Ryo Sakuma
June 28, 2021
100

# Cool Math for Hot Music 輪読会 Sec.16 Part 1

June 28, 2021

## Transcript

1. HASH 2021-06-28
16. Nerves (Part 1)
Cool Math for Hot Music ྠಡձ

2. ໨ඪ
1. άϥϑΛ֦ுͨ͠Β຺ମͩͶɺతͳཧղ
2. ͜ΕΛཧղ

3. େ·͔ͳྲྀΕ
1. άϥϑͱʮͭͳ͕Γʯ
2. ୯ମͱʮͭͳ͕Γʯ
3. ຺ମͱʮͭͳ͕Γʯ
4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ

4. 1. άϥϑͱʮͭͳ͕Γʯ

5. 1. άϥϑͱʮͭͳ͕Γʯ
• લͷ 15 ষͰ΍ͬͨʮάϥϑʯ͸௖఺Ͳ͏͠ͷʮͭͳ͕ΓʯΛҙࣝͨ͠΋ͷ
• ٯͷݟํͱͯ͠ɺ௖఺Λհͨ͠ลͲ͏͠ͷʮͭͳ͕Γʯͱ΋ݟΕΔ

6. 1. άϥϑͱʮͭͳ͕Γʯ
• ઢ෼ʢ1 ࣍ݩʣΛ఺ʢ0 ࣍ݩʣͰʮͭͳ͛Δʯͱάϥϑ͕Ͱ͖Δ
• ࢛໘ମʢ3 ࣍ݩʣΛࡾ֯ܗʢ2 ࣍ݩʣͰʮͭͳ͛Δʯͱ Fig. 16.4 Έ͍ͨͳਤܗ͕Ͱ͖Δ
• খ͞ͳਤܗʢ୯ମʣͷʮͭͳ͕ΓʯͰେ͖ͳਤܗΛߟ࡯͢Δ → Ґ૬زԿֶ

7. 2. ୯ମͱʮͭͳ͕Γʯ

8. 2. ୯ମͱʮͭͳ͕Γʯ
• ୯ମʢsimplex, [ෳ]simplices | simplexesʣpp.129 Def.43
• ఺ɺઢ෼ɺࡾ֯ܗɺ࢛໘ମͷॱʹ 0 ୯ମɺ1 ୯ମɺ2 ୯ମɺ3 ୯ମͱ͍͏ pp.130
• n ࣍ݩਤܗͷʮ୯ମʯ΋ଘࡏ
Hjhornbeck - ౤ߘऀࣗ਎ʹΑΔ࡞඼, CC දࣔ-ܧঝ 4.0, https://commons.wikimedia.org/w/index.php?curid=57828604ʹΑΔ

9. 2. ୯ମͱʮͭͳ͕Γʯ
• ༨ஊɿ༗ݶཁૉ๏
• ݐங෺ͷ଱਒ੑͱ͔ΛධՁͨ͠Γ͢Δ਺஋ܭࢉख๏
• ߏ଄෺Λࡾ֯ܗͱ͔ʹ෼ׂͯ͠ܭࢉ͢Δ͜ͱ͕ଟ͔ͬͨ͸ͣ
By I, Zureks, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2358783

10. 2. ୯ମͱʮͭͳ͕Γʯ
• ͳͥ෼ׂ͢Δʁ
• ؆୯ͳਤܗʹམͱ͠ࠐΜͰɺݩͷෳࡶɾେ͖ͳਤܗΛۙࣅ͍ͨ͠
• ྫɿ୯ମΛͭͳ͛ͯϔϞάϩϏϯͷߏ଄Λۙࣅ
https://www.slideshare.net/Eniod/tutorial-of-topologicaldataanalysispart1basic

11. 3. ຺ମͱʮͭͳ͕Γʯ

12. 3. ຺ମͱʮͭͳ͕Γʯ
• গ͠ผͷΞϓϩʔνͰʮۙࣅʯ͢Δ͜ͱΛߟ͑·͢
• ͜͜ʹԁ͕͋Γ·͢
• ԁ͸఺ͷू߹ X Ͱ͢

13. 3. ຺ମͱʮͭͳ͕Γʯ
• ԁΛ n ຊͷʮހʯʹɺͨͩ͠ॏͳΔΑ͏ʹ෼ׂ͠·͢ʢX ͷඃ෴ɿcovering of Xʣ
• ހ͸఺ͷू߹ C1, C2, ..., Cn
Ͱ͢ʢ͜͜Ͱ C = { C1, C2, ..., Cn } ͱஔ͖·͢ʣ
• ໌Β͔ʹ ∪C = X Ͱ͢ʢDef.43 L.2ʣ

14. 3. ຺ମͱʮͭͳ͕Γʯ
• ހΛ఺ʢ0 ୯ମʣʹஔ͖׵͑ɺࡶʹந৅Խ͠·͢
• ݩʑॏͳ͍ͬͯͨՕॴʢ∩s ≠ ∅ʣ͕͋Ε͹ɺ఺ͱ఺Λઢ෼ sʢ1 ୯ମʣͰͭͳ͗·͢
• ໌Β͔ʹ ∪C = X Ͱ͢ʢDef.43 2 ߦ໨ʣ

15. 3. ຺ମͱʮͭͳ͕Γʯ
• ͜ΕΛඃ෴ C ͷ຺ମʢNerveʣͱ͍͍ɺN(C) ͱॻ͖·͢
• ຺ମ͸୯ମʹʮͭͳ͕ΓʯΛ࣋ͨͤͨू߹Ͱ΋͋Γ·͢ʢந৅త୯ମෳମʣ
N(C)

16. 3. ຺ମͱʮͭͳ͕Γʯ
• ൃలɿ຺ମఆཧ
• ͏·͘ඃ෴Λͱ຺ͬͯମΛ࡞Δͱɺݩͷू߹ X ͱ຺ମ N(C) ͸ʮ࣮࣭ಉ͡ʯˠ ͍͍ۙࣅ
• ࣮࣭ಉ͡ʹϗϞτϐʔಉ஋
N(C)
X

17. 3. ຺ମͱʮͭͳ͕Γʯ
• ຺ମ N(C) ʹؚ·ΕΔ n ୯ମΛ Nn(C) ͱॻ͖·͢
• ྫɿN0(C) ͸ 0 ୯ମʢ఺ʣͷू߹ɺN1(C) ͸ 1 ୯ମʢઢ෼ʣͷू߹
• ͢ͳΘͪɺؔ਺ N1(C) → N0(C) ͸άϥϑʹͳΔʢڭՊॻ pp.133ʣ
N(C)
X

18. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• C Major εέʔϧԻ͔ΒͳΔू߹ X = { c, d, e, f, g, a, b } Λஔ͖·͢
a
b
c
d
e
f g

19. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• C Major εέʔϧԻ͔ΒͳΔू߹ X = { c, d, e, f, g, a, b } Λஔ͖·͢
• Ⅰ ͷ࿨ԻΛ I = { c, e, g } ͱஔ͖·͢
a
b
c
d
e
f g
I

20. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• ͦΜͳ۩߹ʹμΠΞτχοΫίʔυΛ͢΂ͯຒΊ͍͖ͯ·͢
• ͜͜Ͱ C = { I, II, III, IV, V, VI, VII } ͸໌Β͔ʹεέʔϧ X ͷඃ෴
a
b
c
d
e
f g
I
III
V
VII
II
IV
VI

21. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• μΠΞτχοΫίʔυΛ 0 ୯ମͰந৅Խʢthe seven degrees as 0-simplices )
• ॏͳ͍ͬͯͨ෦෼Λ 1 ୯ମͰ݁Ϳʢthe 14 lines for all pairs of degrees that intersectʣ
I
III
V
VII
II
IV
VI
a
b
c
d
e
f g
I
III
V
VII
II
IV
VI

22. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• 2 ॏͷॏͳΓ͸ 1 ୯ମͰܨ͛Δ → 3 ॏͷॏͳΓ͸ 2 ୯ମʢࡾ֯ܗʣͰܨ͛Δ
• ஫ɿ1 ୯ମ Λ 3 ͭܨ͖͛ͨ݀͋ࡾ֯ܗͱɺ2 ୯ମͷ໘ੵ͖ͭࡾ֯ܗ͸ผ
I
III
V
VII
II
IV
VI
a
b
c
d
e
f g
I
III
V
VII
II
IV
VI

23. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• ผͷදݱܗࣜͰ੔ཧͯ͠ΈΔʢલϖʔδͱʮͭͳ͕Γʯ͸ಉ͡ʣ
• ࠨ୺ͱӈ୺͸ʮͭͳ͕Δʯ͸͚ͣͩΕͲɺI ͱ III ্͕Լٯ޲͖ʁ
I
III
V
VII
II
IV
VI
I
III

24. 3. ຺ମͱʮͭͳ͕Γʯ
• Example 39
• Ͳ͔͜Ͱ 1 ճͶ͡Βͳ͍ͱ I ͱ III ͕ʮͭͳ͕Βͳ͍ʯ
• Fig. 16.2 ͰʮϝϏ΢εͷྠʯͱࣔࠦ͞Ε͍ͯΔͷ͸͜Ε

25. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ

26. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Example 39
• ϦʔϚϯͷػೳ࿨੠࿦ "Harmony, Simplified"ʢ1893ʣ
• ౰࣌ͷ࡞ۂՈɿυϏϡογʔʢ1862-1918ʣγΣʔϯϕϧΫʢ1874-1951ʣ
• ౰࣌ͷཧ࿦Ոɿθώλʔʢ1788-1867ʣγΣϯΧʔʢ1868-1935ʣ
I
III
V
VII
II
IV
VI
I
III

27. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Example 39
• ڭՊॻ pp.130ʮୈ 1 ৚݅ͱͯ͠ɺI, IV, V ʹରͯ͠ T, S, D ͱఆΊɺྠֲઢ্Ͱ͸ҟͳΔ TSD Λ༩
͑ΔΑ͏ϦʔϚϯ͸ఆΊͨʯ
• ͔ͨ͠ʹ IV→I→V ͸ S→T→D ͱҟͳΔ
I:T
III
V:D
VII
II
IV:S
VI
I:T
III
্ஈ͔Β

28. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Example 39
• ڭՊॻ pp.131ʮୈ 2 ৚݅ͱͯ͠ɺฒߦͳ౓਺ πY ͸ಉ͡ TSD ΛऔΔΑ͏ʹͨ͠ʯ
• ͓ͦΒ͘ӳޠ൛ݪจ pp.73-74 ͷ Tp, Sp, Dp ͷ͜ͱ
• Tonic parallel, Subdominant parallel, Dominant parallel
I:T
III:D
V:D
VII:?
II:S
IV:S
VI:T
I:T
III:D
্ஈ͔Β

29. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Example 39
• Mazzola ઌੜͷओுʮୈ 2 ৚݅Λ V→III→I ͱॱ൪ʹద༻͍ͯͬͨ͠ΒɺD = T ʹͳͬͯໃ६͢Δ
ΑͶʁ͜Ε͸ϝϏ΢εͷྠ͕޲͖Λ࣋ͨͳͯ͘දཪ൓ରʹͳͬͪΌ͏͔ΒͩΑʯ
• ׂͱి೾ͳ͜ͱݴͬͯΔؾ͕͢Δʢ͜ͳΈʣ
I:T
III:D
V:D
VII:?
II:S
IV:S
VI:T
I:T
III:D
্ஈ͔Β

30. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• ༨ஊɿͦ΋ͦ΋ϦʔϚϯ͸ 1 ͭͷ࿨Իʹ 1 ͭͷػೳΛׂΓ౰ͯΑ͏ͱͨ͠ͷ͔ʁ
• ϦʔϚϯ͸ 1 ͭͷ࿨Ի͕औΓಘΔػೳ͸ߴʑ 3ʮछྨʯͱ͚ͨͩ͠Ͱɺػೳ͕ 3 ͭͱ͸ݴ͍ͬͯ
ͳ͍ɺͦ΋ͦ΋ 1 ͭͷ࿨Ի͕ඞͣಉ͡छྨͷػೳΛऔΔͱ΋அݴ͍ͯ͠ͳ͍ɺͱ͍͏ࢦఠ͕͋Δ
• ػೳ͕ 3 ͚ͭͩɺಉ͡ػೳ͸ަ׵ՄೳɺͳͲ͸ޙͷཧ࿦ՈʹΑΔ֦େղऍ͔ʢౡԬ࿨੠ͳͲʣ
• ࢀߟɿSoundQuestʮػೳ࿨੠ͷมભ ❶ ϦʔϚϯͷ“ݪ࡞”ʯ
I
III
V
VII
II
IV
VI
I
III

31. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• ؓ࿩ٳ୊
• ௕͞ n ͷ chain graph |n| ͱ຺ମͱͷؔ࿈
• ඃ෴ C1(n) = { {0, 1}, {1, 2}, ... , {n - 1, n } }
• Chain graph |n| ͷ edge ͱ຺ମͷ 0 ୯ମɺ1 ʙ n - 1 ͷ௖఺ͱ 1 ୯ମ͕ରԠ
0 1 2 i - 1 i i + 1 n
n - 1
n - 2
... ...
... ...

32. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Ұൠʹ Ck(n) = { {0, 1, ..., k}, ... , {n - k, ..., n - 1, n} }
• ྫͱͯ͠ C3(16) = { {0, 1, 2, 3}, {1, 2, 3, 4}, ..., {13, 14, 15, 16} } ͷ຺ମΛߟ͑Δͱɺ
࢛໘ମΛࡾ֯ܗͰషΓ߹Θͤͯʮͭͳ͛ͨʯΑ͏ͳਤܗʹͳΔ

33. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Ұൠʹ Ck(n) = { {0, 1, ..., k}, ... , {n - k, ..., n - 1, n} }
• ྫͱͯ͠ C3(16) = { {0, 1, 2, 3}, {1, 2, 3, 4}, ..., {13, 14, 15, 16} } ͷ຺ମΛߟ͑Δͱɺ
࢛໘ମΛࡾ֯ܗͰషΓ߹Θͤͯʮͭͳ͛ͨʯΑ͏ͳਤܗʹͳΔ

34. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
• Example 40
• ͜͏͍͏͜ͱʁʢࣗ৴͕ͳ͍ʣ
IM7
IIIm7
V7
VIIm7-5
IIm7
IVM7
VIm7
IM7
IIIm7
V7
VIIm7-5
IIm7
IVM7
VIm7