# Pythagorean system |

**Pythagorean system** – formulated according to the method of the Pythagoreans mathematical. expression of the most typical frequency (height) relations between the steps of music. systems. Other Greek scientists have empirically established that 2/3 of a string stretched on a monochord, vibrated, gives a sound exactly a pure fifth above the base. tone, “arising from the vibration of the entire string, 3/4 of the string gives a quart, and half of the string – an octave. Using these quantities, Ch. arr. fifth and octave values, you can calculate the sounds of diato-nich. or chromatic. gamma (in fractions of a string, or in the form of interval coefficients showing the ratio of the oscillation frequency of the upper sound to the frequency of the lower one, or in the form of a table of vibration frequencies of sounds). For example, the scale C-dur will receive in P. s. the following expression:

According to legend, P. s. first found practical. application in tuning the lyre of Orpheus. In Dr. In Greece, it was used to calculate the pitch relationships between sounds when tuning the cithara. On Wednesday. century, this system was widely used for tuning organs. P. s. served as the basis for the construction of sound systems by theorists of the East. Middle Ages (for example, Jami in the Treatise on Music, 2nd half of the 15th century). With the development of polyphony, certain important features of P. s were revealed: the pitch intonations of this system well reflect the functional connections between sounds in melodic. sequences, in particular, emphasize, enhance semitone gravitation; at the same time, in a number of harmonics. consonances, these intonations are perceived as too tense, false. In a pure, or natural, system, these new, characteristic harmonics, were identified. warehouse tendencies of intonation: it is narrowed (in comparison with P. s.) b. 3 and b. 6 and extended m. 3 and m. 6 (5/4, 5/3, 6/5, 8/5, respectively, instead of 81/64, 27/16, 32/27 and 128/81 in P. s). The further development of polyphony, the emergence of new, more complex tonal relationships, and the widespread use of enharmonic equal sounds further limited the value of the phonatory s; it was found that P. s. – an open system, i.e., that in it the 12th fifth does not coincide in height with the original sound (for example, his turns out to be higher than the original c by an interval called the Pythagorean comma and equal to about 1/9 of a whole tone); therefore, P. s. cannot be used for enharmonics. modulations. This circumstance led to the appearance of a uniform temperament system. At the same time, as shown by acoustic research, when playing instruments with a non-fixed pitch of sounds (for example, the violin) otd. intonation P. s. find application within the framework of the zone system. Diff. cosmological, geometric the ideas that arose in the process of creating P. s have completely lost their meaning.

*References:* Garbuzov N. A., Zonal nature of pitch hearing, M.-L., 1948; Musical Acoustics, ed. Edited by N. A. Garbuzova. Moscow, 1954. Ancient musical aesthetics. Intro. essay and collection of texts by A. F. Losev, Moscow, 1961; Barbour JM, The persistence of the Pythagorean tuning system, “Scripta mathematica” 1933, v. 1, no 4; Bindel E., Die Zahlengrundlagen der Musik im Wandel der Zeiten, Bd 1, Stuttg., (1950).

*Y. H. Rags*