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

Ryo Sakuma
July 04, 2021
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Cool Math for Hot Music 輪読会 Sec.16 Part 2

Ryo Sakuma

July 04, 2021
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  1. 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
  2. 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
  3. 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
  4. 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
  5. 2. δάβάͰ࡞ۂͯ͠ΈΔ • ॎํ޲ʹ͸ 1 ͭͷ૊ʹؚ·ΕΔ δάβάύλʔϯ͕ฒͿΑ͏ʹ • ԣํ޲ʢ࣌ؒํ޲ʣʹ͸ผͷ ύλʔϯ૊͕ฒͿΑ͏ʹ഑ஔ

    • ͨͩ͠ɺશମͱ্ͯ͠ߦ͢ΔΑ͏ δάβάͷଋશମΛ܏͚Δ • ͲΕ͘Β͍܏͚͍ͯΔʁ ࢖͍ͬͯΔટ๏͸ʁ ͍·͍ͪෆ໌
  6. 3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ू߹ X ͷඃ෴ C ͸ 2X ͷ෦෼ू߹

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