Strings, Vibrational modes
Overtones and just tempering.

Consider a taut string (or wire) fixed at both ends; a piano string for example, or a violin string: we'll ignore the effects of the bridges that happen to exist in both instruments. The string set in motion can vibrate in a number of distinct ways called modes of vibration (or eigen modes or eigen functions). Generally, when the string is set in motion, it actually vibrates in some weighted linear combination of these modes. But first it's easier to look at modes individually.

The modes of vibration are related to the nodes or fixed points of the vibration; that is, excluding the end points which are given. In the fundamental mode, there are no nodes. The entire string arches up and down with the maximum displacement from its position of rest being at the center. The maximum upward outline of the string and the maximum downward outline of the string form what is called the envelope of vibration. As the string vibrates in this mode is causes compresional waves in air that we perceive as sound. The frequency of that sound is directly related to the frequency of vibration of the string, so we can speak in terms of the string vibrating rather that in terms of the sound and still be talking about the sound. Say the fundamental mode of vibration is set up to be the pitch called middle C on the piano.

The envelope of the second mode of vibration (sometimes called the first overtone), has arches, above and below which meet at the end points and at the center of the string. There is then a node of the vibration at the center of the string, and the node has divided the string into two equal parts. The vibration produced is almost like cutting the string in half and starting over. The difference being that when the right half of the string in up, the left half is down. It turns out that the frequency of vibration will be doubled; this corresponds also to a C pitch but an octave above middle C; we'll call this C' and the apostrophe as a postfix operator that raises the designated pitch by an octave.

Violinists and flutists often use a technique called "playing in harmonics", which suppresses the fundamental mode of vibration and therefore sounds and octave higher. Describing how a flutist accomplishes this is hard, but describing what a violinist does is easy: the pinky is extended so that it reaches into the middle of a stopped (or open string) and rests ever so lightly there to produce the node and thus suppress the fundamental mode.

The envelope of the third mode of vibration has three arches, above and below, and 2 nodes, which divide the string into 3 parts equal in length. The actual vibration can be described as

               up     down   up
             |      .      .      |
              down     up    down

                   G  =  (3/2) C

The envelope of the fourth mode of vibration has four arches, and three nodes dividing the string into four equal parts.

               up     down   up    down
             |      .      .      .     |
              down     up    down   up

Clearly one can continue dividing the string by an increasing number of nodes, thus generating the overtone series for any given pitch. From a musical point of view there is a point of diminishing returns and one needs to consider only a finite number of overtones; 16 is not a bad place to stop, from a purely practical standpoint; I will list the first 20 pitches associated with a fundamental of C.

NB (September 1, 2000): In the following, regarding the names of the pitches, the actual method of constructing the C major scale, and the tuning of a physical keyboard instrument in just temperament, there are some very misleading and incomplete statements. Be wary. I hope to have these problems rersolved soon, along with more careful definitions of: overtones, harmonics, and (upper) partials. Please bear with me.

   C C' G' C'' E'' G'' Bb'' C''' D''' E''' 

   1 2  3  4   5   6   7    8    9    10   
                       *

   F''' G''' A''' Bb''' B''' C'''' C#'''' D'''' D#''''E

   11  12    13   14    15   16    17     18    19    20
   *                   *               *

        * Bad w.r.t. well tempering 7 is flat relative to keyboard Bb
          In fact, all above 11 except the C at the 16th overtone
          are not all that good relative to an equal tempered
          keyboard.

The first observation that can be made about the overtone series is that its lower elements together form a MAJOR seventh chord, lending weight to the words of taxonomic distinction "major" and "minor". A major chord is somehow more natural in the context of simple vibrating things.

If we tune the C major scale according to the overtones of C, using also the overtones of the F (a perfect fifth below) and G (a perfect fifth above).

                     C  =  (1)    C
                     D  =  (9/8)  C
                     E  =  (5/4)  C
                     F  =  (4/3)  C
                     G  =  (3/2)  C
                     A  =  (5/3)  C
                     B  =  (15/8) C
                     C' =  (2)    C

Keyboard and tuning to C, we are also ok for the harmonically closest keys of G major (with only F#) and F major (with only Bb). The next Generate from F and from G F F' C' F'' A'' C'' Eb'' From this we get the ratios for A and F given above 1 2 3 4 5 6 7 G G' D' G'' B'' D'' F''

The just tempering scheme ulimately based on the perfect fifth runs into problems in constructing a circle of fifths because the circle does not close. The gap or discrepancy in frequency that appears at 12 consecutive fifths and 7 octaves down is known as the Pythagorean Comma.

The various schemes to reconcile the Pythagorean Comma, that is to close the circle of fifths are called cyclic temperaments. Well or equal tempering is one of them. There have been other schemes to divide the octave not only into 12 parts but also into 5, 14, 16, 19, 31 and 53 parts. These are all cyclic temperaments. The ancient Greek tuning was not cyclic and is one of the linear termperaments. Various linear temperaments have been used, throughout musical history under circumstances where the music was not essentially harmonic but linear; hence there was no need to define intervals of relative consonance and dissonance. [Helmholtz 1877] , appendix XX.

There is a mathematical story to tell associated with this construction of overtone series that centers around a very important theorem of Fourier. Cast into the current context the theorem says that any vibration that the string is capable of can be expressed by a suitable addition of the fundamental modes each with some weighting coefficient. The actual mathematical theorem takes into account that the vibrational modes can be given in terms of sinusoidal functions (sine and cosine) of trigonometry. Even among mathematicians, Fourier's theory is also called "Harmonic Analysis".

What is so bad, harmonically speaking if we approximate musical notes with an equal temperament?
It is clear that each string of a piano will have its set of overtones whose fequencies are determined by simple ratios of whole numbers. To get the G above C as above, multiply the frequency of C by (3/2)-1.500. The equal tempered G that is now there would use a factor of 1.498. So now the the equal tempered G and its overtones will beat with those of the C, where this would be minimized if the G were tuned using the 1.5 factor. Beats happen when two tones are played together that are just slightly different in frequency. The double angle formulae from trigonometry are:

        sin( A + B )  =  sin( A ) cos( B ) + cos( A ) sin( B )

   and then also

        sin( A - B )  =  sin( A ) cos( B ) - cos( A ) sin( B )

   Adding these two formulas gives

        sin( A + B ) + sin( A - B )  =  2 sin( A ) cos( B )

   Change variables by letting

        u  =  A + B,  v  =  A - B

   so that inverting and solving for A and B in terms of u and v

        A  =  (1/2)(u + v)
        B  =  (1/2)(u - v)

   Then substituting in the last trigonometric equation gives

        sin(u) + sin(v)  =  2 sin[(1/2)(u + v)] cos[(1/2)(u - v)]

   The expression sin[(1/2)(u + v)] is a sine wave with a frequency
   that is the average of the two frequencies u and v.  If u and v
   are close then (u-v) will be very small and the factor
   cos[(1/2)(u - v)] modulates the average sine wave with a
   frequency that is low with respect to the average frequency.
   The amplitude of the sine wave, hence it's loudness swells
   and dimishes.  This swelling and diminishing of loudness is
   called beating.

   The more beating going on, the less consonant is the perception
   to the ear.  Therefore, equal temperament makes fifths and fourths
   (the inversions of fifths) less consonant.  Since the very basis
   of harmony is the interval of the fifth, some of the consonance
   of all harmony has been compromised by equal tempering.

   Question: If the close frequencies u and v are both above the
   range of human hearing, while their difference is within the
   range of human hearing, will the modulating beat frequency be
   heard as a sound?

   Answer: yes.

With the advent of computers and computer software powerful enough to handle the digital and analog manipulations of sound, the music of the future, providing there is one, can be free of the equal temperament that has been imposed on western music by the piano keyboard and still allow free modulation to maintain natural harmonic relations when wanted, and at the same time allow for music that can also be expressed using the rich nonharmonic linear language that has been created in many other cultures. Should this revolution come about, music may itself still not be universal, but at least it will have a much richer and universal alphabet.