Guitar Harmonics – Equivalent Notes
I thought I would resolve for myself some aspects of the harmonics produced by a vibrating string, and so their possible use in harmonic and melodic structures in music.
First, a brief physics lesson.
When a string vibrates under tension it produces a tone that is a composite of a fundamental and its harmonic series – number of secondary waveforms whose relative amplitude and frequency to the fundamental defines the timbre of the sound heard. This can be affected by many factors – string material (e.g., nylon, steel), thickness, tension, density, guitar body shape, material and design etc.
See the article above for an overview of these basic topics for this diagram to understand how the harmonic series is generated for any standing wave fundamental – a string or an air column:
The above vibrations – increasing in whole number integers as the fundamental is divided corresponds to particular notes in the diatonic major scale. This relationship and the actual musical notes are what I want to identify in this article.
The first harmonic is the easiest to understand. The open string “fundamental” note length is halved so the string now vibrates at half the original length, so at twice the frequency. It is an octave above the fundamental, so the same note but twice the pitch.
In guitar related terms, for example, the low A string tuned to concert pitch vibrates at 110Hz, so the first harmonic at the 12th fret of the A string would vibrate at twice this – 220Hz.
The Hz values for Concert A are helpful to remember as they double at easy to remember numbers that don’t have fractions of a Hertz in them.
The lowest E note on a guitar at concert pitch is 82.41Hz shown against the piano below:From this diagram, you can see a mathematical relationship between actual frequencies in Hz and the note value – C, D, E etc. – also, and it’s important to understand that the human ear perceives the relationships of frequency and relative pitch as diatonic major scale notes naturally (well, Western culture at least). This may be due to natural human psychoacoustics or cultural evolution.
For example, taking the note A4 at 440Hz, then looking at the value of the next E4 – a musical 1/5th -above it – it is seen that the value of E is almost exactly 660Hz which is 440Hz + 220Hz or a continuation of a lower note value (A2) added to A4 to get E4.
The frequency numbers will never multiply and correlate exactly as the human ear perception is logarithmic and the diatonic scale is based on an octave divided by 12 semitones, so a semitone value is not a whole number as explained above – 2 to the power of 1/12th.
It is enough to see that the main musically harmonic notes of the diatonic scale – 1st, 3rd, 5th etc. – have a strong multiple relationship to a fundamental note, wherever you start from. This is a harmonic series.
Looking at a vibrating string again:
- If the open string is A2 at 110Hz, then the
- The first harmonic is A2 x 2 = A3 at 220Hz. Guitar harmonic at fret 12
- This makes the 2nd harmonic a value of about 110 x 3 = 330Hz or close to 329.6Hz = E3 = musical 1/5th. Guitar harmonic at fret 7
- This makes the 3rd harmonic a value of about 110 x 4 = 440Hz or A4. Guitar harmonic at fret 5
- This makes the 4th harmonic a value of about 110 x 5 = 550Hz or C#5 = musical major 1/3rd. Guitar harmonic at fret 4
- This makes the 5th harmonic a value of about 110 x 6 = 660Hz or E5. = musical 1/5th. Guitar harmonic near fret 3
- This makes the 6th harmonic a value of about 110 x 7 = 770Hz. Closest to G5 or dominant 1/7th.
It gets difficult to ring a harmonic on a guitar string here, without amp distortion to help.
The ones that are heard more than others does not mean that some are not present – just more difficult to sound – as shown in the video below:
You can see that predominately, the harmonics in the video example follow the diatonic major scale for A major (for the A string) and so any open vibrating string.
Starting at the harmonic at the 4th fret the Major 1, 3, 5, 1, 2, 3.
On the guitar the home key is E minor so you need to know what note values each harmonic has on each string and choose them accordingly if you are to use them in a musical piece because of that predominantly “major” harmonic relationship.
As a basic example of this, you can play the D, G and B strings as a Gmajor triad, then play their harmonics at the 12th fret, to get another Gmajor, then play them at the 7th fret to get a D major triad, and again at the fifth fret to get a Gmajor triad again. Ok so far.
The problem now comes at the 4th fret harmonics as tabled above, as they produce a major 3rd of G which is a B major triad. This is not in the key of diatonic G major, as the 3rd of G major is a B minor chord.
The harmonics at the 4th fret though, are the same notes as those at fret 4, so you can play a B major chord and chime harmonics at fret 4, on the D, G and B strings to fit.
It is now a question of exploring and documenting for yourself which harmonic notes sound over each fret for a particular string, but if, like me, you don’t think in notes names through having no formal musical training, then you can make a note chart of harmonics to fretted notes, and memorise them, knowing that you would have to jump around string harmonics depending on major or minor key, or just learn specific harmonics parrot fashion that will work for a particular chord or solo.
An obvious example for guitar would be for a 12 bar blues jam in A major. This enables the first to fourth harmonics to be used over the A, D and E strings for each key.
Other ways to use harmonics to great effect have been developed such as Lenny Breau’s style, where an open note is followed by a harmonic rung on the next note of a particular chosen scale.
Eric Johnson shows some great examples of harmonics on his instruction videos:
Another is the classical method where the harmonic is played an octave over the same fretted note or chord shape using the thumb or pick to pick the string while a finger tip touches the string above the correct fret.
There are whole chord sweep styles using the edge of the palm to sweep across the fretted chord approximating its shape also. These require practice to effect well.
These harmonics may only work well for particular keys and for certain scales for obvious reasons – mostly whether they contain open strings note values or not so a note rings on nicely while the next is played.
Certain FX combinations may work well also in highlighting the generally lower volume of some harmonics, such as compression, distortion, and chorus FX.