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Cool Math for Hot Music 輪読会 Sec.16 Part 2
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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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Transcript
HASH 2021-07-05 16. Nerves (Part 2) Cool Math for Hot
Music ྠಡձ
ඪ 1. ಈػͱ͍͏ߏʹ຺ମͷࢹΛ࣋ͭ 2. ͜ΕͷงғؾΛཧղ
େ·͔ͳྲྀΕ 1. Hammerklavier ͱδάβά 2. δάβάͰ࡞ۂͯ͠ΈΔ 3. ຺ମͷදݱྗͱָේͷෳࡶੑ
1. Hammerklavier ͱδάβά
1. Hammerklavier ͱδάβά • "Hammerklavier": Beethoven Op.106 • 4 ָষߏͷϐΞϊۂɻୈ
1 ָষ Allegro ιφλܗࣜ • Ͳ͏Βୈ 1 ָষͷௐੑɾಈػ C#dim7 ʹࢧ͞Ε͍ͯΔΒ͍͠ • ͱ͍ͬͯ Reference [72] Mazzola ઌੜͷຊͳͷͰɺWikipedia ͳΒ [ಠࣗݚڀ] λάҊ݅ • ͳ͓ɺ[72] υΠπޠ
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 ͕༠ಋ͞ΕΔͷࣗવʢͱओு͍ͯ͠Δʣ • पظ 3 ͷδάβάύλʔϯ͕ Hammerklavier ͷಈػ܈ʹݟͯऔΕΔɺͱ [72] Ͱओு͍ͯ͠ ΔΒ͍͠
1. Hammerklavier ͱδάβά https://slidetodoc.com/concepts-locaux-et-globaux-premire-partie-thorie-objective/
1. Hammerklavier ͱδάβά https://www.geogebra.org/m/YCZa8TAH
2. δάβάͰ࡞ۂͯ͠ΈΔ
2. δάβάͰ࡞ۂͯ͠ΈΔ • Beethoven C#dim7 Ͱ 12 ԻΛ 3
Իͣͭʹ͚ͨʢҰ୴ओுΛೝΊΔʣ • ൃలͤͯ͞ C#aug Ͱ 12 ԻΛ 4 Իͣͭʹ͚ͯΈΔʢϝγΞϯୈ 3 ટ๏తʣ • 4 ԻͣͭͳͷͰɺδάβάͷपظ 4 ʹͳΔ
2. δάβάͰ࡞ۂͯ͠ΈΔ • Beethoven δάβάʢͷઢܗมʣͷ্ʹશͯԻ͕ ฒΜͰ͍ͨ • ൃలͤͯ͞ɺෳຊͷδάβάͷଋΛߟ͑ͯΈΔ • Ճ͑ͯɺ1
ຊͷδάβάʹ෦తʹԻΛฒΔ
2. δάβάͰ࡞ۂͯ͠ΈΔ • ॎํʹผʑͷδάβάʮύλʔϯʯ ΛฒΔ • ্࣌ؒ࣠ʹ 1 ͭͣͭݟ͍͖ͯɺಉ࣌ؒ͡ ্࣠ͷॎํͰҰ൪্ʹ͍ͬͯΔԻͰ
ಈػΛߏ͍ͯ͘͠ʢΒ͍͠ʁʣ • ը૾ಉ͡ۂͷ Main Theme ͷߏʹͭ ͍ͯݴٴ͍ͯ͠Δࢿྉ • "Finally, we take the configuration's top notes within C major." • ಈػͷߏ "configuration"ʢδάβά ύλʔϯͷฒํʣͱટ๏ʹґଘʁ The Future of Music: Towards a Computational Musical Theory of Everything
2. δάβάͰ࡞ۂͯ͠ΈΔ • पظ 4 ͷδάβάʢͷԟ෮ʣͷʮඃ෴ʯΛ࡞Γɺδάβάύλʔϯͷ຺ମΛ࡞Δ • δάβάͷʢยಓʣपظ 4 ͕ͩɺԻΛׂΓͯΔॴ
3 Օॴʹ͍ͯ͠Δʢཧ༝ෆ໌ʣ
2. δάβάͰ࡞ۂͯ͠ΈΔ • Exercise 5
2. δάβάͰ࡞ۂͯ͠ΈΔ • ॎʹฒΔδάβάύλʔϯͷ Έ߹ΘͤΛ຺ମʹԊͬͯ࡞Δ
2. δάβάͰ࡞ۂͯ͠ΈΔ • ॎʹฒΔδάβάύλʔϯͷ Έ߹ΘͤΛ຺ମʹԊͬͯ࡞Δ • ͨ͘͞Μ࡞Δ
2. δάβάͰ࡞ۂͯ͠ΈΔ • ॎํʹ 1 ͭͷʹؚ·ΕΔ δάβάύλʔϯ͕ฒͿΑ͏ʹ • ԣํʢ࣌ؒํʣʹผͷ ύλʔϯ͕ฒͿΑ͏ʹஔ
• ͨͩ͠ɺશମͱ্ͯ͠ߦ͢ΔΑ͏ δάβάͷଋશମΛ͚Δ • ͲΕ͘Β͍͚͍ͯΔʁ ͍ͬͯΔટ๏ʁ ͍·͍ͪෆ໌
2. δάβάͰ࡞ۂͯ͠ΈΔ • ϐΞϊͷӈखͱࠨखͦΕͧΕʹผͷ δάβάଋΛ༻ҙ • ͳΜͱͳ͘ԣ࣠ํͷ։࢝ҐஔΛ ͣΒ͍ͯ͠Δؾ͕͠ͳ͘ͳ͍
2. δάβάͰ࡞ۂͯ͠ΈΔ • Ͱɺ͏·͍ײ͡ʹΤΠϠ͢Δͱۂ͕Ͱ͖ΔΒ͍͠
3. ຺ମͷදݱྗͱָේͷෳࡶੑ
3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ू߹ X ͷඃ෴ C 2X ͷ෦ू߹
• ຺ମ N(C) 2C = 22^Xͷ෦ू߹ʢॏͳΓΛ࣋ͭू߹Ͳ͏͠Λ·ͱΊͨΓ͢Δʣ N(C) X ٿڥอશͷͨΊεϥΠυΛ࠶ར༻͍ͯ͠·͢
3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ू߹ X ͷඃ෴ C 2X ͷ෦ू߹
• ຺ମ N(C) 2C = 22^Xͷ෦ू߹ʢॏͳΓΛ࣋ͭू߹Ͳ͏͠Λ·ͱΊͨΓ͢Δʣ • ಉ͡Α͏ʹ຺ମͷ຺ମΛߟ͑ΕɺͲΜͲΜࢦͷݞ͕͍ͬͯ͘ → ߴ͍දݱྗΛ࣋ͭ N(C) X ٿڥอશͷͨΊεϥΠυΛ࠶ར༻͍ͯ͠·͢
3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ָේͷଊ͑ํ 1ɿԻූͷཏྻͱݟΔ • ͭͳ͕ΓΛߟྀ͠ͳ͍ͷͰɺͦΕͧΕͷԻූ͕ݽཱͨ͠ 0 ୯ମʢʣͱͯ͠ࢄࡏ͢Δ͚ͩ •
Ҡௐͯ͠ಉָ͡ේಉܕʢ࣮࣭ಉ͡ʣʹݟ͑ͯ͠·͏
3. ຺ମͷදݱྗͱָේͷෳࡶੑ • ָේͷଊ͑ํ 2ɿશͯͷԻΛؚΉɺ຺ମతͳ 1 ຕͷਤͰߟ͑Δ • ϐονؒͷͭͳ͕Γɺಈػͱల։ͱͷͭͳ͕ΓɺௐͲ͏͠ͷͭͳ͕Γɺͷͭͳ͕Γɺetc. •
͍Ζ͍Ζͳͭͳ͕ΓΛදݱͨ͠ਤͱͳΓɺ༷ʑͳύϥϝʔλΛࡏͤͨ͞ͷʹͳΔ • ͦͷΑ͏ͳਤҰମͲΕ͘Β͍ଘࡏ͢Δʁ
3. ຺ମͷදݱྗͱָේͷෳࡶੑ https://slidetodoc.com/concepts-locaux-et-globaux-premire-partie-thorie-objective/