth that is closer to a multiple of 3 with having an awkward part of the scale that the music generally has to avoid.