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

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

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

4. 1. Hammerklavier ͱδάβά

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

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

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

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

9. 2. δάβάͰ࡞ۂͯ͠ΈΔ

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

11. 2. δάβάͰ࡞ۂͯ͠ΈΔ • Beethoven ͸δάβάʢͷઢܗม׵ʣͷ্ʹશͯԻ͕ ฒΜͰ͍ͨ • ൃలͤͯ͞ɺෳ਺ຊͷδάβάͷଋΛߟ͑ͯΈΔ • Ճ͑ͯɺ1

    ຊͷδάβάʹ͸෦෼తʹԻΛฒ΂Δ

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

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

    3 Օॴʹ͍ͯ͠Δʢཧ༝ෆ໌ʣ

14. 2. δάβάͰ࡞ۂͯ͠ΈΔ • Exercise 5

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

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

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

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

19. 2. δάβάͰ࡞ۂͯ͠ΈΔ • Ͱɺ͏·͍ײ͡ʹΤΠϠ͢Δͱۂ͕Ͱ͖ΔΒ͍͠

20. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ

21. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ू߹ X ͷඃ෴ C ͸ 2X ͷ෦෼ू߹

    • ຺ମ N(C) ͸ 2C = 22^Xͷ෦෼ू߹ʢॏͳΓΛ࣋ͭू߹Ͳ͏͠Λ·ͱΊͨΓ͢Δʣ N(C) X ஍ٿ؀ڥอશͷͨΊεϥΠυΛ࠶ར༻͍ͯ͠·͢

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༻͍ͯ͠·͢

23. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ָේͷଊ͑ํ 1ɿԻූͷཏྻͱݟΔ • ͭͳ͕ΓΛߟྀ͠ͳ͍ͷͰɺͦΕͧΕͷԻූ͕ݽཱͨ͠ 0 ୯ମʢ఺ʣͱͯ͠ࢄࡏ͢Δ͚ͩ •

    Ҡௐͯ͠ಉָ͡ේ͸ಉܕʢ࣮࣭ಉ͡ʣʹݟ͑ͯ͠·͏

24. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ָේͷଊ͑ํ 2ɿશͯͷԻΛؚΉɺ຺ମతͳ 1 ຕͷਤͰߟ͑Δ • ϐονؒͷͭͳ͕Γɺಈػͱల։ͱͷͭͳ͕ΓɺௐͲ͏͠ͷͭͳ͕Γɺ࿨੠ͷͭͳ͕Γɺetc. •

    ͍Ζ͍Ζͳͭͳ͕ΓΛදݱͨ͠ਤͱͳΓɺ༷ʑͳύϥϝʔλΛ಺ࡏͤͨ͞΋ͷʹͳΔ • ͦͷΑ͏ͳਤ͸ҰମͲΕ͘Β͍ଘࡏ͢Δʁ

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