Harmonics have been a part of traditional cello technique for more than 200 years, with some of the earliest string treatises dedicating attention to them. Since the last quarter of the 20th century, however, the use of harmonics has expanded enormously, so much so that entire compositions are written with harmonics as the central organizing principle. They provide composers with a multitude of possibilities in timbre and texture.
How do harmonics work?
Harmonics occur when a string is touched lightly in a precise location (called a “node”), such that a standing wave is induced. The nodes are located at whole-number fractions of the string length:
Dividing the string in this way has a multiplying effect on the frequency, and yields the following pitches, known as the harmonic series:
Harmonic Series of the C-string
This shows the first twelve harmonics, including the fundamental pitch. Theoretically the series continues ad infinitum, but in practice we are rarely asked to play beyond the 8th harmonic. Here’s a video showing the first 8:
It is important to know the pattern of pitches in the harmonic series, at least through the 8th harmonic, because with that knowledge you can play any pitch using either open or stopped harmonics, as you will learn in the lessons that follow. This table shows that pattern:
As the first diagram above shows, and as this table reinforces, you can activate these harmonics by touching the nodes at either end of the string. That is, you can either find them where they would be if you fully stopped them (up in thumb position) OR at the symmetrical place on the string, moving closer and closer to the nut. The odd-numbered nodes, particularly the 5th harmonic, can be found multiple places on the string. In performance you will sometimes make these choices yourself, and in other cases composers will specifically indicate the desired node in the notation.
Open and Stopped Harmonics: Notation
First, a few words about terminology. You are probably accustomed to the terms ‘natural‘ and ‘artificial‘ to describe the two general varieties of harmonics. These terms are somewhat inadequate and misleading — there is nothing “artificial” about an artificial harmonic. As you will see, these two seemingly different types of harmonics are only different in practice, but identical in theory. In their wonderful text The Contemporary Violin: Extended Performance Techniques, authors Patricia and Allen Strange suggest the words ‘open‘ and ‘stopped‘ as substitutes for ‘natural’ and ‘artificial,’ respectively.
Notation of harmonics is not completely standardized, but certain norms have arisen over time. Click the section headings below to see examples of natural and artificial harmonics notation.
Notes whose sounding pitch is the same as the stopped pitch usually have a small circle over the notehead. This notation has been used for many years.
Diamond noteheads are used to show nodes where the sounding pitch is different than the stopped pitch would be. The diamond shape indicates a light touch at the specified location. Usually the string is denoted by a Roman numeral or with text as shown here:
Sometimes the diamond noteheads are filled in to differentiate between half-notes and quarter-notes. This notation seems to be gaining popularity and is especially helpful when the sounding pitches (and rhythms) are not given.
Stopped harmonics operate similarly to open harmonics, except that now we use one finger (usually the thumb) to stop the string, creating a shorter string length and resulting in a new harmonic series. Another finger activates a harmonic node. The stopping of the string limits the available nodes to those that can be reached by the 3rd and 4th fingers.
The most common of these is a “touch-fourth” harmonic, for which we stop the string with the thumb and touch the string a perfect 4th above. This yields a pitch two octaves above the stopped note, as the table above indicates. Touch-fifth and touch-third harmonics are increasingly common in modern scores.
Composers generally notate stopped harmonics with a normal notehead for the stopped pitch and a diamond notehead on the node to be touched. It is not uncommon for the sounding pitch also to be notated for the sake of clarity. Performers have differing opinions about this practice, as some see the additional clutter on the page as a counterweight to the benefits.
This example shows touch-fourth harmonics with and without the sounding pitch notated:
Touching other nodes (the 5th, the major 3rd, the minor 3rd) will yield the other pitches in the harmonic series, as described in the table above.