MUS 111

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.  

FYI, 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.


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 the loudest.  

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.


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 partials


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 length.

Frequency ratio:  (compare to the harmonic series, above)
2:1  octave (twice the vibration rate; 1/2 the vibrating length
3:2  fifth
4:3  fourth
5:4  major third


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 tuning


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.)

  Pythagorean Tuning Just Intonation Meantone Temperament Well Temperament Equal Temperament
C 0 0 0 0 0
C# 114 92 76 94 100
D 204 182 193 196 200
Eb 294 294 310 298 300
E 408 386 386 392 400
F 498 498 503 502 500
F# 612 590 579 592 600
G 702 702 697 698 700
G# 816 792 773 796 800
A 906 884 890 894 900
Bb 996 996 1007 1000 1000
B 1110 1088 1083 1090 1100
C 1200 1200 1200 1200 1200

Comparison of various tuning systems in (modern) cents


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.

Pythagorean Tuning

  • 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?

Just Intonation

  • Play the primary triads:  C, F & G.  Compare chromatic-relationship triads.

  • Compare the various sizes of whole steps:  C-D, D-E, F-G.

Meantone Temperament

  • 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.

Well Temperament

  • Experiment by playing pieces in various keys to determine a "personality" for various key "colors."

Equal Temperament

  • The octave is divided into twelve half steps of equal size.

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