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Harmonics

Ever since I was a kid with a toy steel string guitar, I've been fascinated by harmonics: the ethereal sound they produce, the oddly high pitches, always at odds with the natural voice of the instrument, and the way they tend to ring and ring, lasting much longer than conventionally played notes. And once I began to discover the math behind them, the actual intervals they produce, then they offered no end of additional fascination. It always startles me when I encounter a student who doesn't get harmonics, who doesn't know what notes they produce or how to articulate them effectively. Harmonics seem like one of the most compelling things about the instrument-how can the students have gotten that far along as players and still not have explored harmonics? In this post I'll explain all about them: the notes they produce, their intonation, and how to best produce them. My wife, a singer and music educator, has always called them "pings," as in 'can you add some pings to that line?' Hopefully after reading this and experimenting some, you'll be able to better produce the pings in your own playing.

WHERE/WHAT NOTE
Harmonics occur at locations determined by mathematics: if you divide the string exactly in two, that spot, or "node," (at the 12th fret) is the second overtone (the second harmonic--we count the "fundamental," or open string, as the first); it produces a note one octave above the fundamental. Each successive harmonic produced on that string is another step up the overtone series. 

Divide the string into three equal sections, and at those nodes, you find the third overtone (the third harmonic), at a pitch one octave and a perfect fifth above the open string. The node falls on the 7th and 19th frets. 

Divide the string into four equal sections and the two outer nodes are the fourth overtone (fourth harmonic), producing a note two octaves above the open string. (The middle node is the same as the first harmonic, whose signal is stronger and drowns out this one). They fall on the 5th fret and over the sound hole. 

Divide the string into FIVE equal sections, and you produce the fifth overtone (the fifth harmonic); the pitch will be two octaves and a major third above the open string. Their locations are at the 4th, 9th and 16th frets (and at a location approximately over the rosette). 

Go on, go two farther along this path: divide the string into SIX equal sections and you will hear your sixth overtone (sixth harmonic), pitched at two octaves and a fifth above the open string. For practical purposes, it only sounds in one spot: somewhat above the 3rd fret. 

Finally, divide the string into SEVEN equal sections to find your seventh overtone (seventh harmonic), sounding two octaves and a minor seventh above the open string. This one works best, too, in one spot only, slightly below the 3rd fret. 

This set of divisions/locations/pitches is uniform for all the strings. Knowing basic intervals and the information listed above enables you to understand, globally, every natural harmonic on the guitar.

INTONATION/TUNING
As natural harmonics are our instrument's native way of expressing the overtone series, the pitches are not "tempered" to coincide with our modern scale. Tempering is the assortment of pitch compromises necessary to make playing in all twelve keys possible (remember J. S. Bach and his "Well Tempered Clavier"-- an advert for the new tuning system then coming into vogue?) The pitches produced by playing harmonics are therefore almost all "out of tune" relative to the notes in the tempered scale we play. Some are closer than others. Some are quite far off. 
This problem is the reason for the confusion over using harmonics to tune the guitar. Note the pitch deviations in the chart below (from Wikipedia):

Below is a comparison between the first 31 harmonics and the intervals of twelve-tone equal temperament (12tET), transposed into the span of one octave. Tinted fields highlight differences greater than 5 cents (1/20th of a semitone), which is the human ear's "just noticeable difference" for notes played one after the other (smaller differences are noticeable with notes played simultaneously).


The harmonics which produce octaves above the fundamental are perfectly in tune, showing a deviation of 0. All others show some variance. The third overtone is close, at +2 cents. Close enough to tune to (using the conventional 5th fret harmonic, 6th string to 7th fret harmonic, 5th string method). The first of these two is perfectly in tune, being two octaves above the fundamental. The second, though, is a bit sharp, so making it match will force the string a tiny bit flat. Two cents flat is no problem, generally speaking, but if you use this method from string to string, all the way across the instrument, then the multiplier effect comes into play, with each string ending up two cents flatter, with an end result that the 1st string ends up 10 cents flat, a quite noticeable variance, and a common source of confusion and frustration among guitar players. As a rule, the only harmonics it is productive to use when tuning are the ones that produce octaves above the open string: those on the 5th and 12th frets.  

The Major Third
The tempered major third is quite sharp when compared to the naturally occurring major third - it is 14 cents too high. This variance is easy to hear so we naturally want to tune our major thirds lower to fix this. Doing so renders the instrument out of tune, though, so it's important to understand and learn to live with this inflated interval. (Chords voiced with the third on top are especially problematic, as the third is so exposed: first position D major is constantly, incorrectly adjusted to lower that harsh F#, leaving the first string flat). 

One of my favorite excercises to illustrate this issue is to tune the fifth string, 2nd fret to perfectly match the 6th string, 7th fret harmonic; tune the 4th string so the 2nd fret E matches the 6th string, 12th fret harmonic; then tune the 3rd string, 1st fret G# to perfectly match the 4th harmonic on the 6th string (4th fret). Play this 4-note chord. It sounds sublimely perfect and richly resonant. Now play first position G major (with open 3rd and 4th strings): it is wildly out of tune. Extend the excersise. In addition to the tuning adjustments above, now tune the 3rd-fret d on the 2nd string to perfectly match the 6th string, 7th harmonic (below the 3rd fret). It is quite low. And yet, played together with the other notes, it sounds gorgeous. It is truly the harmony of the spheres- the intervals that exist in nature. And they are produceable on the guitar in this way. Now, play any other open position chord. It sounds awful. And so, we use equal temperament. It doesn't sound as good as the "music of he spheres," but it allows us to play more than one chord, a useful development!


EXECUTION: NATURAL HARMONICS
We play natural harmonics in two ways: touch the node with the left hand while articulating with the right hand, or do both with the right hand. In the case of the first method, there are several elements of the execution worth pointing out. The most critical is the location of the right hand when it plays the string. I see students doing this incorrectly all the time, and not understanding why their harmonics are not speaking. 

The key is to play closer to the bridge than the corresponding node. For instance, for the second harmonic: touch the seventh fret with a left hand finger. Now play the harmonic, plucking the string at the 19th fret (the corresponding node). Nothing comes out. Move the right hand towards the bridge, even just over the rosette, and the harmonic speaks. Try it with the fourth harmonic. Touch at the 4th fret. If you try playing with the right hand point of contact over the rosette, you'll get nothing. Move towards the bridge, and it speaks perfectly. Therefore, awareness of the location of the right hand relative to the nodes in question is necessary for natural harmonics to work. 

The second issue I see misunderstood is the length of the left hand contact during the execution of a harmonic. There is an optimum length of time to remain touching the node before releasing contact to let the note ring. Often players leave the finger touching too long, muting the harmonic before it even starts. 

Finally, the left hand pressure used must be correct for the specific node and the desired volume. A loud harmonic requires considerable pressure on the node before releasing, just shy of pressing the string down all the way, but enough to withstand the force of a forte attack. Similarly, if the intent is to play a very quiet harmonic, the left hand needs press only quite gently. This agreement between the hands is a critical part of execution.

EXECUTION: ARTIFICIAL HARMONICS
As the right hand is solely responsible for the execution of artificial harmonics, technique is especially important. For the treble strings, we use the index finger to touch the node while the 'a' finger plucks the string. For the basses, we again use the index finger to touch the node, but now use 'p' to pluck, as the 'a' finger causes unwanted scraping sounds on the basses. The critical issue is the distance between the node and the location of the pluck; the more the better. This is an often misunderstood issue, so it is worth underlining: it is necessary to stretch the right hand a bit to get as much distance as possible when playing artificial harmonics; otherwise volume will be severely limited:

Right

Wrong


Right
 

Wrong


Perhaps obviously, whatever fret is played with the left hand is then mirrored 12 frets higher with the right hand to produce the octave harmonic. Less commonly done on the guitar, but perfectly useable (and the default method on bowed strings), is to use a higher harmonic for the node while playing artificial harmonics. This node, especially in upper positions, is often reachable with the left hand. For instance, press down on the fifth fret of the 1st string with the first finger while touching lightly at the tenth fret with the fourth finger. With the right hand, play strongly, near the bridge. The harmonic is an A, two octaves above the fifth fret fundamental. Or slowly roll a first position G chord, but when you play the first string G, play instead an artificial harmonic, now with the right hand, at the tenth fret, to hear a D, an octave and a fifth above the third-fret fundamental.

NOTATION
One of the maddening things about harmonics in the literature is that different composers choose to notate them in different ways, leading to considerable confusion. Much has been written about the especially confusing notation in the Villa Lobos scores. Other composers have added their own idiosyncratic approaches, further muddying the waters. As a player, I much prefer to see the actual pitch the composer wants. I find their notions of how to play the harmonics of limited value, as with other fingering advice, since I probably know better all the  harmonics options available to me at any given point better than they do. So a note to composers: write the sounding pitches!!

To summarize, acquaint yourself with the pitches harmonics on your guitar produce. Learn well which ones are out of tune and how much, to demystify tuning with them. Work on awareness of the optimum place to play on the string for each node, so you get the most out of each harmonic. Increase your sensitivity to pressure and duration of left hand contact when playing natural harmonics. Be sure to open up the right hand sufficiently for clear articulation of artificial harmonics. 

Most of all, ENJOY harmonics. They are one if the most beautiful sounds we can produce. And while it's easy to take them for granted, non-guitarists are captivated by them. 

We should be too. 




Sarah Boyson's Senior Recital

Benjamin Verdery at Oberlin

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