Autor: SAVA KOSTIĆ, Gimnazija, Zaječar

Mentor: DRAGANA SEKULIĆ PILIPOVIĆ, profesor matematike, Gimnazija, Zaječar

REZIME

Iako na prvi pogled matematika i muzika nemaju dodirnih tačaka, muziku možemo da objasnimo uz pomoć matematike i malo fizike. Svi zvuci se dele na tonove i šumove. Tonovi su prijatni zvuci koje karakteriše periodičnost promena u sredini kroz koju se prostiru, dok su šumovi neperiodičnog tipa. Svaki ton ima svoju jedinstvenu frekvenciju, a s obzirom da ljudsko uvo čuje samo određene frekvencije, svi tonovi koje možemo čuti imaju frekvenciju od 20 Hz do 20 000 Hz. Zapravo, u muzici postoji vise štimova, tj. načina da se tonovi pravilno rasporede u odnosu na svoju visinu i boju, a najčešće se koristi kamerton, odnosno ton a koji ima frekvenciju 440 Hz. Ponovljeni ton nakon, niza od 12 polustepena, iste boje, ali različite visine naziva se oktava i oktava bilo kog tona ima duplu vrednost  frekvencije za oktavu višeg i polovičnu vrednost frekvencije za oktavu nižeg tona. Znajući da oktava ima tačno 12 polustepena (najmanji razmak između dva susedna tona), razliku u frekvenciji između dve octave podelićemo brojem 12 i dobićemo konstantu koja ima vrednost jednog polustepena i uz pomoć ove konstante može se izračunati frekvencija svakog tona. U ovom, kao i u predstojećim primerima možemo videti da matematikom, kroz malo fizike, možemo opisati muziku, intervale, akorde i sl.

Ključne reči: matematika, muzika, zvuk, ton, interval.

SUMMARY

Although at first glance mathematics and music don't have in common anything, we can explain music with the help of mathematics and a little of physics. All sounds are divided into tones and noises. The tones are pleasant sounds characterized by periodic changes in the ambience through which they  spread, while the noises aren`t  periodic. Each tone has its own unique frequency, but considering that the human ear hears only certain frequencies, all the tones we can hear have a frequency of 20 Hz to 20 000 Hz. In fact, in music there are several tunings, ie. ways to arrange tones correctly in relation to their pitch and timbre, and usually is used camertone, tone that has  a frequency of 440 Hz.  Repeated tone after  a series of 12 semitones, with the same timbre but different pitch is called octave, and any tone octave has double frequency value for a higher octave and a half-value frequency for lower tone octave. Knowing that octave has exactly 12 semitones (the minimum distance between two adjacent tones), the difference in frequency between two octaves we will divide with number 12, and get a constant that has a value of one semitone and with the help of this constant can be calculated frequency of each tone. In this, as in the upcoming examples, we can see that with mathematics, through a little of physics, we can describe the music, intervals, chords, etc.

Keywords: mathematics, music, sound, tone, interval