Cool Math for Hot Music 輪読会 Sec.16 Part 2

Transcript

1. HASH 2021-07-05
   16. Nerves (Part 2)
   Cool Math for Hot Music ྠಡձ
   

   View Slide

2. ໨ඪ
   1. ಈػͱ͍͏ߏ଄ʹ຺ମͷࢹ఺Λ࣋ͭ
   2. ͜ΕͷงғؾΛཧղ
   

   View Slide

3. େ·͔ͳྲྀΕ
   1. Hammerklavier ͱδάβά
   2. δάβάͰ࡞ۂͯ͠ΈΔ
   3. ຺ମͷදݱྗͱָේͷෳࡶੑ
   

   View Slide

4. 1. Hammerklavier ͱδάβά
   

   View Slide

5. 1. Hammerklavier ͱδάβά
   • "Hammerklavier": Beethoven Op.106
   • 4 ָষߏ੒ͷϐΞϊۂɻୈ 1 ָষ Allegro ͸ιφλܗࣜ
   • Ͳ͏΍Βୈ 1 ָষͷௐੑɾಈػ͸ C#dim7 ʹࢧ഑͞Ε͍ͯΔΒ͍͠
   • ͱ͍ͬͯ΋ Reference [72] ͸ Mazzola ઌੜͷຊͳͷͰɺWikipedia ͳΒ [ಠࣗݚڀ] λάҊ݅
   • ͳ͓ɺ[72] ͸υΠπޠ
   

   View Slide

6. 1. Hammerklavier ͱδάβά
   • "Hammerklavier": Beethoven Op.106
   • C#dim7 ͸൒Ի 3 ͭ෼ͷରশੑ
   • ϝγΞϯͷʮҠߴͷݶΒΕͨટ๏ʯୈ 2 ൪ͱ
   ಉ͡Α͏ͳ 3 ൒ԻͷରশੑΛ࣋ͭ
   • Ώ͑ʹ C#dim7 ͔Βपظ 3 ͷδάβάύλʔϯ
   ͕༠ಋ͞ΕΔͷ͸ࣗવʢͱओு͍ͯ͠Δʣ
   • पظ 3 ͷδάβάύλʔϯ͕ Hammerklavier
   ͷಈػ܈ʹݟͯऔΕΔɺͱ [72] Ͱओு͍ͯ͠
   ΔΒ͍͠
   

   View Slide

7. 1. Hammerklavier ͱδάβά
   https://slidetodoc.com/concepts-locaux-et-globaux-premire-partie-thorie-objective/
   

   View Slide

8. 1. Hammerklavier ͱδάβά
   https://www.geogebra.org/m/YCZa8TAH
   

   View Slide

9. 2. δάβάͰ࡞ۂͯ͠ΈΔ
   

   View Slide

10. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • Beethoven ͸ C#dim7 Ͱ 12 ൒ԻΛ 3 ൒Իͣͭʹ෼͚ͨʢҰ୴ओுΛೝΊΔʣ
    • ൃలͤͯ͞ C#aug Ͱ 12 ൒ԻΛ 4 ൒Իͣͭʹ෼͚ͯΈΔʢϝγΞϯୈ 3 ટ๏తʣ
    • 4 ൒ԻͣͭͳͷͰɺδάβάͷपظ͸ 4 ʹͳΔ
    

    View Slide

11. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • Beethoven ͸δάβάʢͷઢܗม׵ʣͷ্ʹશͯԻ͕
    ฒΜͰ͍ͨ
    • ൃలͤͯ͞ɺෳ਺ຊͷδάβάͷଋΛߟ͑ͯΈΔ
    • Ճ͑ͯɺ1 ຊͷδάβάʹ͸෦෼తʹԻΛฒ΂Δ
    

    View Slide

12. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • ॎํ޲ʹ͸ผʑͷδάβάʮύλʔϯʯ
    Λฒ΂Δ
    • ্࣌ؒ࣠ʹ 1 ͭͣͭݟ͍͖ͯɺಉ࣌ؒ͡
    ্࣠ͷॎํ޲ͰҰ൪্ʹ৐͍ͬͯΔԻͰ
    ಈػΛߏ੒͍ͯ͘͠ʢΒ͍͠ʁʣ
    • ը૾͸ಉ͡ۂͷ Main Theme ͷߏ੒ʹͭ
    ͍ͯݴٴ͍ͯ͠Δࢿྉ
    • "Finally, we take the configuration's top
    notes within C major."
    • ಈػͷߏ੒͸ "configuration"ʢδάβά
    ύλʔϯͷฒ΂ํʣͱટ๏ʹґଘʁ
    The Future of Music: Towards a Computational Musical Theory of Everything
    

    View Slide

13. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • पظ 4 ͷδάβάʢͷԟ෮ʣͷʮඃ෴ʯΛ࡞Γɺδάβάύλʔϯͷ຺ମΛ࡞Δ
    • δάβάͷʢยಓʣपظ͸ 4 ͕ͩɺԻΛׂΓ౰ͯΔ৔ॴ͸ 3 Օॴʹ͍ͯ͠Δʢཧ༝ෆ໌ʣ
    

    View Slide

14. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • Exercise 5
    

    View Slide

15. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • ॎʹฒ΂Δδάβάύλʔϯͷ
    ૊Έ߹ΘͤΛ຺ମʹԊͬͯ࡞Δ
    

    View Slide

16. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • ॎʹฒ΂Δδάβάύλʔϯͷ
    ૊Έ߹ΘͤΛ຺ମʹԊͬͯ࡞Δ
    • ͨ͘͞Μ࡞Δ
    

    View Slide

17. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • ॎํ޲ʹ͸ 1 ͭͷ૊ʹؚ·ΕΔ
    δάβάύλʔϯ͕ฒͿΑ͏ʹ
    • ԣํ޲ʢ࣌ؒํ޲ʣʹ͸ผͷ
    ύλʔϯ૊͕ฒͿΑ͏ʹ഑ஔ
    • ͨͩ͠ɺશମͱ্ͯ͠ߦ͢ΔΑ͏
    δάβάͷଋશମΛ܏͚Δ
    • ͲΕ͘Β͍܏͚͍ͯΔʁ
    ࢖͍ͬͯΔટ๏͸ʁ
    ͍·͍ͪෆ໌
    

    View Slide

18. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • ϐΞϊͷӈखͱࠨखͦΕͧΕʹผͷ
    δάβάଋΛ༻ҙ
    • ͳΜͱͳ͘ԣ࣠ํ޲ͷ։࢝ҐஔΛ
    ͣΒ͍ͯ͠Δؾ͕͠ͳ͘΋ͳ͍
    

    View Slide

19. 2. δάβάͰ࡞ۂͯ͠ΈΔ
    • Ͱɺ͏·͍ײ͡ʹΤΠϠ͢Δͱۂ͕Ͱ͖ΔΒ͍͠
    

    View Slide

20. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    

    View Slide

21. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    • ू߹ X ͷඃ෴ C ͸ 2X ͷ෦෼ू߹
    • ຺ମ N(C) ͸ 2C = 22^Xͷ෦෼ू߹ʢॏͳΓΛ࣋ͭू߹Ͳ͏͠Λ·ͱΊͨΓ͢Δʣ
    N(C)
    X
    ஍ٿ؀ڥอશͷͨΊεϥΠυΛ࠶ར༻͍ͯ͠·͢
    

    View Slide

22. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    • ू߹ X ͷඃ෴ C ͸ 2X ͷ෦෼ू߹
    • ຺ମ N(C) ͸ 2C = 22^Xͷ෦෼ू߹ʢॏͳΓΛ࣋ͭू߹Ͳ͏͠Λ·ͱΊͨΓ͢Δʣ
    • ಉ͡Α͏ʹ຺ମͷ຺ମΛߟ͑Ε͹ɺͲΜͲΜࢦ਺ͷݞ͕৐͍ͬͯ͘ → ߴ͍දݱྗΛ࣋ͭ
    N(C)
    X
    ஍ٿ؀ڥอશͷͨΊεϥΠυΛ࠶ར༻͍ͯ͠·͢
    

    View Slide

23. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    • ָේͷଊ͑ํ 1ɿԻූͷཏྻͱݟΔ
    • ͭͳ͕ΓΛߟྀ͠ͳ͍ͷͰɺͦΕͧΕͷԻූ͕ݽཱͨ͠ 0 ୯ମʢ఺ʣͱͯ͠ࢄࡏ͢Δ͚ͩ
    • Ҡௐͯ͠ಉָ͡ේ͸ಉܕʢ࣮࣭ಉ͡ʣʹݟ͑ͯ͠·͏
    

    View Slide

24. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    • ָේͷଊ͑ํ 2ɿશͯͷԻΛؚΉɺ຺ମతͳ 1 ຕͷਤͰߟ͑Δ
    • ϐονؒͷͭͳ͕Γɺಈػͱల։ͱͷͭͳ͕ΓɺௐͲ͏͠ͷͭͳ͕Γɺ࿨੠ͷͭͳ͕Γɺetc.
    • ͍Ζ͍Ζͳͭͳ͕ΓΛදݱͨ͠ਤͱͳΓɺ༷ʑͳύϥϝʔλΛ಺ࡏͤͨ͞΋ͷʹͳΔ
    • ͦͷΑ͏ͳਤ͸ҰମͲΕ͘Β͍ଘࡏ͢Δʁ
    

    View Slide

25. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
    https://slidetodoc.com/concepts-locaux-et-globaux-premire-partie-thorie-objective/
    

    View Slide