B. SVAROG *

                                                                        

                                                                  The scriptures of quantum evolution was recorded in a language                                                                 of numbers at the origin moment of our Universe

 

                                                                                                         Timothy Leary‚ The History of Future

 

 

 

 

         An Octave is a main musical term which marks a coincidence of sounds‚ which frequencies correlate as 2:1. Together with that octave is "eight" (in Greek ὀκτώ).

 

         It’s not difficult to understand‚ that the main characteristic of a consonance is two sounds with  ratio of frequencies 2:1 which is incorporated in the nature of a harmonic curve:  its  full period (2p)  contains two half-phases (p)‚ in two full periods four half-phases and so on‚ what gives  under these conditions a coincidence of fluctuations in phase points:

 

                                              

 

 

 

         Together with that a harmonic curve describes oscillation - a simplest type of motion in nature, and mathematically is an elementary periodical function: the curves of other dependencies may to be reduced to sums of the infinite line of harmonicas.

 

         How does the division of interval maximum consonance appear in music 2 - octave - into intermediate intervals? The answer to this question is kept in ancient histories known in musical building as quint system‚ also named after Pythagoras. It has historically occurred‚ that all things are to be found in Greece‚ however there are no grounds to suppose‚ that its principle was not mentioned earlier in Egypt‚ Babylon‚ India and China.

         String fluctuations generate partial tones (overtones) which correlate as round numbers 1:2:3:4:5... The quint division is based on the second harmonic and following the octave consonance with ratio of frequencies 3:2.  How it is proved by experience with strings of different length‚ the simpler in the numeric expression is this ratio‚ the more pleasing are their fluctuations to hear and the fuller is the synergy (phenomena of resonance) between them.

 

         Let’s name an octave interval do - do¹. The step‚ referring to the tonic do with frequency 3:2 is identified as quint and is traditionally marked sol. From the determination of the interval of an octave follows, that the ratio of upper do¹ to the sol will make herewith 4:3‚ and this harmonic ratio is known as a quart or a turn quint. If we make three strings sound‚ adjusted in do‚ sol and do¹ simultaneously‚ then, immediately three intervals, an octave 2:1‚ quint 3:2 and quart 4:3 will be present in a chord. The ancient knew‚ that quint and quart are reversible, that is to say symmetrical comparatively to octaves‚ and if sol is a quint to do‚ than fa is a quint to do¹‚ and back fa comparatively to do forms a quart‚ but sol is a quart to do¹.

         This proportion can be presented in the manner:  do: fa: sol : do¹  = 1: 4/3 : 3/2 : 2‚ and it means the first harmonic pattern of an octave, the so-called  adjustment of an Orpheus harp. The present expression contains all known in antiquities correlations like arithmetical, geometric and harmonic proportions, as well as the principle of gold section. Its two average members are herewith correlating as 9:8‚ that is to say form an interval of a natural whole tone of Fig. 2. Thereby‚ adjustment of Orpheus harp  means an algorithm (gnomon)‚ by means of which on the grounds of the correlation of the first four members of natural row (Greek τετραξ)  all musical grades are to be established.

 

         How is this possible? The following building is well familiar to musicians as "quint circle". In order that a quint of the first four-in-hands, in its turn, forms an octave‚ it is sufficiently to double the length of a corresponding string.  The Quart in the octave sol-sol¹ remains do¹‚ but the quint becomes step re¹:                                                                                                                                                                                                                                                                                                                   

                                                                                             

          If we take fa in the octave do-do¹ as a grade with the value 1‚ than a quint from fa to do¹ = 1 x 3/2 = 3/2 we are to lower twice‚ so that it would  come to one octave with fa‚ coinciding with a tonic do:

                                                                             do = 3/2 : 2 = 3/4.

         Then sol is defined as a second quint from the value of do 3/4:

                                                                            sol = 3/4 x 3/2 = 9/8.

         Further‚ re¹ is a quint from the sol:

                                                re¹ = 9/8 x 3/2 = 27/16‚ and value 27/16 also must be reduced twice‚ as far as the newly received grade has exceeded an upper border of the octave  do¹:

                                                                             re = 27/16 : 2 = 27/32.

 

         Between do and re, as well as between fa and sol lies an interval of a whole tone:                                                                                                                          27/32 : 3/4 = 9/8.

         The fourth quint, on the count, is built from the step re:

                                                                                     27/32 x 3/2= 81/64,

 and carries the name la‚ relates to the same octave:                

         Between la and sol there is an interval of a whole tone:

                                                                                    81/64 : 9/8 = 9/8.

         The fifth on the count quint (mi) is built from la:

                                                                                     81/64 x 3/2 = 243/128‚

and must fall in one octave with la:

                                                                           mi = 243/128 : 2 = 243/256.

                                                                                                      

 

 

         Mi forms a whole tone gap with re:

                                                                                              243/256 : 27/32 = 9/8.

         The sixth on the count quint‚ which is built from the note mi and lies in one and the same with her octave‚ is identified as si:

                                                                                      si = 243/256 x 3/2 = 729/512.

                                                                                                      

 

          Between la and si there is also an interval of a whole tone:    

                                                                                           729/512 : 81/64 = 9/8.

 

 

         We see‚ that a quint algorithm acts absolutely automatically and uniformly.  The results received are brought together in Table 1.

         The two last quint steps mi and si have formed new intervals inside the octave:  ratio

                                     fa / mi     1 : 243/256 = 256/243, and

                                  do¹ / si   3/4 : 729/512 = 256/243 correspond to an interval of leymma‚ named otherwise semitone.

          Leymma – in Greek is "impassability"‚ and means a cessation of filling an octave gap by intervals in whole tone. Really‚ the interval 2 (octave) can contain only 5 intervals 9/8 and 2 intervals 256/243:

                                                                     (9/8)⁵ x (256/243)² = 2.

 

          Such natural filling of an octave gap (2) by whole tone intervals with formation of seven-step sequence  tone-tone-s/ò-tone-tone-tone-s/ò is identified as a diatonic (major) gamma of main musical notes:

 

 

 

 

                     Six intervals (3¹- 3⁶) of Tables 1‚ received by 1-6 quints (as well as their turning‚ complementing them for the octave) contain a base to all musical buildings. The turn prima (3⁰) corresponds to the octave (2) itself.

         Continuation of the quint algorithm will force new steps to “wedge” into whole tone gaps between the main notes:

                                      VII quint is 3⁶/2⁹ x 3/2 = 3⁷/2¹⁰, as far as it falls on the above octave (exceeding value do¹=3/2), this number must be divided by 2, that gives 3⁷/2¹¹;

                                               VIII quint                   37/211 õ 3/2 = 38/212 >3/2, so 38/212 : 2 = 38/213;

                                      IX quint             3⁸/2¹³ õ 3/2 = 3⁹/2¹⁴;

                                      X quint              3⁹/2¹⁴ õ 3/2 = 3¹⁰/2¹⁵>3/2, so 3¹⁰/2¹⁵ : 2 = 3¹⁰/2¹⁶;

                                      XI quint             3¹⁰/2¹⁶ õ 3/2 = 3¹¹/2¹⁷;

                                      XII quint          3¹¹/2¹⁷ õ 3/2 = 3¹²/2¹⁸>3/2, so 3¹²/2¹⁸ : 2 = 3¹²/2¹⁹.

         All the following steps after the first seven (formed from 0 to 6 quint) are marked as changed (increased or lowered) main notes. The seventh quint is a semitone lower (2⁸/3⁵) to sol: 3²/2³ : 2⁸/3⁵ = 3⁷/2¹¹, so it is identified a sol-flat (sol^b). The eighth quint is lowered by the semitone re, since 3³/2⁵ : 2⁸/3⁵ = 3⁸/2¹³‚ and corresponds to re^b. The ninth is also exactly - la^b (3⁴/2⁶ : 2⁸/3⁵ = 3⁹/2¹⁴), the tenth - mi^b (3⁵/2⁸:2⁸/3⁵=3¹⁰/2¹⁶), the eleventh - si^b (3⁶/2⁹ : 2⁸/3⁵ = 3¹¹/2¹⁷). At last, the twelfth quint falls on a lowered half-tone interval of the seventh, as far as 3⁷/2¹¹ (sol^b): 2⁸/3⁵ = 3¹²/2¹⁹. Following the accepted indication we are to note this step as a sol‚ twice lowered by the semitone (sol^bb).

 

 

 

 

        

 

        

         In Pythagorean natural build two semitones (copy-flat) do not give in the interval an equally whole tone (2⁸/3⁵ õ 2⁸/3⁵ = 2¹⁶/3¹⁰ < 9/8)‚ so the last grade does not comply with fa ¹1 - its value is not in accuracy 1‚ that is to say, it is not equal to the interval of prima (3¹²/2¹⁹=531441/524288 =1.0136432 >1)‚  but it exceeds it by the value of Pythagorean comma (marked by us as D).     

         Hence‚ the twelfth quint upwards from the initial step generates a new small interval 3¹²/2¹⁹ in fà ¹ 13 (Fig. 5)‚ numerically corresponding to a difference of the whole tone (9/8) with two semitones 3²/2³ : (2⁸/3⁵)² = 3¹²/2¹⁹, that makes approximately 1/8,69 of a whole tone (in other words, 1.0136432^8,69 = 9/8). The formation of micro intervals ("commas") in a quint build creates a numerical basis of the system of natural micro chromatics. The full filling of an interval of an octave with intensified steps gives a natural (12 or 17-step) chromatic gamma - Figs. 6‚7. Thus, the ascending quints (clockwise a circle) form flat notes ()‚ and the descending (counter-clockwise) - increased notes (#).

 

 

 

        

 

 

 

 

 

                   It is obvious‚ that the quint process is not limited by that. If it is to be continued‚ all steps with quints upwards will be repeated  with raising by comma (+D) with each coil of a spiral‚ and when moving  quints downwards (consequent values are multiplied by 2/3‚ and also are resulted in an initial octave) with decrease (−D) in each cycle of twelve. We let the reader check it personally. Iterative division of a piece do-dî¹ by harmonious numbers is infinite‚ and the concurrence of position of two various numbers never occurs.

 

                   We shall note the basic moments‚ invariant for the concrete values received by a quint algorithm - it's sufficient that their numbers went in consequent order*^*:

 

1. Each 12 consecutively taken numbers form a natural chromatic gamma with the subdivision of an octave into 12 semitones

 - Fig. 6;

 

2. 17 consistently taken steps form a 17-step chromatic scale of 12- semitone intervals with enharmonically unequal raised and lowered steps‚ divided by comma intervals  (D) - Fig. 7;

 

 3. Each thirteenth step in the period of 12 quints locks the octave cycle‚ generating a shift by microtone interval ("comma"D) - Fig. 5;

 

4. Every diameter‚ conducted through the opposite steps of 12-sectioned circle‚ marks an axis of mirror symmetry and two poles‚ comparatively to which the binary pattern of Octave (Figs 4‚9) is shown;

 

5. Each 7 following on the order numbers,  separated by the diameter (Figs. 10) form a seven-step (diatonic) gamma with five whole tone and two semitone gaps of one of seven accepted in antiquity tunes (or corresponding them tonality), that  means septenary pattern of Octave:

 

                                                                          

 

                           

6. All the above said characteristics are determined by a pentatonic (quinary) octave cycle: five quints upwards result in reduction of a source note by a semitone‚ or rising by semitone when moving a quint downwards - Figs. 5‚8;

 

7. Further on‚ a duodecimal order is divided into "quadrants"  and "trigones" of small and big tierces‚ used in the music as the base of its accord and harmonic building - Fig. 11;

 

8. And finally‚ all above described properties come from the relation of odd and even (3ⁿ : 2^m)‚ originally lain in a quint algorithm (Figs. 6,9).  The Octave as an abstract object does not depend on the nature of a sound, and we have the right to consider it as a manifestation of the law of numbers. Similar principles of numeric organization are also found in DNA - a universal life code.

         The presence of commas in a quint circle has been known for years and served a constant irritating factor for musical theorists. The formation in the natural build gammas with comma-displaced notes seemed to them inconvenient‚ as well as the fact that an octave is not closed, and two natural semitones do not compose in accuracy a whole tone. Already a Greek philosopher Àristoxen‚ a pupil of Aristotle‚ offered a temperation or fission of the octave into equal intervals. But this rationalistic idea was realized only in the second half of XVII century‚ when each semitone has been declared equal exactly 100 cents‚ or. This enables‚ infinitely rising on the stairway of quints‚ unchangeably to go back to its beginning‚ as we can see on the engraving by Ìauriz Escher‚ and what is proved by Johann-Sebastian Bach’s îrgan compositions - Pic.1.

 

         But those who stood closer to the bases of the quint system‚ could clearly realize its fractal possibility, the prove of this is found in Plato’s Thimeus (36).

         Let us continue iterations whorl by whorl in the duodecimal circle. It is not difficult to make sure‚ that in the fifth circle there will be an increase by 4 commas‚ that exceeds an interval of a semitone (3.85 D). Then a quint step ¹54 =12õ4+6  mi will become a semitone higher‚ that is to say "will be turned" into fa‚ and a quint spiral will cross  itself for the second time (first rapprochement we saw in ¹13). At that time‚ as shows a calculation‚ the "join" is more full - a new microinterval (s) is 6.5 times less than a Pythagorean comma (D)‚ but "spiral" whorls have become much broader.

         Continuing from new values we have the right to expect‚ that in the seventh circle of a cycle in 53 quints incrementations are reached value D‚ and once again the "join" will occur with the value ¹ 1 fa. Herewith, the step ¹54 + 53x6 = ¹372 will reach (and will exceed by some microtone fission) value of ¹13 fa^D. If ¹13 comes in compliance with ¹ 372‚ than ¹1‚ obviously‚ answers 372-12 = ¹ 360‚ so the following "return" occurs on this number exactly‚ as evidenced by computer modeling of Pythagorean harmonic numbers.

         Not being afraid to bore a reader‚ we do not refuse the pleasure to conduct a calculation "on fingers" of the following join. As far as the interval of octave 2 contains 51 microtones of I order plus one microtone of II order‚ than incrementations of commas with each 12-step cycle for 51õ12=612 numbers will cover an interval in 51D‚ but for getting a small interval s‚ as we know‚ we need another period of 53 quints. Having added to ¹1ôà 51 cycles of 12 numbers each and one of 53 numbers‚ we get sought for a step ¹1+612+53 = ¹666 - the closest value after 1 amongst more than 16000 harmonic numbers (its interval forms 1/15878 part of an octave) - Fig. 12. So‚ being busy with wholly harmless deal‚ we, by accident have touched an "eschatology" subject.

 

         54 is an amount of "double and triple numbers"‚ provided in Òhimeus as a base of dividing the Octave of Cosmos:

 

                                                                   

 

 

360 is well known too - degrees of circumference are counted like this so far, but why is a «number of Antichrist" mentioned in this case? Answer can be concluded in the fact that 666 is no less suitable than other patterns of an Octave available for presentation of temporal cycles:

 

                            "making the sky, He creates an eternal image, moving from a number to a number, which we have named time" (Thimeus‚ 36b).

 

         Time is the "matter" - both usual and odd - on the affirmation of all mystics it is a principle of fractal multiplication of the uniform Being.

        

 

 

 

         Each of Octave periods corresponds to a gamma with the appropriating microtone fission‚ supporting the main fractal property, creates cycles inward cycles‚ repeating again and again the same numeric patterns in the top-down order of scales‚ like well-known fractal of Mandelbrot.  In this sense we are to speak of "internal octaves"‚ put one in another in the same way as matrioshkas. The period in 665 quints performs as a distinctive attractor‚ repeating the initial drawing of twelve steps in its interval - Fig. 13.

 

 

 

 

 

 

 

 

 

 

 

 

         Periods of higher orders may be represented by uppers as forming modules‚ where the connecting link is a cycle of 665:

 

                                                       

  

At the end of XVI century a French scientist Zhozef Skaliger has conceived to create a chronology‚ which was in the best way coordinated with the astronomical and historical data at that moment‚ and has offered the so-called Julian period in 7980 years‚ having got just "in the bull's eye" as far as Skaliger era is still, by the way, being used in chronological calculations‚ - is just based on the period 665 (7980 = 665 õ 12).

  Scaliger‚ as well as Saint Johann has hardly ever based on the direct knowledge of the Octave‚ but its traces go much deeper. Carrying the name of Pythagoras the quint system, in terms of the depth of its generality is an immortal monument, as well as a universal code of mankind: even if to assume‚ that the knowledge of it would be ever lost‚ it inevitably would be re-opened again. The strange "mystical" principles put by the ancient in the bases of the nature and used in astrology and the count of time - pair jin-jang, three gunas‚ five elements‚ eight directions and twelve signs of Zodiac‚ Chinese sexagenary cycle‚ mysterious I ching‚ Zolkin of ancient Maya and Indian eras - all of them anyhow find in the Octave their numerical prototype.

 

 

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