The Oldest Magick
The Miraculous Coincidence
Lew Paxton Price
Many years ago the most ancient one who teaches us all, in Her dispassionate yet devoted way, provided a demonstration that I still see as proof of the existence of the Eternal Intelligence-or at least the miracles of long-chance coincidence. I attempt to show it to the reader. It may be too subtle for some of you to grasp at first, but don't give up. You will see it all someday.
Look at the circular tempered scale. It is composed of the roots of two, so that frequency increases as one moves counterclockwise about the circle, and doubles (goes up one octave) with each complete revolution. We can use any place as the starting point, any key, or any fraction of a step between keys, and this phenomenon still occurs; so any frequency that we assign as our starting point will do. We can divide a circle into twelve sections, one for each note, and number them from zero to twelve (overlapping the numbers so that the same section we call zero is also twelve). We assign the pieces certain roots of two as multipliers according to how far each piece is from the starting point.
Thus, we assign the zero to one piece as our reference point. The number one piece will be one-twelfth of the way around from the zero point, so we assign it the twelfth root of two (the number which, when multiplied by itself twelve times, gives us a product of two). The number two piece is one-sixth of the way around, so we assign it the sixth root of two (the number which, when multiplied by itself six times, gives us a product of two). The third piece is one-fourth of the way around, so we assign it the fourth root of two; and so on all the way around. Piece number twelve is the same piece we started with, so we are twelve-twelfths or all the way around; thus we assign it the first root of two (the number two itself) as a multiplier.
0is assigned 2 = 1.0000
1is assigned 2 = 1.0595
2is assigned 2 = 1.1225
3is assigned 2 = 1.1892
4is assigned 2 = 1.2599
5is assigned 2 = 1.3348
6is assigned 2 = 1.4142
7is assigned 2 = 1.4983
8is assigned 2 = 1.5874
9is assigned 2 = 1.6818
10 is assigned 2 = 1.7818
11 is assigned 2 = 1.8877
12 is assigned 2 = 2.0000
If we choose any frequency as the starting frequency (the one at the zero point), we can use the multipliers to arrive at the correct frequencies for the notes in the octave beginning with the starting frequency and going to the octave above it. For instance, if we use the frequency of 440 Hertz (the modern standard for the note A above middle C), we can find the frequencies of the notes from A above middle C to the A above that A:
Reference Multiplier Frequency Note
440 x 1.000000 = 440.000 A
440 x 1.059463 = 466.164 A#
440 x 1.122462 = 493.883 B
440 x 1.189207 = 523.251 C
440 x 1.259921 = 554.365 C#
440 x 1.334840 = 587.330 D
440 x 1.414214 = 622.254 D#
440 x 1.498307 = 659.255 E
440 x 1.587401 = 698.456 F
440 x 1.681793 = 739.989 F#
440 x 1.781797 = 783.991 G
440 x 1.887749 = 830.609 G#
440 x 2.000000 = 880.000 A
This tempered scale produces precisely the same frequencies for all the notes as long as we start with a frequency we know to be in the scale. For example, if we were to begin with the frequency for C (523.251 Hertz) we would compute the following:
Reference Multiplier Frequency Note
523.251 x 1.000000 = 523.251 C
523.251 x 1.059463 = 554.365 C#
523.251 x 1.122462 = 587.330 D
523.251 x 1.189207 = 622.254 D#
523.251 x 1.259921 = 659.255 E
523.251 x 1.334840 = 698.456 F
523.251 x 1.414214 = 739.989 F#
523.251 x 1.498307 = 783.991 G
523.251 x 1.587401 = 830.609 G#
523.251 x 1.681793 = 880.000 A
As you see, the frequencies for the notes do not change even though we started with a different note. This is typical for the tempered scale. And we could have chosen a 60 note scale or even a 360 note scale as the Chinese once did. All that is necessary is dividing the circle into 60 or 360 parts. Then we would have used roots of two again to arrive at the proper frequencies. For instance, we can compute the second notes of these two scales: the 60th root of two is 1.011619 and that times 440 is 445.113; and the 360th root of two is 1.001927 and that times 440 is 440.848. We could choose a scale with an infinite number of notes; then playing one octave would then sound like sliding a finger up the vibrating string of a violin, or moving the slide on a slide trombone. And with each full revolution around the circle, we would move up exactly one octave. This is the type of scale we call tempered.
Now consider the human hearing mechanism. Apparently the human ear and brain are not capable of working with logarithmic scales such as the tempered scale. But the natural shifting of frequencies caused by the planets in motion is logarithmic, like the slide trombone. The fact that life forms do not seem to respond to the logarithms of nature calls into question the theory that the cycle sensing mechanisms and internal clocks of life forms are tied to nature's frequencies. (Planetary motion through the zodiac is considered to create local environmental phenomena, via ionosphere/magnetosphere electromagnetics). What is really happening?
The musical decoding mechanism of the human ear and brain is based upon frequency comparisons in which frequency peaks coincide often. When the peaks of one waveform coincide often with the peaks of another waveform, the notes of the waveforms are said to be harmonious. The notes C and G, for instance, are harmonious because every second peak of C coincides with every third peak of G. The ratio of the frequencies for C and G is . This fractional or ratio system for notes looks like this:
Number of Ratio of Note to Decimal
Note in Circle Reference Note Multiplier
0 1/1 1.000000
1 20/19 1.052632
(Pythagorean Scale)
1 16/15 1.066667
(Just Scale)
2 9/8 1.125000
3 6/5 1.200000
4 5/4 1.250000
5 4/3 1.333333
6 45/32 1.406250
or 1.4143
7 3/2 1.500000
8 8/5 1.600000
9 5/3 1.666667
10 9/5 1.800000
11 15/8 1.875000
12 2/1 2.000000
The notes formed by the Pythagorean and just scales do not fall into the best place for a note between 0 and 2. The ear would normally prefer a note with a ratio of lower numbers such as . However, because the ratio forms a note that is closer to the place between 0 and 2, the ear prefers the Pythagorean ratio over the just when playing the twelve-note scale.
Actually, the numbers in the ratio are so large as to be heard as inharmonic. As all the notes in the scale should be harmonic with one another to some degree, this note should be compared with notes other than the reference note. If it is measured as of the ideal (tempered frequency) for note 1, it would have a multiplier of 1.4127. If it were measured as of the ideal note 2, it would have a multiplier of 1.4031. Or as of the ideal note 3, it would have a multiplier of 1.42704. As all these combinations need to work with one another, an average might be best for our use. This average is about 1.415846.
We can now compare the system that the human ear prefers to the natural or logarithmic system (tempered scale).
Multiplier Multiplier
No. (ear) (Tempered) Difference %
0 1.000000 1.000000 0 0
1 1.052632 1.059463 .006831 .64
2 1.125000 1.122462 .002538 .23
3 1.200000 1.189207 .010793 .91
4 1.250000 1.259921 .009921 .79
5 1.333333 1.334840 .001507 .11
6 1.415846 1.414214 .001632 .12
7 1.500000 1.498307 .001693 .11
8 1.600000 1.587401 .012599 .79
9 1.666667 1.681793 .015126 .90
10 1.800000 1.781797 .018203 1.02
11 1.875000 1.887749 .012749 .68
12 2.000000 2.000000 0 0
Note that the percent difference between the humanly sensed scale and the logarithmic (tempered) natural scale is almost negligible. The human ear/brain sensing/decoding mechanism for cycle sensitivity is very nearly the same as the theoretically perfect sensing/decoding mechanism. This, to me, is miraculous.
Now we can go a step farther and look at the ratios in our humanly sensed scale. The rule is, the fraction that is the ratio between harmonic frequencies must have a low number numerator and a low number denominator. The possible ratios and comments about each follow:
1/1 This simply means that we can hear the same note on two separate instruments and recognize that the note is the same.
2/1 This means we can recognize a note an octave above another.
2/2 = 1/1.
3/1 = 3/2, but an octave higher.
3/2 This is 7th section note, or the one we call the musical 5th.
4/1 = 1/1, but two octaves higher.
3/3 = 1/1.
4/2 = 2/1.
4/3 This is the 5th section note, or the one we call the musical 4th.
4/4 = 1/1.
5/1 = 5/4, but two octaves higher.
5/2 = 5/4, but one octave higher.
5/3 This is the 9th section note, or the one we call the musical 6th.
5/4 This is the 4th section note, or the one we call the musical 3rd.
5/5 = 1/1.
6/1 = 3/2, but two octaves higher.
6/2 = 3/1.
6/3 = 2/1.
6/4 = 3/2.
6/5 This is the 3rd section note, or the one we call the musical minor 3rd.
6/6 = 1/1.
7/1 = 7/4, but two octaves higher.
7/2 = 7/4, but an octave higher.
7/3 = 7/6, but an octave higher.
7/4 This is a note that would go between the musical 6th and the sharp above it. It is probably a legitimate harmonic to a lesser extent but does not fit with our scale.
7/5 This is the harmonic/inharmonic that would fit as the 6th section note and do better than the 45/32. However, it does not fit the interval properly in our scale.
7/6 This is the note that would go just over our musical 2nd. However, it does not fit the interval properly in our scale.
7/7 = 1/1.
This could go on into the higher numbers, but even at the sevens the sound begins to be inharmonic because the seven is too high a number between wave crests to recognize as harmony.
It is possible to tune flutes to scales using such odd notes. The result would be less than pleasing as 7/6 is bettered by 6/5, 7/4 is bettered by 5/3 with 9/5 better fitting the next interval, and 7/5 is actually about what is used for the 6th section note (and it does not sound very harmonic with the reference note).
Today the schools claiming some repository of the old knowledge say that music and astrology are based upon the same phenomenon. The fact that the human mechanism senses harmonies so closely akin to the natural ones found in the circular tempered scale when superimposed upon the zodiac, and the fact that certain angles (aspects) between planets coincide with certain harmonies (which seem to lead to the results one would expect from those same musical chords on the circular scale), tend to convince me that herein lies the miracle found within the cycle sensitivity of life forms. Perhaps the old repositories of knowledge reveal the truth.
1992 by Lew Paxton Price | 
