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

Transcript

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

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2. ໨ඪ
   1. άϥϑΛ֦ுͨ͠Β຺ମͩͶɺతͳཧղ
   2. ͜ΕΛཧղ
   

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

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4. 1. άϥϑͱʮͭͳ͕Γʯ
   

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

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

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7. 2. ୯ମͱʮͭͳ͕Γʯ
   

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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
   

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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
   

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

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11. 3. ຺ମͱʮͭͳ͕Γʯ
    

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

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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
    

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

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

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

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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
    

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

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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
    

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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
    

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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
    

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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
    

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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
    

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

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25. 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    ... ...
    ... ...
    

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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
    

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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
    

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

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