Ryo Sakuma
July 04, 2021
49

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

July 04, 2021

## 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/