Cello Natural Harmonics
This page is a sub-page of the article “Cello Harmonics“.
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, 7 etc equal parts. Because we don’t actually fold the strings, we will call these points at which we touch the string lightly to sound the harmonics, “nodal points”. 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). The following chart shows the harmonics we will obtain as we divide the four cello strings into increasingly smaller equal sections. This progression is called the harmonic series and always, for any string on any instrument, follows the same interval sequence of 8ve + 5th + 4th + major third + minor third + minor third + tone:
To divide the string into two equal sections, we only have one possible nodal point at which to touch it: the mid-string harmonic, one octave above the open string. But to divide the string into three equal sections, we have two possible nodal points to choose from: a higher and a lower one. Likewise, to divide the string into 4, 5, 6, 7 and 8 equal sections we will have 3, 4, 5, 6 and 7 nodal points respectively to choose from over the entire length of the string. To understand this better, let’s look at the natural harmonics on the G string and see which notes correspond to their different nodal points:
We will choose which point (node, finger location) to use for each harmonic according to the characteristics of the passage that we are playing (see below).
A NATURAL CHOICE: THE HIGH ROAD OR THE LOW ROAD ?
As seen above, for every natural harmonic except for the mid-string one, we can achieve the same harmonic note (pitch) in several different fingerboard positions on the same string. The following example shows this phenomenon for the harmonic that is produced when we divide the string into five equal sections:
For this reason, composers are often not sure how to notate their harmonics (where to play them on the fingerboard) in order for them to sound the best. In these cases, we will 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 fingering exactly but we may be able to improve on their suggestion by finding the same harmonic note in another position or on another string. Here is a popular children’s tune that, when played with natural harmonics has at least three different possible fingering configurations:
Sometimes where the composer suggests a natural harmonic we might prefer to use an artificial one, but at other times the situation might be the exact opposite.
It is very much a characteristic of 20th-century music to make use of the natural harmonics in the lower half of the fingerboard. But even in Romantic and Classical virtuoso repertoire (where the harmonics used are almost always notated in the higher fingerboard region), we might sometimes choose to play them in the lower half of the fingerboard in order 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 bravura!
The mid-point of the string, situated one octave above the open string, is like a mirror in the sense that the progression (series) of harmonics that sound as we go away from this point is identical no matter whether we go upwards or downwards. But even though the harmonic series is mathematically identical on both sides of the mid-point on any string, in practical terms the differences between the harmonics in the lower half and those in the higher half of the fingerboard are enormous, which is why it is helpful to subdivide our natural harmonics into those of the upper and lower halves of the fingerboard. But before we do that, let’s look at some other characteristics that are common to natural harmonics all over the fingerboard.
INTONATION PROBLEMS: IMPOSSIBLE TO CORRECT
No matter in which part of the fingerboard we play our harmonic, the tuning of a natural harmonic cannot be corrected without retuning the string, so if our string is out of tune …… tough luck (we are doomed) !! And even if our string is perfectly in tune, 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 deviation means that 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. 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 sound flat.
When we play this harmonic in the lowest fingerboard position, the point at which we will need to touch the string is also a little flatter than the notated pitch. Try the following Mary’s Harmonic Lamb example and we will see that the first finger needs to be more separated from the second finger than if we were playing the stopped notes:
If we are playing this note up high in any circumstance where it is more than just a short passing note, we will have to resist the temptation to play it as a harmonic because of this intonation problem:
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.
A LITTLE TRICK WITH NATURAL HARMONICS IN ANY FINGERBOARD REGION: EASY FINGER SUBSTITUTIONS
We can easily and imperceptibly change the finger on our natural harmonics, which, if we are careful, keep on ringing during the finger change. This allows us to use them as stepping stones to a new position, which can, at times, be very useful:
Because it is the natural harmonics in the higher fingerboard regions (from the mid-string point upwards) that are the most frequently used, we will start with them, before moving to the lower region harmonics.
NATURAL HARMONICS IN THE HIGHER FINGERBOARD REGION
Going upwards from the mid-string harmonic all of the commonly-used harmonics sound stronger, clearer, and more easily than those in the lower half. This is why when Shostakovich writes the following passage in his Cello Sonata, we will always play it with the glissando in the upper half of the fingerboard rather than in the lower half (try both to see the difference).
Above the mid-string harmonic, the natural harmonics not only speak more easily and sound more resonant than their lower region equivalents, they also sound at the same pitch as would sound if we were to stop the finger which means that the harmonic version can be used easily as an alternative to the stopped note. Some rarely-used harmonics of the upper half are however an exception to this “rule” and these are shown below. In the first line, we are using the middle nodal point of those that divide the string into five sections (5 being a prime number, all of these harmonics sound at the same pitch). The second line shows the harmonic that sounds when we divide the string into seven equal sections. Because 7 is, like 5, a prime number, all of these harmonics also sound at the same pitch, no matter where on the fingerboard we play them. These are very rarely-used harmonics, not only because they are difficult to sound but also because they are very out-of-tune (flat).
SOME GREAT THINGS ABOUT THE UPPER-HALF HARMONICS:
1: SOUND QUALITY, RESONANCE AND MARGIN OF ERROR
There is something very beautiful, expressive and unforced about the resonance of these “upper-half” natural harmonics: we don’t need to vibrate to make the notes ring and they are less inclined to scratch than either stopped notes (especially the high ones), open strings, or the same harmonic sounded in the lower fingerboard region. The extraordinary resonance of the simpler harmonics (the ones with the longest sections of vibrating string) compensates for their lack of vibrato to such an extent that they can often be used in situations where we might normally expect to use the power of a stopped note, such as in the climactic moment of an explosive fireworks display:
The above examples use the high natural harmonics for short intense climaxes. In music of the Classical Period, the “classical” purity of the natural harmonic’s resonance allows us to sometimes use them even on longer, culminating (highest) notes of a phrase, something we would be very unlikely to do in music of the Romantic Period:
Natural harmonics – especially the simplest harmonics – are wonderfully tolerant and forgiving creatures not only for sound quality but also for intonation. Their positional margin of error 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, see below) automatically in tune, unless of course our open string is out of tune. Because of these two factors – great natural resonance and great positional tolerance – natural harmonics give both our left-hand and our right-hand many beautiful, comforting moments of relaxation and security. Natural harmonics really are like occasional piano keys distributed over the fingerboard!
All of the above principles (advantages) apply especially to the lower, simpler natural harmonics – those with the longest sections of vibrating string. We need to be aware that as we divide the string into smaller and smaller equal parts however, the length of the vibrating string sections becomes shorter and thus the harmonics become more and more “fragile”: less resonant and more difficult to both sound and find.
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, 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. For example, we would normally do two consecutive shifts only in the slowest passages, but if those consecutive shifts are up to and down from the mid-string harmonic then there is no problem at all. Not only is there no problem: those shifts to and from the harmonic are also actually golden opportunities for the hand to relax.
The number of tricky situations that the mid-string harmonic gets us out of gives it the category of a “joker” card or a magician’s trick. Let’s look now at some more 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 stratospheric 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.
The above example uses the midstring harmonic to get us up into and down out of the Thumb Region but the midstring harmonic is equally useful as a safe haven, and a stepping stone into and out of the Intermediate Region. The Intermediate Region is our equivalent to the singer’s “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 (from both above and below) gives us a much safer entry into this no-man’s-land than a shift to a normal stopped note would do.
An extensive section looking at shifting up to and down from the midstring harmonic can be found via the following link:
2.2: FOR ITS UNEQUALLED RESONANCE IN BROKEN DOUBLESTOPS
Even when we may not particularly need its intonation and sound security, using the mid-string harmonic can give a wonderful enriching effect in arpeggio intervals across two strings in which its extreme resonance continues even after we have left it, its sound thus continuing to “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 are especially appreciable in the high regions of the fingerboard where, unlike in the lower regions, the harmonics sound at the same pitch as if we were to stop the string. 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. If we do a shift to a natural harmonic (instead of to a stopped note), we give ourselves a safe 100%-secure target and we also give ourselves extra time to get our hand comfortable in the new position. They are infallible reference points, giving us moments of safety and security when we are leaping all over the fingerboard, thus helping us to get comfortable and orientated in the poorly-mapped and dangerous quicksands of the stratosphere.
Of course, in thumbposition, using harmonics on any finger except the thumb requires that the thumb be lifted off the strings so that the harmonic can sound, but this thumb removal gives a delightful, helpful release of tension, once we get used to doing it (see Thumbposition: Thumb On How Many Strings?)
The above examples are scalic (move stepwise) with only the occasional use of the harmonics. We can also play arpeggios, using only natural harmonics. This is a wonderful, “user-friendly” way to get to know the higher registers. These harmonic series of natural harmonics are often used by composers to give a spectacular virtuosic fireworks display, commonly at the end of a virtuoso piece. In these types of passages – almost always loud – the brilliance and resonance of the short, articulated harmonics create a sound world more like sparkling diamonds than the ethereal, ghostly, otherworldly sounds that we usually associate with passages in harmonics:
But 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 on the fingerboard, and getting used to the typical fingering patterns of the one-string natural harmonic 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. At the top of this section dedicated to Natural Harmonics In The Higher Fingerboard Region, can be found a link to a page of exercises which could be very useful for developing this skill.
3A: LANDMARKS WITH A TRAP. FROM HARMONIC TO STOPPED = A DANGEROUS TRANSITION
When shifting up to the higher fingerboard regions, we will often shift to the harmonic version of a note, if that exists. Shifting to the harmonic, rather than to the stopped note, makes our shift more secure and gives the note a beautiful resonance: this is the idea of the “landmark”, the safe reference point. Unfortunately, this moment of easy ringing and intonational benevolence is 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”:
- 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).
- 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 of 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.
These two ideas for intonation security are particularly useful in thumbposition 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 thumbposition, 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 fifth above the octave (or on the next harmonic, two octaves above the open string), 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 (stopping the harmonic before the note finishes sounding) we can make the best of all the advantages of harmonic use without sacrificing our post-harmonic intonation security. We can even shift to the thumb with extraordinary intonation security:
We can also use this identical technique in downward shifts:
Here below are some links to material for working on this little trick of finding notes initially with a natural harmonic, and then subsequently stopping the note before continuing:
Playing A Note First As A Harmonic, Then Stopping It: REPERTOIRE EXCERPTS
4: 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.
5: LOW-PRESSURE TRILLS
Surprisingly, fast trilling from (over) a natural harmonic is actually easier than trilling from a normal (stopped) note. This is because neither the harmonic base note (the base of the trill) nor the trilling finger needs to be properly stopped: we can basically trill with a 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 and see ….
SOME COMPLICATIONS ASSOCIATED WITH THESE “HIGH REGION” NATURAL HARMONICS
HARMONICS IN CHORDS AND DOUBLESTOPS: ONLY ONE FINGER WITHOUT PRESSURE
The intonation security and resonance that natural harmonics give in the higher fingerboard regions can be especially useful in chords and doublestops but it can take a while to get used to stopping one string (or two) with normal finger pressure while only pressing lightly (for the harmonic) on the other string. The following example uses the final bars of Saint Saens’ “Allegro Apassionato”, in the original key and then transposed, to illustrate this:
The secret seems to be to separate the placement of the fingers, placing (stopping) first the fingers that need to be properly stopped, and only then placing the harmonic finger. The following link opens a page of exercises and repertoire excerpts that deal with this little peculiarity:
Let’s look now at the natural harmonics in the Neck Region:
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, especially the French impressionists (most notably, Ravel, in his orchestral music, but also Debussy to a lesser degree).
The following links open up material for working on our natural harmonics in the neck region:
COMPLICATIONS ASSOCIATED WITH LOWER-REGION HARMONICS
Even though our left hand is in the comfortable part of the fingerboard, natural harmonics in the neck region suffer from quite a few complications:
1: PITCHES DON’T CORRESPOND TO STOPPED NOTES
In contrast to the high fingerboard region harmonics, natural harmonics in the lower fingerboard region (Neck Region) do not sound at the same pitch as the stopped note in the same position. What’s more, their pitch progression goes in the opposite direction as we are used to (the pitches get higher as our hand nears the scroll)
Because we make frequent use of the harmonic on the third finger in “fourth position” (7th position with chromatic numbering) the already-complex pitching system becomes even stranger and therefore even more difficult to remember:
These complications can be seen, heard, and felt when we play the following simple children’s song in natural harmonics in this low region. While the song sounds simple, it is not simple to remember how to play it because the fingerings have absolutely no relation to the pitches which sound:
2: DIFFICULTY OF MAKING THE NECK-REGION HARMONICS SPEAK
Probably because our left hand is so far from the bow, natural harmonics in this fingerboard region don’t sound as strongly or as clearly as the harmonics of the same pitch up higher on the fingerboard. In the high-fingerboard region, we will often go up the harmonic series to the 6th harmonic above the mid-string octave (which sounds three octaves above the open string). But in the lower fingerboard region, we very rarely go any further than the 3rd harmonic below the mid-string octave (which sounds 2 1/3 octaves above the open string). If we try and continue up the harmonic series in this fingerboard region, not only do the harmonics not speak reliably but also the “nodal points” don’t correspond to in-tune notes.
If the harmonic doesn’t sound, we have a lot of trouble knowing whether our finger is too high or too low, and, unlike in the higher fingerboard regions, stopping the harmonic finger slightly doesn’t help.
3: TOTAL HAND RELAXATION AND EXTENSIONS: TWO OPPOSING WORLDS
With no need to stop the string, the hand is so relaxed that the fingers tend to relax together so much that even our normal finger spacings feel like they are extended (and our simple extended position seems to require much more conscious effort than when we are actually stopping the string)
Here below is a curious example of a situation in which we might want to make a very unconventional use of a neck-region harmonic. This situation is quite common when transcribing violin music for the cello because this doublestop corresponds to the violin’s open A-string with the note one octave higher on their E-string (in their “first position”):