Doc. PhDr. Elena Letňanová /SK/ - pianist, theorist, teacher, STU, Bratislava

Pythagorean Numerical Ratios and Rational Elements in Musical Piece „Alice“ by Slovak Composer Juraj BENEŚ

  1. Pythagoras (6th Century B.C.) inspired the birth of harmony in music and
  2. The area of musical thinking and
  3. Discovery of numerical ratios, fragments, 2:1, 3:2, 4:3, 5:4, etc. which represent the basic intervals in music (the octave, the fifth, the fourth, the major third, the minor third, the big second, the smal second).
  4. The interval in music represents the distance betwen two tones counting the lower tone clockwise toward the higher one.
  5. Pythagorean ratios, expressed in music as intervals, were accumulated in the suprapositions and created later in the development of music the major and minor musical chords especially in the era of Renaissance, Baroque, Classicism, Romanticism, Impressionism, later even the musical clusters in the 20th Century, and have been commonly used in the contemporary music until the present.
  6. Pythagoras set for the long time the paradigma of the consonances and the dissonances with his idea of numerical ratios wheather perfect or imperfect, that is the fragments with endless decimal number (1.33, etc.)
  7. The periods of the 17th, 18th Centuries accepted Pythagorean dissonances e.g. the thirds (tercie) and the sixths (sexty) as consonances in music compositions and theoretical treatises. The intervals of the seconds (sekundy) and sevenths (septimy) the harshest dissonances were gradually accepted as consonances in the 19th Century (just to mention one example for all the seventh chords and diminished sevenths in Beethoven´s Sonata C- major „Waldstein Sonata“, in the 3rd Movement before the recapitulation, and Chopin´s second chords in the Balade A Flat Major, Nr. 3, on the page Nr. 3 and 2 final pages).

8. Discovery of ratios by means of hammers heard in a quary or the dividing of Pythagorean string on his monochord?

8.1 The length of the string - a monochord, musical instrument is more probable means of discoverying the intervals.

8.2 The sounds of hammers - striking the metallic anvils of various sizes,           simmultaneously, or in sequence might be just inspiration to construct the monochord as an experimental instrument. How pythagoras might say that the sounds were pleasant or having a certain pitch?

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

9.Rationalisation versus apeiron. Mathematics versus philosophy:

Pythagoras famous statement „the substance of all things on earth are numbers“ it means not four elements of the world (e.g. water, fire, earth, and air, as his younger colleagues-philosophers thought) is for music crucial and thruthful.

9.1 Simultaneous pleasant sounds expressed in numbers give the two intervals of so called triads and seventhchord (septimové akordy). Two triads sounding together give the nineths (nónové akordy). Further juxtapositions of intervals gives the undecima chords (undecimové akordy, and further three triads of chords give us the polyharmonic dissonances (example in piano compositions of Darius Milhaud c-e-g-h-d-f sharp-a-, the tripple triad). This is no more a harsh dissonance, the similar examples may be found in Bartók´s compositions with the intervals and chords containing the big and small seconds.

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.

9.3 Pythagoras set also the dictate of the music theory over the musical practice. The vocal compositions, sang in the church in the 8th A.D. were constructed as simple musical forms. 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.

9.4 Pythagorean ratios, the row of numbers 1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:16,controlled almost the majority of Medieval theory and music.

The restriction of the interval called „diabolus in musica“ it means the interval of c-f- or f-b (in Slovak f-h) if in fact the interval of raised fourth. That is not the pure fourth, was forbidden to use in music because of its bad fragment, ratio about 53: 46, a hash, unpleasant, and inharmonious sound. The interval of the octave C –c, was actually a singing in unison, that is the most pleasant interval. Similarly the fifth, that is C to g and the fourth, C to f, sounded also pleasantly, harmoniously. This all was set by Pythagorean ratios when dividing the string of monochord as follows:

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.

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).

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 the small third, the number 8, the c represents the octave counting from the main tone of C, the number 9 is the tone „d“, representing the interval of the 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“ is a small second, a very tough dissonance, the number 12 is also a small second, etc.

Slovak composer Profesor 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.    

  1. 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 - the sea waves, 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 a very bad ratio and the dissonance - the Pythagorean 9th, that is C-h, the distance of 9 tones, steps. Is this all composed only subconsciously? The composer put all his knowledge inside of this piece and concerned himself with Pythagorean tuning, and P. coma.


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, maybe the small second, that is a half tone step. When we call or cry or sigh we „sing“ an descending line in half steps, half tones.





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