Mathematical Music Theory 3: Combination Tones

Today we’ll cover one of the most dangerously overlooked consequences of what we’ve been talking about. I say it is overlooked because most people can probably get a degree in music composition without this ever once being mentioned. It should be obvious by the end of the post why it is dangerous for anyone writing music to be unaware of the phenomenon.

Suppose you play two notes at the same time. These are two sound waves, and (in the real world) it is impossible to keep them completely separate so that all you hear are the two sounds. The waves combine. Hence you get a new wave which is some other tone called the combination tone. For this reason they are sometimes called sum tones, or more confusingly difference tones (because the frequency of the tone is the difference of the frequencies).

As before, it is important to note that combination tones are a physical reality. I think that sometimes (when this is taught at all) people write them off as a psychological phenomenon. Maybe it is just your brain filling in something it thinks it should hear. In order to convince you this is not the case, take a look at this video:

He isn’t playing any of those low notes, yet they are the dominant sound. As you can see, with enough patience (and the knowledge I’ll give you below) you can work out how to play melodies entirely using the combination tones.

In terms of the overtone series there is a nice easy way to figure out what the combination tone will be. For example, take a major third by playing C and E at the same time. From the first post we see that the interval occurs as the fourth and fifth tones of the overtone series. All we do is subtract 5-4=1 and find that the combination tone is the fundamental of the series. Thus any two notes that occur sequentially in the overtone series will have a combination tone of the fundamental of the series.

If you take G and E, these are the third and fifth tones and hence the combination tone is the (5 minus 3) second tone in the overtone series and so on. It is quite easy. I suggest that anyone that wants to be a good composer take a simple two voice line in whole notes and work out what all the combination tones are and see if this alters what you thought you wanted.

It is scary that people can be out there composing and entirely unaware of this phenomenon. Think about the danger. You think you are writing certain sounds, but other tones are coming into the writing totally unbeknownst to you. It is even worse than that. Because of the overtone series, you actually get second order, third order, etc combination tones and not just this first order phenomenon.

Here is an example of why this might be important. There are certain intervals that feel stable and others that have tension. Here is a good way to tell which is which. Take the interval of a fifth (a C with a G over it). The combination tone is the same as the bottom note, so the combination tone anchors you to that bottom note and everything feels stable. The interval of a fourth is just the inversion of that (G with a C over it … the same two notes!!), so the combination tone doubles the higher note and you just have this floating middle note which makes it feel less stable.

Composers such as Bach were intimately familiar with this phenomenon. Rather than have it do unexpected things to his compositions, he used it to his advantage. When he wrote two part inventions there would only be two melodies on top of each other, but due to combination tones it sounded much more fleshed out as if many more parts were being played. He would know that in parts where he wanted forward motion he would use unstable forms of intervals and where he wanted resolution he would use the stable forms.

This may seem like some tiny unimportant detail, but it really makes a big difference in how you voice your chords, and takes quite a bit of time and effort to internalize so that you can start to use it effectively.

Author: hilbertthm90

I am a mathematics graduate student fascinated in how all my interests fit together.

4 thoughts on “Mathematical Music Theory 3: Combination Tones”

1. Interesting but not that musical in this example.

2. Wait a minute, please! Most people are iwlling to hear music. Some of them then are willing to play it. And out of these players, I thought some are willing to write it. It probably has to do with one’s mathematical background that one usually first writes math and then start to play with it just as one can first read Goedel, Escher, Bach and then learn about music! Thanks, but I didn’t quite get the stuff although I do like to solve the wave equation and get the eigenvalues right. Maybe, I should go back to the 3 parts later.

3. In electronics school, we were taught that sum and difference tones were only created if the original tones were combined in a non-linear device. Otherwise, you would get the simple algebraic sum. I can demonstrate intermodulation distortion vs harmonic distortion in a circuit. Perhaps the sum and difference tones are an acoustic artifact of resonant room surfaces, or of non-linearities in the ear itself. But in an ideal free space, I don’t think they’d happen. I’m no mathematician, so can you explain it better?

4. There are tons of reasons why they don’t happen in real life situations. I was going to point this out if I continued with another post. For example, everything has to be perfectly in tune which is usually not the case (even pianos have three strings per note which are intentionally the tiniest bit out of tune partially for this purpose). Also, the tone has to be a “straight tone” meaning the instrumentalist can’t use vibrato or waver the tone at all. So as you can see any real life instrumentalist will probably not produce audible combination tones.

The main purpose of understanding combination tones is that it provides a theoretical foundation for things our ear naturally tells us such as the major third feels very concrete and stable yet the minor sixth (i.e. the same interval inverted) feels less stable.

I don’t know this for sure, but I think that in “ideal free space” they should still be produced (even if it is much less pronounced than the two tones separately). This is because mathematically the combination tone is really just coming from the two sound waves combining. I feel like lots of imperfections such as resonant room surfaces may dampen the effect, but it should be noticeable in an ideal situation where nothing prevents the combination from happening.