The harmonic series

The key to it all

Tone is the be all and end all of guitar making, yet few seem to realize how it works. Over 100 years ago Herman Helmholtz published a book that explained it. His work is often quoted and referred to, yet seldom read by guitarists or guitar makers.  “On the sensations of tone” was initially published in 1885. In this long but quite readable tome, Mr. Helmholtz lays out the foundation for our understanding of what constitutes “tone”.  In a nutshell, this is it: The Harmonic series.

For guitarists, I believe the easiest way to envision the harmonic series is to think of a vibrating string, which is the source of all the notes on a guitar. We think of a vibrating string as having one note, say open E or C on the first fret of the the second string. This identifying label is actually just the first in a series of notes that the vibrating string generates. That series of notes is the Harmonic Series. It is generated as the string vibrates in successively smaller sections. The fractional component of the string is the amount the frequency increase by. 1/3 the vibrating area = 3 * the frequency.

This is easily explained if we remember that every note is actually a specific pitch, which is identified in Hertz. A Hertz is the number of cycles per second. So A2 is 220Hz or cycles per second. A3, the octave above A2, is double the cycles per second or 440Hz. The harmonic  series is just the number of cycles per second multiplied by the positive whole numbers (Integers) in order. So in the case of A, we have:

Notice that every time the frequency doubles, an octave over the initial frequency occurs. So every even number ends up being an octave of some previous note.. Every odd multiplier is a new note and double that is it’s octave. So harmonics 1,2,4 and 8 are all the same note in different octaves, as are 3,6 and 12, as are 5,10 and 20.

You may also notice that if you look at the first 8 harmonics, they form a dominant 7th chord. So if the instrument in question radiates each harmonic efficiently, what you hear as the note is actually a chord. Our minds do little tricks on us and we don’t actually hear a chord, but what we do hear is a tone quality, made up of these frequencies. When we hear a real chord, i.e. all of those “notes” played together, we are really hearing each note and some number of each note’s harmonic series all blended together. In the same way that a chord has a “richer” sound than a single note, a note with more harmonics (up to a point) sounds richer than a note with fewer harmonics. These are primarily because they are consonant intervals, at least up to the 7th harmonic.

The higher up the harmonic series you go, the more you encounter harmonics that clash with the fundamental, musically they are dissonant. Let’s continue beyond 8 and see, I’ll leave out the even harmonics as they are just octaves of previously occurring harmonics:

I don’t think we need to go much further, although you can if you want to. The point is that these are all dissonant intervals, and some of them we can’t even identify in terms of the equal temperament scale. The 11th and 13th harmonics are virtually equidistant from the two notes shown for each. When these harmonics are heard they clash with the fundamental and a discernible “beat” can be heard when the single note is played. This is very disconcerting and can make it difficult to even tune the string.

It should be noted in passing that how and where the musician strikes a string has a great deal to do with what harmonics are induced or emphasized. The extent to which the musician makes use of the techniques required to do this has a lot to do with how successful he/she may be in expressing the feel of the music. 

Conclusion:

Every note on the guitar is generated by the string and amplified by the sound box. A string is capable of generating a number of harmonics, the first six being strongly consonant and harmonics above 7 often being dissonant.