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Musical Frequencies and Other Questions

Asked by on August 4, 1996:
In a sphere...why do we ignore the other 3/4 of the sphere and concentrate only on the positive and negative's of the X, Y, and Z axis ? Are the other areas less important? If so Why?

#2 In the mathmatical progression on musical harmonics why do we only busy ourselves with the pleasant tones. Do these others not pose possibilities to doorways not yet opened?

#3 Is mathmatics so sacred that to question its validity is to bring upon oneself damnation and excomunication? Am I missing something here?

I'm going to answer your second question first. The question of what is and isn't a "pleasant" tone, and what possibilities the others offer, is really not a mathematical question at all. It comes down to a matter of physics. I'll try to briefly indicate the physics involved.

Suppose you start with one tone at a certain frequency, and then sound another one along with it. What will the result be? Well, if we think of both tones starting off together at the beginning of their respective pulses, a certain amount of time will elapse before they get back "into synch" with each other again, and during that time there will be an ever-changing sound.

Consider two examples:

  1. A frequency of 200 beats per second, and one of 300 beats per second.

    The second tone will complete three full cycles in the same amount of time (1/100th of a second) that the first one takes to complete two cycles, so there is a changing sound-pattern that lasts for 1/100th of a second, which then repeats itself over and over.

    I've illustrated this in the picture below. The top shows the first wave, the middle shows the second wave, and the bottom shows the combination (sum) of the two.

                           |        ******            ******   |        ******
            First tone:    |     ***      ***      ***      ***|     ***
                           ******            ******            ******
                           |                                   |
                           |     ****        ****        ****  |     ****
            Second tone:   |   **    **    **    **    **    **|   **    **
                           ****        ****        ****        ****        ***
                           |                                   |
                           |                                   |
                           |        *                    ***   |        *
            Combination    |     *** **                **   *  |     *** **
            of the two:    |           *** ******     *      **|           ***
                           |   **         *      *****         |   **
                           ****                                ****
                           |           |     |     |           |
              Time:        0         1/300  1/200 2/300       1/100

    Although the first wave repeats itself after 1/200th of a second, and the second wave repeats itself after 1/300th of a second and again after 2/300th of a second, it's not until 1/100th of a second that the combined pattern starts repeating itself.

  2. For our second example, Suppose that one had a tone of 200 beats per second and one of 303.91 beats per second. If you draw a diagram similar to the above, you'll find that it takes a full 100 seconds (20,000 beats of the first tone and 30,391 beats of the second tone) before the pattern starts repeating itself again.

These have quite a different effect on the ear. In the first case, the combined tone consists of a 1/100th-of-a-second pattern being constantly repeated. The fact that the 1/100th-second fluctuation has a complicated shape (rising all the way up, falling partway, rising a little, falling again, rising all the way, then finally falling all the way) gives the tone richness. The fact that each fluctuation lasts only 1/100th of a second means that it isn't perceived by the ear as causing the sound to change over time; it sounds the same from beginning to end. If you sound the two tones together for 10 seconds, the sound at the end of those 10 seconds will be the same as the sound at the beginning.

However, in the second case, you have an ever-changing sound that lasts 100 seconds before repeating! If you sound the two tones together for 10 seconds, the sound at the end of those 10 seconds may be quite different from the sound at the beginning.

For musical purposes, the first kind of sound has been traditionally judged "pleasing" to the ear, because it's a tone that's rich in texture and always sounds the same. The second will be full of discernible "beats" and other long-term modifications to the sound, that most people judge "ugly".

Mathematics comes into play because, to figure out the length of time it takes before the pattern repeats itself, you need to find when, if both tones start out beginning a pulse together, they next get back in phase with each other (i.e., when they next begin a pulse together). When that happens, tone 1 will have completed some whole number "a" of pulses, and tone 2 will have completed some whole number "b" of pulses, so you'd have

(a)(time taken by one pulse of tone 1) = (b)(time taken by one pulse of tone 2)
a/(frequency 1) = b/(frequency 2)
(frequency 1)/(frequency 2) = a/b.
Thus, if the ratio of the two frequencies is expressible as the ratio of two small whole numbers (as our first example was; the frequencies were in a 3/2 ratio), the combined tone will be a short pattern repeated often. But the larger the numbers a and b have to be, the longer the pattern is.

It is when a and b are small numbers like 1, 2, 3, and 4 that you get the "purest" combined tones with the shortest fluctuations. That is why the standard musical scale is based on intervals such as a 2:1 frequency ratio (one "ocatve"), a 3:2 frequency ratio (one "fifth") and a 4:3 frequency ratio (one "fourth"), along with a 2:1 frequency ratio (one "octave"). One can employ mathematical analysis to figure out other notes of the scale from these.

The standard scale is obtained by starting with one tone and continually taking 3/2 ratios until one gets back to something very close to an exact number of octaves above the original tone. It takes 12 times before you get back, and this gives the standard 12-note system. (One doesn't get back exactly, but one gets back close enough that the ear doesn't notice the difference).

There is no mathematical reason at all why one cannot employ other scales. As I said, it's not a mathematical question what scale one wants to use. The standard one just happens to be the one that involves the most intervals with the "purest" tone (shortest length of the fluctuations in the tone). The same kind of mathematical analysis would apply to any musical system one wanted to consider. The study of the musical tones of our standard scale just happens to be one particular application of the mathematical theory of numbers; it is not itself a part of mathematics. The same mathematics would apply equally well to any other musical system.

Now, for your first question: I really don't know what you're referring to. First of all, there's no natural way to partition a sphere into quarters. You can, however, partition a sphere centred a the origin into eighths, called octants, based on the signs of the coordinates x, y, and z; perhaps that is what you are referring to.

If so, then the so-called "first octant" (where x, y, and z are all positive) is no more special than anywhere else on the sphere. The only reason one might consider it separately is if one is applying some technique where the sign of x, y, or z makes a difference in the precise details of the technique. This happens sometimes in multivariable calculus. In cases like this, one often gives the argument for the first-octant case and then, rather than writing it all out again for the other seven octants, leaves it as an exercise to the reader to make the appropriate changes to do the other cases.

Also, if one wants to give a problem involving, say, part of a sphere, to test a student's understanding of how to handle surfaces with boundaries, one would tend to choose the first-octant part so that the student can focus on the problem at hand and not have the main idea obscured by extra fiddly little details like putting all the minus signs in the right place.

But, mathematically, the first octant is no more special than any other part of a sphere.

As to your third question: mathematics is concerned with what is and isn't true. The entire basis of mathematics is asking questions, and coming up with answers that decide the question beyond any possibility of doubt through strict, logical argument. So there's nothing at all wrong with the concept of "questioning" in mathematics; it's what the subject is all about.

However, it's also important to realize that mathematics is a matter of fact and proof, not of opinion; one cannot argue against a proven fact, unless the proof is incorrect and one can demonstrate where the incorrectness lies. It's also important to realize that a lot of what masquerades as "mathematics" really is not, but rather is a combination of mathematics together with some assumptions about some real-life things. Perfectly valid mathematics can sometimes be used to draw incorrect conclusions about real-life situations, through error in what one was assuming about the situation. Questioning those conclusions is not at all the same thing as questioning the validity of the mathematics.

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