ToninWorkon the harmonie seriesand you'lI releasea wealth of eolour from your instrument without effort.Pedro de Alcantara explains how Wsdone

BUOW the heavenlyspheres: an ancien1Greek traditian taughtthat each planetpraduced a nateand that the relative

distances af the planetsfram the sun defined theharmonie series inmusic

Summertime, end you're sittlng in a swinging porchbench, moving gently back and forth with the smallesteffort of your legs. A breeze blows across your face,and you feel contented and fulfilled, in sync with life' svery rhythms. Wouldn' t it be nice if producing big,beautiful sounds on the violin were just as easy?

A force of nature exists that allows you this ease atthe instrument - as long as you' re at one with it. It' sthe acoustie phenomenon called the harmonie series.It's an oscillatory phenomenon similar to the swinging

The harmonie series

Almost everynote eversung, played or struekis a

eombination of severalfrequeneies(see figure 1). Take

the eelJo'sopen C string. TheC ilself- ealJedfhe fundamental

- is the prevailingsound. Indeed,iYsso pofentfhat most listeners

won't notieeany othersoundsat firsl. Veta number of other pitehes

- ealJedeitherovertones,partialsor harmonies- are produeedabove

the C everytime you play il. Togetherwith the fundamental theseform

the harmonie seriesand eaeh one is given a number whieh shows how

fast it vibratesrelativeto the fundamental. Thefirst harmonie is the funda-

mental itself; the seeondharmonie, an oetaveabove the fundamental,

vibratesaf twiee the frequeney;fhe third, a fifth higherthan the seeond,

vibratesat three times the frequeneyand so on.

In fheory,the seriesis infinite; In praetiee,it beeomesinstrumentalJy

and auralJyinsignüieanfafterthe first dozen or so harmonies.Onee

you attune your ear to the existenee01the series,you'lJ hear the

seventhharmonie - a note around B flat if your fundamental is

a C, whieh will sound out of tune eomparedwith the B flat on

a piano. Asyou go fartherup the series,other partialswill

also sound 'out of tune' if you measurethem against

equal temperament;theseare marked with

arrows in example2.

of a pendulum or the waves a stone makes when

thrown into water. Pluck the string or drop the stoneand the energy is released without any further effort.As with a pendulum, poise is more efficient thanbrawn in setting off the mechanism. Time your play-ing skilfully - as you do when you swing that porchbench - and you'lI have tremendous tonal energy atyour disposal, without effort or strain. But first yourear must discover the phenomenon.

Sit at a well-tuned piano with the lid open. Withoutsounding it, press the C below middle C with your rightindex finger and hold it down. Now play the C below it(the equivalent of the cello's open string) with the leftindex finger, with a short, loud, accented articulation(see example 1).As the note dies down you'lI hear theC immediately above it ring out, almost like an echo.

Example1hold silently play simile

iI}:, ..11

It may sound for sever:.alseconds, depending on thepiano and the acousties of the room. Try the otherpartials, each in turn, then several in combination.They'lI appear like an eerie, soft choir heard in thedistance. Holding down a cluster of notes with yourright hand, strike and release two adjacent notes acouple of octaves below the cluster (C and C sharp, forexample). You'lI hear a variety of consonances anddissonances as the heavenly choir grows in complexityand subtlety.

Play the low C again, on its own this time, strikingit with any number of artieulations - soft or hard,sustained or short. Having awakened your ear,you should now perceive at least some of the

The Strad January 2005

.. .. - ----

- ./'

~~Flgure1

f

upper partials that make up that one note. The partialsare always present; hearing them takes some training,but in time you'll detect their presence in many musiealsituations - and in non-musical ones, such as the

whirring of a fan or the rumbling of a refrigerator.Most instruments are constructed around the

harmonie series. Trombones don't sound their funda-

mental note, their lowest note is the second harmonie.

There are seven positions of the slide, whieh cover thenotes up to the third harmonie, one fifth higher. Theharmonie series extending from these seven notes isthe source of the instrument' s entire pitch range.Similarly, through a combination of fingering andembouchure techniques, flautists sounQ'many of theirnotes as partials of the instrument's fundamentalpitch. In a pipe organ each stop has its own character-istie overtones or partials - giving it an individualtimbre - but there are also higher-pitched stops,pitched at octaves, fifths, tenths and so on, whieh havetheir own partials. This exaggerated manipulation ofthe harmonie series - partials of partials, if you like -gives the organ its characteristie sonority.

On a bowed instrument, you can sound the naturalharmonies of an open string by pressing lightly onprecise spots, called nodes, along the string andbowing or plucking it. If you look at figure 1, the nodesrepresent the plaees along the string where it is notmoving. You ean also sound artificial harmonics bystopping any note down normally and pressing lightlya fourth or a fifth higher with another finger of the

The strad Jonuary 2005

left hand, in eaeh ease ereating a different harmonie.You ean play the same piteh on different strings, usingnatural or artificial harmonies as appropriate, givingthe instrument a tremendous eolour range.

Modern composers have often exploited theharmonie series as a eompositional tool- using thepiano as deseribed earlier, for instanee. But the effeetof the harmonie series ean be notieed in eompositionsfrom mueh earlier. The sixth harmonie of any piteh isthe major third above the fundamental, but twooetaves higher. It's c1early audible on the low notes ofa cello or bassoon. When playing pieces in minor keys,this harmonie c1ashes harshly with the tonality'sinherent minor thirds. You ean hear this by playing theehord in example 3 on a well-tuned grand piano. TheE flat of the C minor ehord c1ashes against the E, thefifth harmonie of the bass C.

Example3

The harmonie series provides a mathematiealunderpinning for tonal musie' s vital interplay ofeonsonanee and dissonanee. The frequeney ratio ofany two notes in the harmonie series is simply the ~

Example2 t bJ1 qo. -&ABOVEtwowaysofillustratingtheharmonseries.Learningto exp

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 harmoniesiskeyto-& controllingyourtone

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1

f-&

:

-& \:e:/

ratio of their positions within their series.Consonances are based on the simple ratios - 2:1,4:3, 5:2 and so on - whereas dissonances are based

on more unwieldy ratios, like 11:7.The developmentof Western musie, from plainchant to organum (twovoices sounding fourths, fifths and octaves) to modaland tonal counterpoint and beyond, largely corre-sponds to a climb up the harmonie series, eachevolution demonstrating the willingness of composersand audiences to accept the validity of ever morecomplex intervals.

We're now ready to work on the harmonie series,using the cello's strings and pitches for our exercises.If you're a violinist, violist or bassist, transpose themas needed.

. Practise drawing slow bows near the bridge on openstrings, executing crescendos and diminuendosanywhere along the bow. This basie stroke is essentialfor most of our exercises (see example 4).

Example4

MV MV MV

,,}:MV~' . . . . . . .

- - '-"-e-'-" -e- <::: >-<::: >-

. Find and execute each string's natural harmonies,one at a time and in order, starting with the instru-ment's lowest string. You can produce naturalharmonies at either end of each string. For this exercise,play the ascending harmonics as they approach thebridge, not the nut. Find a comfortable left-hand posi-tion for each harmonie and play it without hesitation ortimidity, varying the dynamies at will (see example 5).Memorise the series for each open string and, overtime, for all pitches.Example5

-=:::: ::::=-IV

This is easier than it seems, as the sequence of intervalsdoesn't change. Once you've grasped it you canderive it afresh for any note. The first six harmoniescreate a major chord that your ear recognises as asingle entity, cutting down the amount of informationyou need to process.. Play the lowest open string's second harmonie (theoctave) four times in a row, with a crescendo; follow it

with a short, ringing open string. You'll soon be able toperceive the harmonie as a component of the openstring sound. Repeat the exercise with the first sevenpartials of each open string. Your ear will seize somepartials more readily than others - the third harmonie(octave plus fifth), for instance, resonates brilliantly onmost string instruments. The sixth harmonie (twooctaves and a fifth) is given a helping hand as theunbowed section of the string, between the bridge and

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"LEF'polspir4Iho'keelon~bee

the tailpiece, is normally set up to be a sixth of thebowed string length, giving it a natural frequencyof six times that of the open string and so enhancingthe sixth harmonie. 500n you'll become able to hearseveral partials simultaneously whenever you drawan open string, as in example 6.

Example 6

J =80- '-' '-'=fp IV -=fpIV

. Play an open string with a short, ringing up bow;follow it immediately with the octave, played softly,not as a natural harmonie but as a stopped note on thenext string up, with a focused down bow. The octavewill blend with the resonating partials of the openstring, creating a beautiful effect in whieh the octaveand its fundamental complement each other. Do thesame for other harmonies, playing them as stoppednotes where they're brightest, as in example 7. Varythe dynamics and speed of the exercise, alternatingamong several partials.

Example7V M

IV III

V M

IV IIV MIV 1

f.\

~sft p -==::::=- sft p -==::::=- sft p -== :::::::==-

Create a simple improvisation using the first fewpartials of each open string. Connect the noteswith an elastic legato and make sure that the changefrom note to note doesn' t hinder the resonance ofthe whole instrument.

50 far we've sounded a pitch and worked upwards;now we'll take a pitch and work downwards. We canregard any one note as the second harmonie of a ~

MV

f.\ f.\ f.\ f.\ f.\ f.\ f.\o 0 0 0 0 0 0

")=..

-e-... ae

fundamental an oetave below it or the third harmonie

of a fundamental a twelfth below it, and so on throughthe harmonie series.

. Play the cello' s open D string and think of it as aseeond harmonie. The D an oetave below, in first posi-tion on the C string, is its fundamental. Now playaversion of the previous exercise (see example 8).

Exomple8v11

r;IV

o

~u«fi p -=== ===-- «fi p -=== ::::=-

You ean expand this and use the first few partials ofany open string as trampolines for ereating new funda-mentals, adding low Es and Bs and Fs to your soundpalette (see example 9).

Exomple9 p cresc.-e-

-e-

f ===--p cresc.

. Bring together pitches that are one of the following:open strings; seeond, third, fourth or fifth harmonies ofan open string; either of the fundamentals for whieh anopen string is a seeond harmonie (D and A). All thesepitches - some belonging to the harmonie series of twoor more open strings - resonate beautifully on string

instruments. Play seales and improvisations aroundthem, as in example 10 (see top; the arabie numeralsrepresent the partials, the roman indieate the strings).. Certain notes - D flat and E flat, for instanee - aren't

supported by vibrant fundamentals. Take a few pitehes

liGHT 0 spiragraph ofharmonics. In fime. Ihefree resonances 01 Iheharmonie series will

inform every nole ofevery piece you play

"

\.

66

from the previous exercise and improvise a shortmelody passing from supported pitehes to theseunsupported ones. Shaping the left hand and usingvariations in bow pressure and speed, find ways toadjust the resonanee of these unsupported pitches. It'sas if some notes were natural ehildren to open-stringparents, others adopted. Your job is to tend to theadopted ones until they resemble the blood family.These unsupported notes are marked with circles inexample 11.

Exomple11

. Take any mainstream piece - Chopin' s Cello Sonata,Baeh's partitas or Hindemith's Viola Coneerto no.3'Der Schwanendreher'. Notiee how many pitehes inany passage belong in eaeh of these eategories andhow freely they resonate. Choose a phrase, addfermatas to the relevant pitehes and exeeute your ownmeditations on the intimaey between fundamentalsand harmonies. In time, the free resonanees of the

harmonie series will inform every note in everypassage (see example 12).

Exomple12Cello Sonata In G mlnor op.65, Chopln

112 IV III 2 FirstmovementIII3 4 IV3 11

~dolce

These exercises have startling effeets. If you synehro-nise your playing with the vibrations of the harmonies,you ean produee a big, beautiful sound with greateconomy of means, mitigating or eliminating physiealeffort. Vibrant harmonies give the sound a penetratingquality that earries far from a eoneert stage, even inpianissimo dynamies. The instrument itself appears toimprove and, indeed, you ean easily produee the truevibrating qualities of your violin or cello if only you'let it ring'. The aural knowledge gained from exploit-ing the harmonie series enhanees the execution of tonalmusie, as the player starts sensing the dominant-to-tonierelationship between the third harmonie and thefundamental, for example.

Above all, the simplest aets of musie making beeomejoyful expressions of a marvellous natural phenomenon:the release of energy through oseillatory patterns, setinto powerful motion with virtually no effort. Bowingan open string beeomes akin to reeeiving the oeean'sembraee and being endlessly anointed by warm,gentle waves. If that doesn't make you want topractise, nothing will. (;'I

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Exomple10I IV 114

IV 11 III I IV 11 III2 4 11 2 IV III4 I 2 III IV III6 I 31 0 1 0 2 1 IV 3 1 -e- III3 5 IV 6 113 5 8 IV 4 IVlO

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e e .... e ..e .. e

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v r;

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