SOUND FUNDAMENTALS & TUNING SYSTEMS
in brief ...
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The transmission of sound occurs as sound waves,
produced by vibration, travel through the air. A single wave
consists of a two-part vibration cycle. The point of increased
pressure caused by plucking a string, for example, is called compression,
and the point of decreased air pressure as the string moves to the
opposite side of the cycle is called rarefaction. A
complete vibration cycle occurs when a vibrating body completes these
two movements and again passes through the point of rest.
a medium, such as air, is required in order for sound waves to
travel. In a vacuum jar, sound does not travel in the absence of
air. An alarm clock in a vacuum jar cannot be heard if the air
has been removed.
DURATION & SOUND ENVELOPE
As any scholar might guess, duration is simply
the length of time a sound or tone is sustained. The sonic
history of a sound is described in terms of its ADSR envelope,
i.e., its attack, decay, sustain and release. Along with other
acoustical properties, the ADSR envelope helps us identify what we are
listening to--a piano or a soprano?? Amazingly, if a tone's
attack and release are obscured, e.g.,, via using the volume slider on
a synth, you may be surprised to discover how difficult it is to
identify the timbre being played.
The harmonic series is a naturally-occurring
phenomenon that is truly an acoustical, as well as a mathematical,
marvel. Any time a pitch is sounded, we are actually hearing a composite
sound which consists of that particular fundamental pitch plus
its many partials or harmonics. Partials are
typically inaudible to the human ear, but they do exist. The ear
identifies the fundamental as the specific pitch in that it is
An interesting experiment is to highlight partials via
sympathetic vibration on a good acoustic piano. Lightly depress
a key representing a partial for a given fundamental. Depress
the key slowly so the note does not sound, but only releases the
hammer from that particular string(s). Continue to depress that
key. Now, firmly play the fundamental note and quickly release
it. The pitch of the partial should sound (via sympathetic
vibration) when the fundamental is released. Also try notes
"near" the partial. Observe that they do not
vibrate sympathetically as do the partials.
SOURCE OF PARTIALS
Where do partials come from? Partials are
produced because the vibrating body, whether it is a string or some
other medium, is not only vibrating as a whole but in smaller sections
as well--halves, thirds, fourths, etc. Of course, these
secondary vibrations are more subtle than the fundamental vibration,
which is why partials are softer and, therefore, difficult to
hear. The following example illustrates nodes, which are
points of zero disturbance. By lightly touching a
vibrating string at a node point, its corresponding partial is
heard. Touching a vibrating string in the middle (diag. B)
divides the string in half and produces a pitch one octave higher (2nd
partial). Touching the node point in diagram C produces a pitch
an octave and a fifth higher (3rd partial), etc. Try this on a
violin or guitar string ...
Vibration pattern of a fundamental & first two
APPLICATION OF THE HARMONIC SERIES IN BRASS
How does a trumpet player manage to play more
notes than three valves would seem to indicate? The
answer once again lies in the harmonic series. By relying on the
same principles described above for string harmonics, brass players
are able to utilize subtle shifts in embouchure and air pressure to
play various upper partials on any given fundamental pitch. This
basic principle also applies to woodwind instruments, where an octave
key, register key, or shift in air stream (flute) allows access to
higher pitches within the given harmonic spectrum. Moving the
fingers does the rest. (Compare to the harmonic series diagram
above. On brass instruments, the fundamental is a
so-called pedal tone.)
Having taken a re-e-eal quick look at the
properties of sound, we are now ready to take a brief look at a few
historical tuning systems ... another fascinating topic. Our
present-day equal tempered scale has certain disadvantages, but
the advantages seem to have outweighed most drawbacks for the past few
hundred years. In order to better understand and appreciate
"equal temperament," we will first take a look at a couple of the
earlier systems. So you are aware, tuning systems are not
few in number, and not a recent development. Dozens upon
dozens of tuning systems have been proposed and utilized to varying
degrees throughout history.
Yep, named after our Greek friend, Pythagoras (c. 500
B.C.). Philosopher/mathematician Pythagoras taught that music
and arithmetic were not separate, and that the understanding of
numbers was key to understanding the whole spiritual and physical
universe. Not surprisingly, through the measurement of vibrating
strings, Pythagoras or his followers developed ratios to represent
intervals. The general principle is: if the frequency
of a tone is X, the frequency of its octave is 2X; the frequency of
the fifth (above the octave) is 3/2X, etc. The frequency of
vibration of a string is inversely proportional to its vibrating
|Frequency ratio: (compare to the harmonic
2:1 octave (twice the vibration rate; 1/2 the vibrating
5:4 major third
HOW OUR SCALE WAS DERIVED
By using the ratio 3:2, the pure (beatless)
fifth, Pythagoras enabled construction of the diatonic scale
through successive fifths when reduced to a single octave. Chromatic
pitches may be obtained simply by extending the succession of
fifths. At this point, a rather significant problem
arises. Because of the Pythagorean scale derivation utilizing
the pure fifth, enharmonic pitches are not equivalent! In
other words, F# does not sound the same as Gb, for
example. As we will discover under equal temperament, our
present-day tuning system utilizes a fifth which is slightly out of
tune ... Pythagorean tuning becomes less palatable as music
becomes more chromatic.
Diatonic/chromatic scale derived from Pythagorean
SOME FAMILIAR TUNING SYSTEMS
The comparisons below indicate measurements in cents.
The cent is equal to 1/100 of a semitone, or half step, in equal
temperament. For example, the octave contains 1200 cents, or 100
cents representing the distance between each half step. This
measurement was not available until the mid-nineteenth century, but
has become a standard unit of acoustical measurement, and is most
helpful in comparing tuning systems and intonation.
Following the system attributed to Pythagoras, the
just intonation model will be that of Bartolomeus de Pareja. The
meantone temperament model will be the one proposed by Pietro Aron.
And the well tempered system will be that of Italian organist Vallotti.
It has been conjectured that Vallotti's 1/6-comma scheme, or some
variation thereof, was the system which Bach utilized for The
Well-Tempered Clavier (the set of pieces in each key designed to
display the benefits of well temperament). (Comma is the
term for a specific tuning error.)
Comparison of various tuning systems in (modern)
SYNTHESIZER SETTINGS FOR TUNING SYSTEMS
The synthesizer settings below are designed to emulate
the designated tunings. This experiment would not be so easy
without digital technology!
The numbers represent deviations in cents for each
note. +12 means twelve cents higher than our present pitch for
that note. -08 means eight cents lower than our present pitch
for that note.
Play the pure fifth, from C to G.
Compare to equal temperament tuning, which is more narrow by two
cents. Other fifths are pure as well, except for the wolf.
Play the wolf, from G# to Eb.
Pretty nasty, eh?
Play the primary triads: C, F & G.
Compare chromatic-relationship triads.
Compare the various sizes of whole steps:
C-D, D-E, F-G.
Note that this temperament utilized the tempered
fifth, each narrowed by five cents. Meantone fifths are 697
cents. Pure fifths are 702 cents.
Locate the pure (beatless) major thirds.
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