20.500.12556/RUL-99059 Matematika v glasbi magistrsko delo Mathematics in music V magistrskem delu začnem z matematiko glasbe v preteklosti. Najprej predstavim harmonična razmerja, ki so jih poznali antični Grki. Nato opišem Fibonaccijeva števila, zaporedje in zlati rez ter s skladbo prikažem primer uporabe Fibonaccijevega zaporedja. Zlati rez je uporabil tudi Stradivarius pri izdelavi svojih violin. V nadaljevanju definiram transpozicije in inverzije, njihovo uporabo prikažem z Bachovo fugo. Nato matematiko povežem s plesom. Najprej z linijskim kontra plesom, katerega najboljši matematični model je diedrska grupa, nato pa še z ljudskimi plesi, ki jih lahko matematično opišemo s permutacijami, negibnimi točkami ter kompatibilnimi zaporedji. Nadaljujem z opisom diedrskih grup in višinskih razredov. Nato definiram delovanje grupe na množici ter durove in molove trozvoke. Sledi poglavje o PLR-transformacijah, v katerem podrobno opišem njihovo delovanje in primere uporabe na grafih, torusu in v Beethovnovi Deveti simfoniji. Nato pokažem, da sta grupi T/I in PLR dualni. Za zaključek navedem primere uporabe tonske mreže in PLR-transformacij. In my master thesis I start with the history of mathematics in music. First, I introduce the harmonic relationships that were already known by the Ancient Greeks. Then, I describe Fibonacci numbers, Fibonacci sequence and the golden ratio and then I present the use of Fibonacci sequence with a piece of music. The golden ratio was also used in violin construction by Stradivarius. After that, I define transpositions and inversions and present their usage with the Bach's fugue. Then, I connect mathematics with dancing. First, I mention the contra dancing for which the best mathematical model is the dihedral group, and later, I continue with folk dances which can be mathematically described by using permutations, fixed points and compatible sequences. Next, I continue with the description of dihedral groups and pitch classes. Then, I define the group acting on sets and major and minor triads. After that, I continue with PLR-transformations and precisely define their effects and usage in graphs, torus and in Ludwig Van Beethoven's Ninth Simphony. Then I prove that the groups T/I and PLR are dual. Finally, I name some examples of usage of tone-network and PLR-transformations. matematika v glasbi Fibonaccijevo zaporedje transpozicije in inverzije linijski kontra in ljudski ples diedrska grupa višinski razred T/I-grupa delovanje grup durovi in molovi trozvoki PLR-transformacije PLR-grupa Neo-Riemannova teorija mathematics in music Fibonacci sequence transposition and inversion contra dancing and folk dances dihedral group pitch class T/I-group group acting major and minor triads PLR-transformations PLR-group Neo-Riemannian theory true false false [M. Drnovšek] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2017-12-23 07:16:29 2017-12-23 07:16:32 2024-05-30 12:14:24 0000-00-00 00:00:00 2017 0 Ljubljana 0 VI, 51 f. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 87024 18212697 8.pdf 8.pdf 1 EE5ADEB5464059E2FEF604DDCC5AA346 1893712a9907fc28a4fd06b41b8b722f83a6ebce3ecd231225d3dc02f2be7a9a 7cc6d99f-a1b4-11eb-a523-00155dcfd717 20.500.12556/rul/e876d497-c5fd-4826-abac-1975d5e7972d https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=108408 Fakulteta za matematiko in fiziko 0 0 0