NMR ISSUE 5

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The Oldest Magick

Five: The Eternal Scale

Lew Paxton Price

There it was. She had done it again. Why did it work? Why did the Creator have such a law? And wasn't it was wonderful that she, Ashar, the current shaman for the clan of the spirits-who-speak, the one whose great grandmother first to make the proper lengths for tuned pipes, would now be the one to discover the unit length?

The idea had come to her last night after a dream. She had been climbing a ladder of bamboo and each rung was shorter as she climbed. She would take her measuring stick and laid it against each rung as she passed and each rung would be a precise multiple of the measuring stick's length. She had laid there in the dark remembering the dream over and over so that she would not forget it. Then she had meditated upon it, as her mother had taught her to do.

Then, that morning, she had taken a length of straw and laid it beside each of the lengths of pipes in her musical instrument. The first time she did not use the correct length of straw. Nor did she the next few times. She began to make the straw shorter and shorter, on the twelfth attempt she succeeded. Now each pipe was almost precisely an even multiple of the straw's length.

In the distant past, the people of the clan had made many lengths of pipes. Some did not sound as if they belonged. Others definitely did not belong with a particular set of pipes. Some sets of pipes did not fit well with the other instruments. All those that fit seemed to use pipe lengths that were exact multiples of the length of her unit straw . . ..

The human ear determined what sounded best. This, over a period of time, determined the musical scale in its fundamental state. Before the discovery of what we call harmony, its principle was at work determining our musical scale. Each note of the scale has a simple mathematical relationship to the note that begins the scale.

The first thing that Ashar's clan had discovered, about what we call panpipes, was that pipes of the same size sounded about the same way. The length of one pipe related to another of the same length we call a ratio of one to one, or a fraction of 1/1. The numerator (top) and the denominator (bottom) of the fraction are the lowest possible number, the number we call one.

That morning Ashar compared pipes which sounded similar, though one was definitely higher than the other. She discovered that the low pipe was twice as many `unit straws' long as the higher pipe. The lower pipe was sixty unit straws long while the higher pipe was thirty. This we call an octave. A pipe half as long as another is one octave higher and the ratio of the pipe is 2/1. Two is the next higher number after one, so both the numerator and the denominator are still very low numbers.

In any set of six pipes, the pipe that sounded almost the same as the lowest pipe was forty unit straws long. This is a ratio between low pipe and high pipe of 60/40, which reduces to 3/2. Again, both three and two are low numbers. And there was a pipe forty-five straws long (ratio of 60/45, or 4/3), one of thirty-six straws (ratio of 60/36, or 5/3), and one of forty-eight straws (ratio of 60/48, or 5/4).

These were the pipes that sounded best together because the human ear hears the frequency ratios that have low numbers as harmonic. Even when notes are not played simultaneously, the most obvious melodies are those composed of harmonic notes. So Ashar had a set of six pipes. The lowest was twice as long as the highest. The next lowest was 4/5 of the length of the lowest. The length ratios in order, from low to high, were like this: 1/1, 4/5, 3/4,

2/3, 3/5, and 1/2. In other words, the second pipe was 4/5 of the first in length. The third was 3/4 of the first in length. The fourth was 2/3 of the first in length. The fifth was 3/5 of the first in length, and the last (octave note) was 1/2 of the length of the first.

If we begin with middle C, the notes in this musical scale would be middle C, E, F, G, A, and the C above middle C. This is called a pentatonic scale (one that has five notes) because it is composed of five notes: C, E, F, G, and A (the C above middle C is not a separate note, it is the same C on a different level). The numbers of unit straws that fit these notes are 60, 48, 45, 40 and 36. The basic numbers that constitute the mathematical solution for the basic unit straw length are 3, 4, and 5 (3 x 4 x 5 =60, which is evenly divisible by all three numbers).

We could say that C equals 60, E equals 48, F equals 45, G equals 40, and A equals 36. All the notes are harmonics of C and the lowest note on the most primitive set of pipes is the dominate note—the note that establishes the key. The dominant here is C, so each tune played in the key of C would repeat C over and over again. The melody and would begin and end with C.

Interestingly, similar musical scales would probably evolve anywhere in the universe as long as the sensory organs involved (e.g., the ear) can recognize fundamental mathematical relationships. They always will if they can distinguish a loud sound from soft one (i.e., changes in pressure). It is the beat of these changes, these peaks and valleys of energy, that allow the ear to hear musical ratios.

Sixty was the mathematical base of the oldest culture we know, the Sumerians of the area about the Tigris and Euphrates Rivers. From the Sumerians we have the hour divisible into 60 minutes and the minute divisible into 60 seconds. We have each degree of the circle divided into 60 minutes of arc and the minute of arc divided into 60 seconds of arc. Sixty was a sacred number to the ancients. It was a product of 3, 4 and 5 (3 x 4 x 5 =60). When multiplied by the perfect number (six), 60 became the number of degrees in a circle (3 x 4 x 5 x 6 =360, or 60 x 6 =360). Six is the perfect number because it is the only number which is both a product and sum of the same numbers (1 x 2 x 3 =6 and 1 +2 +3 =6).

Ashar noticed that the pipes she held were not a regular pattern from the longest to the shortest. There was a noticeably large difference between the first and second and between the fifth and sixth pipes. Perhaps an additional pipe between the first and second and between the fifth and sixth pipes would be a good idea. She set to work . . ..

Sixty was all right for a unit straw multiple for basic pentatonic panpipes, but there were gaps between C and E, and between A and C that were too large. Although not exactly harmonically appropriate, the closest ratio that could be used for what we call and is 8/9, and the closest ratio for what we call B is 8/15. When these two notes are filled in, we have the major diatonic scale in C. This is the scale we sing as: do, re, mi, fa, so, la, ti, do.

The unit straw (unit length) for this new diatonic scale is based on 360. By dividing the low pipe into 360 equal lengths, we determine a new unit that applies to all the notes. Now C equals 360, equals 320, E equals 288, F equals 270, G equals 240, A equals 216, and B equals 192. We have a hint that the musical scale might best be expressed with a circle, as we shall see in the next issue.

© 1988 by Lew Paxton Price