HARMONICS
4d!
! NORMAN LUDWIN
Why is this Important?
2!
No musical instrument produces a pure single tone. In the 1860’s, the German scientist Herman
Helmholtz discovered that what the human ear perceived as a pure pitch was actually a composite of
many different tones. What the ear hears as the basic pitch is called the fundamental, and the other
pitches are called overtones (or harmonics).
“All musical instruments produce composite tones, consisting of many pure tones, called
harmonics, produced simultaneously.”
Harvard Dictionary of Music
The fundamental, or lowest frequency is louder than the other notes. The resulting quality, or
timbre, of an instrument is created by the overtones sounding simultaneously with the fundamental.
That is why each instrument has an unique sound, and why for instance the oboe has more overtones
than the flute, and therefore we say the oboe has a richer sound than the flute.
The harmonic series lays the framework on how sound is produced on all the musical instruments,
from the brass to the strings.
The harmonic series also dictates why some chord voicings or spacing (the vertical placement of
notes in relation to each other), sound clear and bright and others unclear and blurred..
Harmonic Series
3
¨ Harmonic Series - C Fundamental
¨ An illustration of the harmonic series as musical notation, based on a fundamental of the
pitch C. The numbers above the harmonic indicate the number of cents it deviates from
tempered tuning. Red notes are sharp, and blue notes are flat.
Harmonic Series
4
¨ When notes are placed together at the low end of the harmonic series, they will sound muddy
because each voices harmonics rub against each other in our ears.:
¨ Chords that follows the spacing of the harmonic series will produce a bright, clear sound.
¨ Close position voicing refer to the distance between the various notes of a chord. The chords
are usually within one octave; and sound best when written in the higher strings, flutes,
trumpets, and saxophones.
¨ Open position voicing occurs when the chord tones are not written
together. There is usually a distance of more than an octave from the
lower to upper voices.
¨ Examine this spacing example from Gustav Holst’s Saturn from
The Planets; note the open spacing in the low and upper voices.
Equal Temperament
5!
Just intonation tuned every note in a scale from a progression of pure perfect fifths. This was
quite satisfactory for most musical practices up to the 1700s.
n Notes were derived naturally from the harmonic series
n The intervals had a purity and stability when sounded
n This tuning worked as long as you remained in one key, otherwise you had to retune
your lutes, keyboards, or viols
n This also meant that notes in tune in one key would would be out of tune in other keys
Equal temperament begin around 1700 as music began to modulate from one key to another,
as opposed to remaining in only one key.
u Equal temperament divides the octave equally into twelve intervals
u Some intervals are expanded and others contracted
u If the piano was not tempered one octave would contain 77 different pitches…
obviously a compromise was needed.
Fundamental and First Harmonic
6!
You can demonstrate on the piano how the higher overtones vibrate sympathetically to produce
the composite of the fundamental and its overtones.
When you hold down a key on the piano, you are removing the dampers from only that note,
which isolates those strings and allows them to vibrate freely.
If you hold down middle C, which is C4:
C4!
C3!
and strike the note C3 one octave lower:
you can hear the C4 strings vibrating sympathetically.
Thus C5 is our fundamental and the C4 is the first harmonic.
1 The 1st harmonic is also a C - same note name
2 Sounds one octave higher than the fundamental
3 String is divided into two parts
Second–Third–Fourth Harmonic
7!
The 2nd harmonic is the fifth of C or G:
1 First different interval G4!
2 Root and dominant
3 This interval establishes our tonal system
4 String is divided into three parts
The 3rd harmonic is another C (an octave above the first harmonic) or two octaves above
the fundamental:
1 Four parts vibrating
C5!
The 4th harmonic is the interval of the third:
1 Five parts of the string vibrating E5!
2 Notice that the intervals are getting smaller
3 We have used the intervals: octave-fifth-fourth-third
4 These intervals inverted form a triad, the cornerstone of our Western music…
symphony, rock, jazz or hymns.
Fifth–Sixth-Seventh-Eighth Harmonic
8!
The 5th harmonic is an interval of a fifth, another G, one octave higher than the third
harmonic G.
1 Six parts of the string vibrating G5!
The 6th harmonic is between A and Bb, also called a blue note…(meaning an
expressive sound made by bending or sliding notes)
Bb5!
The 7th harmonic is another C:
C6!
The 8th harmonic is a D, a second:
D6!
1 When all these five pitches are altered they become a pentatonic scale: (C,D,E,G,A)
2 This is most common scale in the world
The series continues but we will stop here.
String Harmonics
9!
¨ String harmonics are played by lightly touching the string at certain select points,
called nodes.
¨ The Fundamental is the open string, for example “G”:
¨ First Harmonic is lightly touching the string at the octave (will sound one octave higher
than the open string, or the same pitch you are touching):
¨ The second harmonic is lightly touching the string at the fifth node {on a G string touch
D} (will sound one octave and a fifth higher than the open string, or one octave higher than
the note you are touching);
Touching! Sounding!
¨ The third harmonic is lightly touching the string at the fourth node {on a G string touch
C} (will sound two octaves higher than the open string, or one octave and a fifth higher than
the note you are touching);
Touching! Sounding!
String Harmonics
10!
¨ The fourth harmonic is produced by lightly touching the string at the third node {on the G
string touch B} (will sound two octaves and a third higher than the open string, or two
octaves higher than the note you are touching):
Touching! Sounding!
¨ You can see the pattern which should help you learn it:
¤ First harmonic is at the octave
¤ Second harmonic is at the fifth
¤ Third harmonic is at the fourth
¤ Fourth harmonic is at the third
¨ As the harmonics get higher, the intervals are getting smaller.
VIOLIN NATURAL HARMONICS
! "
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#˙ ˙
&c
11!
I
touch sounds:
!
octave harmonic touch sounds: touch sounds:
! "
w ˙ ˙ ˙
˙ #˙ ˙
II & w
sounds:
!
octave harmonic touch sounds: touch sounds: touch
! "
w ˙ ˙ ˙
III &w ˙ #˙ ˙
sounds:
!
octave harmonic touch sounds: touch sounds: touch
! "
& w ˙ ˙ ˙
IV
w octave harmonic
˙ ˙ ˙
touch sounds: touch sounds: touch sounds:
VIOLA NATURAL HARMONICS
w !˙
w
˙ !˙ ˙ # ˙ "˙
12!
I Bc
touch sounds:
!
octave harmonic touch sounds: touch sounds:
"
w ˙ !˙ ˙
˙
#˙ ˙
II B w
sounds:
!
octave harmonic touch sounds: touch sounds: touch
! "
w ˙ ˙ ˙
III
Bw ˙ #˙ ˙
sounds:
!
octave harmonic touch sounds: touch sounds: touch
! "
B w ˙ ˙ ˙ ˙
IV
w ˙ ˙
octave harmonic touch sounds: touch sounds: touch sounds:
CELLO NATURAL HARMONICS
w !˙ "
˙ !˙ ˙ #˙ ˙
13! I
?c w
octave harmonic touch sounds: touch sounds: touch sounds:
w ! !˙ "
? w ˙ ˙ ˙ #˙ ˙
II
touch sounds: touch sounds: touch sounds:
octave harmonic
! !˙ "
? w ˙ ˙ ˙
III w ˙ ˙
octave harmonic touch sounds: touch sounds: touch sounds:
! ! "
? w ˙
˙ ˙ ˙ ˙ ˙
w
IV
octave harmonic touch sounds: touch sounds: touch sounds:
BASS NATURAL HARMONICS
Sounding 8vb
!˙ "
w ˙ !˙ ˙ ˙ ˙
14! I
?c w
octave harmonic touch sounds: touch sounds: touch sounds:
w ! !˙ "
? w ˙ ˙ ˙ #˙ ˙
II
touch sounds: touch sounds: touch sounds:
octave harmonic
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? ˙ ˙ ˙ #˙ ˙
III w
octave harmonic touch sounds: touch sounds: touch sounds:
! ! "
? w ˙
˙ ˙ ˙ #˙ ˙
IV
w octave harmonic touch sounds: touch sounds: touch sounds:
! ! "
? w ˙
˙ ˙ ˙ ˙ ˙
w
IV
octave harmonic touch sounds: touch sounds: touch sounds:
Artificial Harmonics
15!
We have just examined natural harmonics, let us know look at artificial harmonics.
The string player is not limited by the natural overtone series when using artificial harmonics.
They can play many more chromatic pitches than are available naturally. The sound is very
similar to natural harmonics, but not as open or as pure sounding.
Look at the next page for the techniques to produce artificial harmonics.
Note: artificial harmonics works best on the violin and to a lesser extent the viola; on cello they
can be used but do not work on the double bass (too large a distance).
Artificial Harmonics
16!
Artificial Harmonics
Touch four: Stop a note and touch a perfect fourth above it
w
& 44 wO MOST COMMON
Produces a note
two octaves higher than the lower note
Touch five: Stop a note and touch a perfect fifth above it
w
& wO
2
Produces a note
one octave higher than the higher note
Touch three: Stop a note and touch a major third above it
w
& wO
3
Produces a note
two octaves higher than the higher note
Further Study
17!
I encourage anyone interested in learning more about the harmonic series and its role in Western
classical music to obtain Leonard Bernstein’s book, or video, entitled “The Unanswered Question…
Six Lectures at Harvard.” He covers this topic in a wonderful lecture as well as talks on aesthetics,
composition, Mozart, Tchaikovsky, Brahms and many other musical matters.
