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Topic: John Keely's
Laws of Harmony
Section: Law of Harmonic Pitch
Table of Contents to this Topic
"Any aggregate in a state of vibration develops in addition to its fundamental pitch a series of vibration in symmetrical sub-multiple portions of itself, bearing ratios of one, two, three, or more times its fundamental pitch."
Commentary July, 1986
What we begin to see through this law is the unfolding of the inner relationships between the evidenced eigenfrequencies of an aggregated mass and the subtler, less evident vibrations of its makeup. Quite clearly we have seen and discussed the vibrations and the fundamental origins of those vibrations. This law indicates there are more vibrations not necessarily perceived as such. We perceive here a series of motions that may be categorized as a quality perpetuating itself by simple number ratios, i.e., a "quantization" (condensation and rarefaction) seems to be taking place. What are being described here are the harmonic pitches that arise from the fundamental pitch of the aggregate. However, in reviewing last months law, "The Law of Cycles," we see that there are also, in any given aggregate, another parallel group of vibrations. These other vibrations, we saw, are developed as BEAT frequencies, summation or resultant tones. See issue 9 for examples.
So here we have a description of two types of frequencies arising from the same fundamental or source of all other vibrations as appear in a given aggregate. Harmonic pitches and summation or difference pitches make up the whole of the vibrating spectrum.
The next step in dealing with these multiple frequencies is to determine how and why any given vibration of frequency arises from another or in this case the fundamental. The answer lies in Music Theory. A fundamental will, when sounded, give rise to its harmonic and the other summation or beat tones. Any sophomore music theory student knows this! The Key question still remains - HOW DOES ONE NOTE BECOME ANOTHER? The creation of harmonics is simply the fundamental dividing itself by two over and over again. The next step is the harmonics sounding together and with the fundamental causing the summation and difference tones. For instance: 2 as a fundamental gives rise to 4 or 8 or 16 as harmonics. 8 minus 2 gives us a difference tone of 6 whose lower harmonic is 3 and several higher harmonics of 9, 12, 18 etc. Where these frequencies match we get higher amplitudes or lower amplitudes as they "mis-match". Below we have a listing done by Sir James Jeans of music theory renown wherein he calculates some of the generated frequencies from a fundamental. The root is C at 64. Ramsay and Hughes both differ from this system.
C 64 x1
C1 128 Harmonic x2
G 192 x3
C2 256 Harmonic x4
C+E 290 Summation
E+G 354 Summation
C+G 320 Summation x5
C-E 34 Difference
E-G 30 Difference
C-G 64 Difference
C3 384 Harmonic x6
G3 576 Harmonic x9
C+E+G 482 Summation
C2+E 418 Harmonic + Summation
E2+C 452 Harmonic + Summation
E2+G 516 " "
G2+E 546 " "
G2+C 512 " " x8
C2+G 448 " " x7
The above list of frequencies is only partially complete. It is revealing to note that all these notes are generated from sounding the single note of C (64 cps). This single fundamental gives rise to its harmonics C1, C2, etc. and all the rest are derived from it "spontaneously".
Therefore, when we say no thing can or does exist in and by itself, the evidence is now before us. All things are related SYMPATHETICALLY. One can even say that all the above notes are ONE, because they have arisen from ONE source, C. A more accurate statement would be that they are from ONE family or ONE origin, but not ONE single separate unit. This evolutionary process of musical notes will be further explored in future issues.
Commentary September, 1989
This law of one of the more recognized and accepted laws in use today. All fields of engineering use it extensively in calculations both derivative and functional.
The use of the word pitch is also accurate as a pitch is a relative frequency where frequency alone is simply a number of cycles per second. Again the use of musical terminology is fundamental to the in-depth study of vibratory phenomena. Being relative is the state of relating to something else - another frequency, the fundamental or key note, in this case.
It is not known when electronic and other engineers began using harmonics in their work. Certainly this usage began some time ago. What is even more puzzling is why hasn't this well-known law of physics been identified as a law? When something always and invariably takes place as a result of a given action it is without a doubt a law. A vibratory occurrence will always and at all times produce these harmonics. They can be calculated by doubling the frequency and the subharmonics can be calculated by halving the frequency. The Electromagnetic Scale commonly seen in school class rooms and engineering departments is based entirely on this concept. These scales begin with 1 cycle per second and doubles at each octave resulting in a sequence of numbers: 1, 2, 4, 8, 16, 32, 64...ƒ.
All of the above is just dandy - what is a vibration anyway? Is it a periodic motion back and forth or to and fro? Or is it an alternating pulsation? It may be either, both and even other modes of periodic changing of state. Various objects in motion and under consideration may behave in different motions. However, what remains the same in all cases is the object changes state from what may be considered a positive state, to a negative state and also the neutral state. For instance a stretched piano or guitar wire may be said to swing back and forth. Upon consideration it may be seen to also shorten and length as the string increases its swing further away from the neutral or rest position. Thus a simple guitar string vibrates in at least two different directions at 90° to each other. As the string moves towards is furthest distance away from the neutral position the ends tend to draw closer together simultaneously (if they were not rigidly fixed they would approach each other).
Another fundamental principle of harmonic creation is that of tension. Increase the tension the string and the frequency also increases. Therefore as the string swings again to its furthest extension away from the neutral position it tends to increase the tension. as it approaches its furthest extension the frequency again changes to a higher rate.
The law of harmonics even applies to gravity. For every 16 feet an object falls its velocity doubles. Would this mean that gravity is a function of velocity? Accelerate an object to a given velocity (escape velocity) and it will escape the pull of gravity. Velocity therefore may be a rate of change from one polarity (state) to another. Polarity is a difference of state, harmonic, dominant (neutral) or enharmonic.
To say that a complex shaped piece of metal vibrates to a given frequency is misleading. It would, however, be correct to say that the piece of metal does have a fundamental pitch. From this fundamental pitch are developed harmonic pitches, enharmonic pitches and a wide assortment of node configurations. The various wave forms and frequencies constitute a chord of pitches. Thus this complex assortment of wave forms constitutes Keely's "chord of mass."
To sum up a little, even though a vibrating aggregate does indeed create a series of harmonic pitches, it also creates a series of enharmonic pitches. These various forms of evolved pitches constitutes the vibratory composition of the aggregate. Harmonic pitches alone do not reflect the true vibratory nature of any aggregate. They but represent an over simplification of the analysis. Small wonder then that "modern" science has done so little when it could do so much more. Their analysis has been confined to harmonic analysis, without recognizing this function as a law of nature, and have lumped all the other noted frequencies derived from their analysis as unrelated mathematically to the fundamental. These other frequencies have been classified with statistical methods alone. Statistical methods are nothing other than various ways to average indeterminate numbers. Averaging does not recognize the individuality nor interrelatedness of a set of numbers. Music terminology and methodology makes no such error. It recognizes a given frequency for what it is and how it may relate to others. Music (SVP) is broad-minded enough that it can give credit where credit is due. If it didn't music (SVP) would have no basis to exist as its very existence is founded on the idea that all things do exist and do have a meaningful relation to all other things.
This may be the salvation for dogmatic and differentiating science. If an object sounds a fundamental it also sounds all of its harmonics and enharmonics as an infinite chord of frequencies. Radionics has long recognized and worked quite successfully with this fact. Since this chord is really an infinite series of pitches it therefore stands to reason that all things have an infinite effect - not just the effects found within a certain and limited spectrum of frequencies.