Natural harmonics occur at the points that would correspond to the “folds” if we were to fold the string, like origami paper, into 2, 3, 4, 5, 6 etc equal parts. They are “magical points”, a little bit like the meridians of the human body for chinese acupuncturists, but infinitely easier to understand (and to find) because their locations are the result of simple mathematics (as discovered by Pythagoras). Let’s look at the natural harmonics of the G string:
SOME GREAT THINGS ABOUT (NATURAL) HARMONICS
1: SOUND QUALITY, RESONANCE AND MARGIN OF ERROR
There is something very beautiful, expressive and unforced about the resonance of natural harmonics: we don’t need to vibrate to make the note ring and they are less inclined to scratch than either stopped notes or open strings. However, as we divide the string into smaller and smaller equal parts, the length of the vibrating sections becomes shorter and thus the harmonics become more and more “fragile” (difficult to both sound and to find).
Natural harmonics – especially the simplest harmonics (the lower ones of the series) – are wonderfully tolerant and forgiving creatures not only for sound quality but also for intonation. The positional margin of error for the simplest (lowest) harmonics especially is such that even if our playing finger is quite a bit too high or too low, the harmonic will still sound (with one notable exception) automatically in-tune, unless of course our openstring is out of tune.
Because of these two factors – great natural resonance and great positional tolerance – natural harmonics give our left hand many beautiful, comforting moments of relaxation and security. Natural harmonics really are like occasional piano keys distributed over the fingerboard! Their natural resonance even allows the right hand to relax somewhat.
2: THE MID-STRING HARMONIC: A MAGICAL GIFT OF NATURE, THE COSMOS, PHYSICS AND PYTHAGORAS
The most commonly used harmonic by far is the mid-string harmonic (an octave above the open string). Perhaps we could refer to the mid-string harmonic as the “Saint of the Fingerboard” because it often merits a level of appreciation approaching religious devotion. If we were to leave a little offering to the mid-string harmonic every time we used it to get us out of a tricky situation, it would soon become very wealthy indeed! With this harmonic (on any string but especially on the higher strings) we can break just about every principle of fingering and get away with it. Let’s look now at some of the special uses we make of this harmonic
2.1: AS A STEPPING-STONE TO GET TO AND FROM THE HIGH REGISTERS
The geographical situation of the mid-string harmonic, right in the middle of the dangerous Intermediate Region and just at the start of the stratosphere (Thumb Region) means that its secure resonance and intonation can often be used as a life-saver, giving us a much-needed safe passage across the dangerous frontier zone (“The Break“) between the Neck Region and the higher fingerboard regions.
2.2: AS AN ABSOLUTELY SECURE REFERENCE POINT IN THE INTERMEDIATE REGION
The Intermediate Region is our equivalent to the singers “break” area, and we are often so grateful for the mid-string harmonic here to give us a moment of forgiveness and security, especially in passages in which we shift to the harmonic. Shifting to the very tolerant and kind mid-string harmonic gives us a much safer entry into this no-man’s land (from above or below) than a shift to a normal stopped note would do.
2.3: FOR ITS UNEQUALLED RESONANCE
Even when we may not particularly need its intonation and sound security, using the mid-string harmonic can give a wonderful effect in “harmonic” passages in which its extreme resonance continues even after we have left it, its sound can thus “accompany” the following notes
3: NATURAL HARMONICS UP HIGH: LANDMARKS TO HELP ORIENT OUR HAND IN THE STRATOSPHERE
The comfort and security of natural harmonics is especially appreciable in the high regions of the fingerboard where, unlike in the lower regions, the harmonics sound the same pitch as if we were to stop the string. We can use natural harmonics as a wonderful, “user-friendly” way to get to know the higher registers. They are user-friendly because they sound so easily, clearly and “in-tune”, without us having to actually stop the note in a fingerboard region where the strings are usually very high above the fingerboard. The high harmonics thus serve us as navigational aids up there in outer space where they are like illuminated beacons along a dark runway, safe resting points in hostile terrain, mountain refuges etc. They are infallible reference points, giving us moments of safety and security when we are leaping all over the fingerboard, allowing us to get comfortable and orientated in the poorly-mapped and dangerous quicksands of the stratosphere.
The harmonic series is also used often by composers to give a spectacular virtuosic fireworks display, often at the end of a virtuoso piece:
But the harmonics are used at other times as a pianissimo, angelic, ethereal stairway to heaven, normally going upwards, but occasionally descending:
Knowing where the harmonics of the natural harmonic series are, and getting used to the typical fingering patterns of these one-string arpeggios up to the top of the fingerboard (and beyond), is part of a great technique, although we rarely need these skills in any music other than concertos and virtuoso showpieces. Here then is a page of exercises to specifically practice this skill:
NATURAL HARMONICS IN THE NECK REGION
Even though the natural harmonics in the neck region are used less than those in the higher regions, it’s still necessary to know and practice them. 20th-century composers often use them – Ravel in particular just loved them!! Often however, composers are not sure how to notate their harmonics (where to play them on the fingerboard) in order for them to sound the best. Therefore we will very often need to use our detailed knowledge of the cello fingerboard to translate the composer’s ideas into practical, playable harmonics. Sometimes the composer just writes a º over the desired pitch (note) and leaves it up to the cellist to find where to play it. At other times they might notate the harmonic exactly, but we can improve on their suggestion by finding the same harmonic note in another position or on another string. Sometimes where they suggest a natural harmonic we might prefer to use an artificial one, and other times the situation might be the exact opposite.
A NATURAL CHOICE: THE HIGH ROAD OR THE LOW ROAD?
We can achieve the same natural harmonic note (pitch) in many different fingerboard positions on the same string:
It is very much a 20th-century characteristic to make use of the natural harmonics in the lower half of the fingerboard. But even in Romantic and Classical virtuoso repertoire, we might sometimes choose to play a natural harmonic in the lower half of the fingerboard in order to to avoid spectacular (and perhaps dangerous) leaps to the stratosphere. Although this is definitely unexciting choreography, sometimes – especially for an audio recording – security might be more important than showmanship!
VIBRATO ON NATURAL HARMONICS
We usually think of natural harmonics as being automatically and obligatorily without vibrato. Surprisingly – and fortunately – this is not necessarily true! By simply applying a little finger pressure to the string, we can actually make quite a nice vibrato on our natural harmonic. This can be a very useful effect, especially in the many pieces that end up high, on a long harmonic. Using this little trick we can start that long final note with all the intonation security that the harmonic gives us, but also with the added colour and expressivity of the vibrato, which we can gradually remove, allowing the piece to finish on a pure, “heavenly” harmonic.
TRILLING ON NATURAL HARMONICS
Surprisingly, trilling on (above) a natural harmonic is actually easier than trilling on a normal (stopped) note. This is because neither the harmonic base note (the base of the trill) nor the trilling finger need to be properly stopped: we can basically trill with the light harmonic finger pressure for both notes of the trill even though the top note theoretically shouldn’t sound cleanly when not stopped properly (see Trills). Try it ….
SOME DISADVANTAGES OF NATURAL HARMONICS
1: DANGEROUS TRANSITION FROM HARMONIC TO STOPPED
Unfortunately, this moment of easy ringing and intonational benevolence is very short-lived. The first stopped note after the harmonic brings us suddenly back to the hard world of real life and this transition can be quite brutally painful: suddenly we may find ourselves playing out of tune, because that beautiful harmonic not only didn’t give us any precise positional feedback, it actually gave us a false sense of security.
There are two ways to avoid this “post-harmonic hangover”:
- when practicing, it is a good idea to deliberately NOT use harmonics: that way our hand learns exactly where the note is. Although harmonics help us a lot in performance, it is more useful to actually practice them as stopped notes (instead of as harmonics) as this is what really gives us the true, exact feedback as to where we are on the fingerboard. Harmonics are too tolerant, their margin or error is too forgiving, and because of that, they don’t really help us develop that precise sense of geography in the upper regions that we so greatly need for playing up high.
- stop the harmonic just before continuing on with the following (stopped) notes: that way we can get some real positional feedback and thus correct our intonation before starting the next note (this is the meaning of the arrows in the following examples).
These two ideas for intonation security are particularly useful in Thumb Position because, when the thumb is up on the fingerboard, we have very little Absolute Positional Sense and thus depend almost entirely on the feedback (sound) from the note we are playing to “tell us” where we are. The playing of harmonics with the thumb itself occurs quite often, because, in Thumb Position, playing harmonics on the other fingers requires the temporary removal of the thumb from the string, and this is not exactly standard practice. When the thumb is on the mid-string octave or, more rarely, on the 5th above the octave, it is very tempting to use the harmonics under the thumb for the extra resonance and intonation security that they give. By following the above advice, we can make the best of these advantages, without sacrificing our post-harmonic intonation security.
We can’t correct the tuning of a natural harmonic without retuning the string, so if our string is out of tune …… tough luck !! Also, even when our string is in tune, some of the higher harmonics sound out of tune (flat). This is surprising for what seems like such a mathematically perfect system but can be verified using any tuning meter. The harmonic that is played by dividing the string into five equal parts is a notorious example of this out-of-tune phenomenon. Even though composers may notate this harmonic as being played on the third finger in the first position, we may prefer to play it (when possible) with the third finger in fourth position, as there we can put the thumb up on the fingerboard to convert the natural harmonic into an artificial harmonic with the same pitch. This has the advantage that now we can correct the intonation. Each group of three bars sounds the identical pitch, but the third bar in each (enclosed in red) is “intonation-adjustable”.