and e

intothecontinuum:

Making use of music theory, group theory, and category theory

From Musical Actions of Dihedral Groups

Abstract:
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.

3 months ago

1. kogiso-annex reblogged this from beyondneptune
2. trxfreely reblogged this from proofmathisbeautiful
3. infiniteuniverseof0dimensions reblogged this from ipiphi and added:

   OMG! My two favorite things: Math nad Music! In a paper!!!! *swoon*

4. circumferences reblogged this from intothecontinuum
5. carynw liked this
6. makeasplash liked this
7. without-a-handrail reblogged this from jdao and added:

   Oh my god. I clicked on this picture and traced through cycles in the first picture and realized how much I miss music....

8. without-a-handrail liked this
9. jellyish28 liked this
10. jailbrokejon reblogged this from proofmathisbeautiful
11. devstep liked this
12. e-x-d-e-e reblogged this from ipiphi
13. laarche reblogged this from fuckyeahgenderneutralstem
14. faintedtowers liked this
15. spellstreble reblogged this from theshoutingendoflife
16. joejbarr liked this
17. thaneventhesun liked this
18. georgeevans reblogged this from georgedarby
19. paymentguide reblogged this from theshoutingendoflife and added:

    oh my god beautiful

20. theshoutingendoflife reblogged this from fuckyeahmath and added:

    Keeping on the topic of math being really fucking cool sometimes.

21. georgeevans liked this
22. georgedarby reblogged this from proofmathisbeautiful and added:

    Making use of music theory, group theory, and category theory From Musical Actions of Dihedral Groups Abstract: The...

23. sound0ne reblogged this from beyondneptune
24. philosophilae reblogged this from proofmathisbeautiful
25. coxinator reblogged this from fuckyeahmath
26. allthelandslides liked this
27. alexlovesclay reblogged this from fuckyeahmath
28. casualkoala reblogged this from intothecontinuum and added:

    reblogging this so i wont lose it

29. ladata reblogged this from proofmathisbeautiful
30. sfweeffrffggh liked this
31. fuckyeahgenderneutralstem reblogged this from vault--girl
32. man-tou liked this
33. man-tou reblogged this from proofmathisbeautiful
34. blindmen6 liked this
35. ilaaljawzawayn reblogged this from intothecontinuum
36. fersherbs liked this
37. kd8ofp reblogged this from intothecontinuum
38. chelseatwat liked this
39. antepenult reblogged this from proofmathisbeautiful
40. miclaus liked this
41. mylifeisborromean reblogged this from proofmathisbeautiful
42. noises reblogged this from fuckyeahmath
43. madzzz reblogged this from fuckyeahmath and added:

    people are so smart!

44. rebop75 liked this
45. isomorphismes liked this
46. macmankev liked this
47. lilouette reblogged this from fuckyeahmath
48. rpwongzone liked this
49. Show more notesLoading...