Cello Harmonics

“Harmonics” refer to the notes that sound when instead of stopping the string firmly with a finger, we just touch it lightly. Harmonics can be divided into “Natural Harmonics” (only one finger touches the string) and “Artificial Harmonics” (a lower finger – usually the thumb – stops the string firmly, while a higher finger just touches the string lightly). Before we talk about the differences between these two types of harmonic, let’s talk about their similarities.

Harmonics take us into a different physical and sound world from normal stopped notes and can teach us a lot, especially about “not forcing”. For both the left hand and the right hand, most passages in harmonics require us to use delicacy and lightness rather than pressure, transforming us from opera singer into child soprano, from bass (or tenor) to falsetto (countertenor). In most (but not all) situations in which we use harmonics, volume, vibrato, density and intensity are replaced by purity, lightness, innocence, and natural resonance. Compare the following versions of Mary Had A Little Lamb: the top stave shows the sounding pitches as played normally (not with harmonics) while the other staves show six alternative fingerings using harmonics of different types (both natural and artificial).

HARMONICS AND THE BOW

For harmonics, we not only can, but probably even need to, use a point of contact somewhat closer to the bridge than for normal (stopped) notes. A harmonic played “sul tasto” (with the bow over the fingerboard) is a harmonic that risks not sounding. In fact, “sul ponticello” doesn’t exist for harmonics: no matter how close to the bridge our bow is, it is quite difficult to get the sound of a harmonic to “break”. Harmonics not only encourage us to play with our bow point-of-contact closer to the bridge, they also encourage us to use faster bow speeds and lighter bow pressure than we would for “normal” notes. The French Impressionists (Debussy, Ravel etc) make very frequent use of harmonics whereas Brahms, Wagner and the other German Romantics basically never use them. What a difference a few hundred km and 50 years can make !

HARMONICS AND LEFT-HAND FINGER PRESSURE

Harmonics give our left hand the unique, wonderful and almost magically effortless experience of playing without finger pressure. The simultaneous combination of total relaxation of finger pressure with the need for tension in other parts of the hand can however create some curious situations:

• the complete absence of finger pressure in natural harmonics can lead us to unconsciously relax the hand so much that we no longer separate the fingers sufficiently to reach the required distances, especially in the case of extensions. Try the “Mary’s Harmonic Lamb” example above (with natural harmonics) to see if this is also true for you. This problem possibly only significantly affects smaller hands
• in the case of artificial harmonics, in order to produce a clean clear harmonic we need to combine the absolute lightness of touch of the higher (harmonic) finger with a very firm pressure of the stopping finger (usually the thumb). This is quite a feat of fine motor skill and coordination.
• a similar situation occurs in those double-stops (or chords) in which we may prefer to use a natural harmonic on one of the notes in order to give both intonation security and added resonance. Using one finger with very light pressure and the other finger(s) with firm pressure does not come naturally and will probably require practice. This subject is looked at in more detail on the Natural Harmonics page:

FINGERING AND NOTATION OF HARMONICS: MULTIPLE POSSIBILITIES, DOUBTS, INCONSISTENCIES … AND ERRORS

One of the complications associated with harmonics is that there are usually several different possible ways (fingerings) to obtain the same harmonic pitch. Not only can we often find several different natural harmonics which sound at the same pitch but we also have the possibility of using different artificial harmonics to play any possible harmonic pitch. And, to make matters even more complicated, not only are there several different ways to finger each harmonic, there are also several different ways to notate harmonics.

Sometimes composers will just write the pitches they want, with the harmonic sign over each note, for example:

This type of harmonic notation requires that the cellist work out the most appropriate fingering, and we will often need to rewrite the notes according to how we would like to finger them. We saw above how this tune (“Mary Had a Little Lamb”) can be played entirely with natural harmonics using three different fingerings (hand positions). But it can also be played using artificial harmonics, also with three different fingering possibilities. If all but one of these cases, the music needs to be rewritten in a way that, instead of writing the actual sounding pitches, we write what we actually need to do with our hand in order to make those pitches, for example: (and see also the other versions with natural and artificial harmonics above)

Rather than notating the true pitches of the harmonics with little circles over the noteheads, composers most commonly try to work out the best fingering for each harmonic, notating the natural harmonics with hollow diamond noteheads at the pitch that would sound if we were to stop the note, and notating the artificial harmonics with the thumb always a perfect fourth below the stopping finger. Another less commonly used notation indicates the sounding pitch above the stopped note of an artificial harmonic (see the Stravinsky example below).

Unfortunately, the large choice of cellistic fingering possibilities means that despite a composer’s best intentions to help us, they often get it wrong and their harmonics are not notated in the most practical cellistic way. If they get it right then we are lucky …. we don’t need to rewrite anything. But in many other cases, even when they specify the fingering, we may need to rewrite the passage or some of the notes in it. The following examples illustrate some of these situations. The top stave reproduces the composer’s original notation …. and below is the cellist’s playable (renotated) version.

For every one of the harmonics in the above example, we will need to find our own fingerings and will end up using a mix of natural and artificial harmonics. Curiously, for the only two harmonics for which Stravinsky indicates a fingering, he suggests an artificial harmonic at an interval of a major third above the thumb, which is a very rarely used artificial harmonic. His suggestion is not at all helpful because we can actually play those two harmonic notes with a very easy-to-find natural harmonic on the neighbouring string! So this is an excellent example of the notation and reading problems we will often encounter in passages using harmonics.

In the following example, we could substitute Ravel’s natural harmonics for our own artificial ones to make the passages easier to play:

REPLACING OUT-OF-TUNE NATURAL HARMONICS WITH ARTIFICIAL HARMONICS

Normally, we would expect natural harmonics to be perfectly in tune, according to all the beautiful mathematical Pythagorean laws. How could dividing the string into perfectly equal proportions (in half, in three, in four, in five etc) not produce perfectly in-tune notes? Well, nature and physics play some tricks on us and as we divide the string into increasingly smaller sections we arrive at some harmonics that are noticeably out-of-tune. This is surprising for what seems like such a mathematically perfect system but can be verified using any tuning meter. The first of these permanently out-of-tune harmonics is the harmonic that is played by dividing the string into five equal parts. This harmonic, played (fingered) in all its possible positions (nodal points), is 15 cents flatter than it should be in an equal-temperament tuning system (in which a semitone is 100 cents). This is especially important when the note needs to be sustained as a chord component (rather than as a quick passing note).

For this reason, if this harmonic needs to be held for any length of time, then even though the composer may notate it as being played by the third finger in the “first” position (second position with chromatic position numbering), it is usually a better idea to play it (when possible) with the third finger in the “fourth” position (7th position). With this fingering, we can put the thumb up on the fingerboard to convert the natural harmonic into an artificial harmonic with the same pitch which has the enormous advantage that we can now correct its intonation. It’s a shame that nobody told Ravel this vital piece of information!

In the following example, the three bars of each group all sound at the identical pitch, but the third bar of each group (enclosed in red) is “intonation-adjustable” (correctable) whereas the other two bars (with the natural harmonic fingerings) always will always sound flat.

The following illustration shows this in another way and includes the higher fingerboard region versions of this harmonic:

When we divide the string into seven sections, the harmonic is even flatter – 31 cents to be exact – which makes it so flat as to be almost useless. Fortunately, this doesn’t matter much as it is also very difficult to sound and, as a result of both these factors, is almost never used.

For a more detailed discussion of both Natural and Artificial Harmonics, click on the following links:

Natural Harmonics              Artificial Harmonics