PYTHAGOREAN NUMERICAL RATIOS AND RATIONAL ELEMENTS IN MUSICAL PIECE „ALICE“ BY SLOVAK COMPOSER JURAJ BENEŚ

Doc. PhDr. Elena Letňanová /SK/ - art historian, concert pianist, musicologist STU Bratislava

Pythagoras´ (6th Century B.C.) famous statement „the substance of all things on earth are numbers“ is for music crucial and thruthful. Not the four elements of the world (e.g. water, fire, earth, and air, as his younger colleagues-philosophers thought).

Pythagoras inspired the birth of harmony in music and musical thinking by discovery of numerical ratios by means of hammers heard in a quary , this might give him the idea of pleasantly sounding tones. How he got the idea of a certain pitch? The hammers were striking simultaneously?

Or did he get the idea of a pitch when dividing the string of his monochord. The sounds of hammers - striking the metallic anvils of various sizes, simmultaneously, might be the inspiration to construct the monochord as an experimental instrument.

I myself experienced the sound of five hammers in the scene of construction the Ramzes´pyramid in Egypt in one documentary film.  The probable proof might be that Pythagoras lived a certain time in Egypt and might have heard this scene too.

His discovery of the so called Pythagoreian numerical row, is actually the row of numbers, the ratios, written as fragments 2:1, 3:2, 4:3, etc. THE WHOLE ROW IS follows 1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:16, originated when Pythagoras divided the string of monochord, the well known row of tones

C, c, g,  c, e, g, b , c, d,  e,  fis,  g,   a,    b,   h,    c  

1, 2, 3, 4, 5, 6, 7,  8, 9, 10,11, 12,13, 14,  15, 16

In the Pythagorean row of numbers, the second „c“ that is the number 2 represents the octave, number 3 represents the fifth-quinta, the number 4 the fourth-quarta, the number 5, that is the tone „e“ represents the third (the major third) , the 6 is a small third, the number 7 represents again the small third, the number 8, that is „c“ represents the octave counting from the main tone of C, the number 9 is the tone „d“, representing the interval of the major second, which was the dissonance in the medieval times, not today. The number 10 ,the tone „e“ represents also the interval of a second, the number 11 – the tone „f sharp“ is a second, a dissonance, the number 12 is „g“ that ius also small second, a vcery tough  dissonance, not today, 14:15:16 are small seconds, dissonances.

The monochord of Pythagoras: The basic three ratios 1:1, 2:1, 3:2, the pure octave, the pure fifth, and the pure fourth, were the most preferred intervals in the early music, Christian and Medieval periods and also the most perfect intervals in theory and then in composed music. Why? It was due to the perfect result of their fragments (the whole number 1, 2 in the first cases, and 1, 5 in the third case).

The ratio 4:3, the interval of the fourth: the result is a number 1.33333, etc. the fragment with endless numbers. Therefore Pythagoras said the interval of the fourth was not so pure as the first three intervals. Nevertheless, the musical fourth was added to the group of consonant intervals, the so called harmonic consonances—the prima, the octave, the fifth and the fourth. The proof can be demonstrated in the Huckbald´s parallel organum from the 8th Century A.D.

The first 4 number or ratios controlled almost the majority of early Medieval theory and music that is (the octave, the fifth, the fourth, the major third). The parallel organum of Huckbald, moved in parallel octaves, parallel fifths and parallel fourths during the whole composition. For us, now, it sounds quiet boring and statical. The octave and the fifth were the perfect fragments therefore the perfect intervals-the consonances. The imperfect fragments of these ratios have endless decimal number like (1.33, etc.), this was then considered as an imperfect interval, that is dissonant.

With this row Pythagoras set for the long time the paradigma of the consonances and the dissonances with his idea of perfect or imperfect intervals according to the mathematical fragments. The restriction of the interval called „diabolus in musica“ the interval of c-f  sharp- or f-b (in Slovak f-h) in fact is the imperfect interval (of the raised fourth) for a long time forbidden to be used in music because of its bad fragment, the ratio about 53: 46, a harsh, unpleasant, and inharmonious sound. The interval of the octave C –c, was actually a singing in unison, that is the most perfect and pleasant interval-the consonance. Similarly the fifth, that is C to g and the fourth, C to f,  sounded also pleasantly, harmoniously then.  Along with this main tone C there are sounding the partial, aliquote or upper tones, which we cannot hear. We hear the main tone C only. There are sounding the parts of the string or all parts (for instance of a clarinet body).

The Slovak composer Juraj Beneš´s composed the 15 minutes long piano piece „Alice Was Getting Very Tired of Sitting on the Bank Next to Her Sister And Having Nothing to Do“ in 1992. Continuoisly connected 4 parts as the whole. The Beneš began his piece with an ascending horizontal line and positive feeling, upholding every listener.       

Beneš theme – tetractys, the number 10, experimented with the idea of the number (as 1 plus 2 plus 3 plus four equalls to 10). Musically left hand 5 plus right hand  5 tones. The first part the composition beginns with the Pythagorean ratios 2223, 3223, in row of 10 tones, grouped as 5 plus. Made by Mr. Beneš deliberately or by emotional state?

 I tried to analyze the use of Pythagorean ratios in this piece and quasi composing the Pythagorean row of aliquotes in the first theme of this piece. The horizontal and slow melody - pieceful, slow, pleasant although perceiving statistically often occurance of intervals of 2, 3, less often the sixths. The row 2223 is the sequence of 3 seconds (small, great, great),and one small third only. This row repeats 3 times. Why? Mathematically 3 is the prime number. The piece is very long, the symphony of Mozart or Beethoven had usually 20 minutes and includes 3 or 4 movements. This piece is just one movement continuously developed with 4 parts.

The the first one which is the most hamonious part prevailing ratios named above.

The second part of the Beneš´s piece sounds more chromatic, with the half steps or tones and uses the whole tones a seconds.

The third part is a jump into a different universe,  a certain 50 seconds lasting organized chaos with many distant aliquotes, never the prime numbers and perfect intervals.

The 4th part - again a return to simplicity, and harmonious ratios, ending with the questioning mark and the dissonance - the Pythagorean 9th. The composer put all his knowledge inside of this piece and concerned himself with Pythagorean tuning.

 P.S.

 The composing is the creation and recreation of the nature. And Pythagoras said, the nature is the number. Although this music is full of rational numbers, the result is a pleasant feeling and emotions. Pythagoras said also that the universe or the movements of planets are full of harmonious ratios, equations? In music I am giving you the proof of the Pythagorean philosophical idea of mathematical rationalisation of the world, relations and things. Carl Stockhausen the significant composer of Germany in the 20th Century, composed an opera in which the movements of all planets were represented by lines of vibrations, various sounds , structures. The most distant planet Pluto was demonstrated by the lowest tone, very long. The planets closer to the Earth sounded like interrupted intervals of fourth, or fifth placed in the middle register of piano, and the others gave the impression of quasi trilling, unpleasant, schrilling sounds of no pitch, sounding in the highest conceivable position. Thus Stockhausen experimented musically with physical fundaments of a mater, sound. At the beginning of the world the Script said, was a word, I say the sound, some kind of interval.

Doc. PhDr. Elena Letňanová

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