†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† No doubt what nature in all stays itself like

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††††††††† †† ††††††Pythagoras

 

 

†††††††† It is considered that a fifty three -steps musical system was for the first time described by N.Mercator (not to confuse with a famous cartographer) at the beginning of the XVIII, but later re-opened by “аnaka and Bosanket. Yet exists a strong basis to supposeВ that it was known much earlier in ancient China (at least once from Chjou dynasty - XIII-III B.C.)В in Front Asia in the time of compilation of Moses booksВ as well as in narrow societies of medieval Europe. Its numeric characteristics could not attract attention of connoisseurs of Cabbala. Aside from thisВ a 53-steps sound row is significantly universal, as far as it combines qualities of tempered, quint (Pythagorean) and "clean" build.

 

†††††††† As it has already been shown earlier, under the consequent procedure of raising by a quint and bringing all forming grades in the initial octave, the first 11 quints (along with є1В taken for source) give seven main and five lowered by the semitone musical notes, but with 12-th quint this harmonic series is repeated with the shift upwards by the micro interval of Pythagorean comma (microtone I order D) В and so in each cycle of 12 quints.

 

†††††††† †As far as an interval of a halftone contains less than 4D -† about 3.85, how shows a calculationВ then in fifty third quint (12 x 4 + 5) a step є 54 will be formedВ which "is connected" with the source step є1 faВ accepted as a unit value. The new positive micro interval of II order s by the value near 1/6.5D† (or 3.6 cent) is obtained - Fig. 1. Thus, the first cycle of 53 quints is terminated.

 

†††††††† Like the first 3 harmonic numbers answer to division of an octave interval into 2 quints and 2 quarts with forming a whole tone interval between them, 5 numbers form sound raw of pentatonic gamma (without semitones), 7 - seven-step (diatonic) gamma with two semitone intervals (mi-fa and si-do¹)В and 12 numbers - twenty-step semitone (chromatic) gamma, consequently taken 53 quints gives a natural 53-step gamma of natural build.

†††††††† 53 stepsВ therebyВ numerologically answer the octave: 5+3=8В and follow its pattern 13: 52 generating quints from the initial number constitute 4 x 13.† The role of fifty three-step system is not limited by its obvious musical value. Fractal-numeric regularity being the basis of harmony can ever be followed as far as remote areasВ as solar-moon cycleВ Mayan sacral calendar Zolkin and I Ching of ancient China.

 

 

†††††††† As far as the increased (#В formed by the retrograde motion of quints) and lowered (В formed by direct motion) steps differs on the comma value in quint buildВ and in the cycle of 53 four micro intervals are formed in addition inwardly each semitoneВ a wholetone gap will be divide into 9 micro intervals ("nonnas") - Fig. 2. Seven of them in accuracy equal D (23.46 cent), but two (between 4Δ⁺ и , and between ^3Δ+ and main notes) are lowered by s and equal 0.85D† (near 20 cents). Such division may be easily transformed into 53-step equal (tempered) build - Fig. 3. Each micro interval δ in it must take the value = 1.0131642 (about 22.64 cents)В differing from the natural micro interval D by a value near 1/29D (0.8 cents).

 

†††††††† †Structure of dividing the оctave interval in the equal 53-step build is presented as follows:

 

 

 

 

 

 

 

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†††††††† These inward harmony steps assign the base of natural microchromatics or so-called "enharmony" (in Greek έναρμόνιον† γένος = "regime kind") build. In consequence of equal dividing of a semitone the 53-step gamma allows to build the eight twelve-steps (4# and 4 ) or four seventeen-steps chromatic (semitone) gammas with enharmony changes between them - Fig. 4. In each of these intermediate grades, just as on a tonic, a corresponding tonality can be formed, but we are not touching here the questions about inflexions and microtone constructions.

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†††††††† All these steps are linked through the relations of clean quints and quarts (3:2 and 4:3)В i.e. may to be made one from another in a certain number of quint moves. The accepted twelve-steps temperationВ as it is well- knownВ is possible only at the condition of reduction all quints by the interval 1/12 part of a Pythagorean comma (D)В what means no other than the refusal of consonants.

 

†††††††† †Take, for example, a table of comparative frequencies for the 12-steps temperated, clean and the 53-step equal build:

 

 

 

†††††††† From this comparative table it is possible to see that the fifty three step musical system contains the clean quints and quarts (difference in the fifth decimal sign), coinciding in this feature with the "clean build". Its minor tercie also corresponds to a minor tercie in the clean build (difference in the third decimal sign). What concerns a major tercie, which "noneuphony" is considered to be the defect of the quint build (and in the 53-step formation it is closer to "tempered", than to "clean" major tercie), †its "purity" can be reached by reduction of an upper or raising of a lower note of the interval by 1 microtone (by δ-alteration).

†††††††† SoВ for the lowered on 1 microtone step mi (mi^Δ-) frequency takes a value †1.248989 comparatively to do, corresponding to an interval of 389 cents, that is practically interchangeable with the "clean" major tercie (386 cents). Alterations by a microtone (23 cents) may be reached by programming (in the сase of electronic instruments) to achieve a "clean" major tercie sound.

 

†††††††† For about three centennialsВ passed after BachВ European musical composition has exhausted its all initially creative potentialВ in the significant measure obliged to a success of the 12- step tеmperation. On the great part of music amateurs an accent of interest exhibit the more and more bias to the alternative directions of "ethnic classicism". At the same time the attempts of widely using of microchromatics have been undertaken since XIX centuryВ and it was one of the main kinds of an ancient and antique music. Creations of all significant composers of the second half of XX century demonstrate a tendency to overcome the oppression of a twelve-sound temperation.

 

†††††††† A 53- step build fills in a main defect of the latter - an absence of consonances, for the art, which nature lies in harmony, becomes a critical factor. Together with that it is based on numeric laws of music, distinguishing in an artificial clean building, which theoretical basis is arbitrarily elected.

 

†††††††† Getting closer to the oriental music (not having a fixed row) in the sound compositionВ and possessing together with that the tempered closed build qualities (differing with the Pythagorean system), the fifty three-steps system enables to use all developed in the West means of musical compositions, at the same time vastly increasing their borders.† The problemВ seems to beВ in the technical entailmentВ but this question does not appear for electronic programmed instruments.

†††††††† One can be possibly sureВ that the efforts, undertaken both on the part of sciences and modern art to remove a West civilization from a spiritual and ecological blind-alley, to restore its third (axial) lost measurement - andВ hereuponВ reassessment of many earlier established valuablesВ† - †do not pass such a forming image of culture element, as basis of its musical system.

 

†††††††††††††††††† Table 2.

††††††††††††††††††††††††††† Frequencies (Hz) of the musical tones for standard la¹ = 440 Hzs in 53-step equal build*^*:

 

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