Cool Math for Hot Music 輪読会 Sec.19

Transcript

1. HASH 2021-07-26
   19. Group Actions, Subgroups, Quotients, and
   Products (Part 1)
   Cool Math for Hot Music ྠಡձ
   

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2. ໨ඪ
   1. Իָͷ Example ʹ͓͚ΔओுΛཧղ
   2. GIS ͷ֓೦Λײ֮Ͱ௫Ή
   3. Fig. 19.4 ͷҙຯΛཧղ
   

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3. େ·͔ͳྲྀΕ
   1. ಋೖɿָཧͷྺ࢙ §2
   2. Lewin ͷཧ࿦ͱ܈࡞༻ §19.1
   3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ §19.2.1
   4. ޙষʹඞཁͳ஌ࣝ §19.1 & §19.2
   

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4. 1. ಋೖɿָཧͷྺ࢙
   

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5. • (1517-1590) G. Zarlino: ࡾ౓ԻఔͷීٴͳͲ
   • (1722) J. P. Rameau: ࿨੠࿦
   • (1739) L. Euler: ࿨ԻͷάϥϑʢTonnetz, Euler spaceʣΛൃ໌
   • (1893) H. Riemann: ػೳ࿨੠࿦
   • (1970) Berklee College of Music ։ߍ
   • (1973) A. Forte: American Set Theory ͷᅘ໼
   • (1987) D. Lewin: Transformation Theory Λఏএ
   • (1990) H. Klumpenhouwer: Klumpenhouwer Network Λൃ໌
   1. ಋೖɿָཧͷྺ࢙
   

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6. • Lewin ҎલɿԻָతର৅ͦΕࣗମΛॏࢹ
   • ྫɿC Major key ʹ͓͍ͯɺG7 ͸υϛφϯτ
   • LewinɿԻָతର৅͕ A → B ͱભҠ͢Δࡍͷաఔɾม׵Λॏࢹ
   • ྫɿC Major key ʹ͓͍ͯɺCM7 → G7 ͷʮաఔʯ͸υϛφϯτม׵
   • ࿨ԻʹݶΒͣɺ༷ʑͳԻָతର৅ͷมԽաఔΛؔ਺ɾ܈ͱͯ͠ߟ͑Δͷ͕ Lewin ͷ
   "Transformation Theory"
   • Lewin ͷߟ͑ํ͸ɺָཧͷྺ࢙ʹ͓͚ΔҰछͷʮύϥμΠϜγϑτʯ
   1. ಋೖɿָཧͷྺ࢙
   

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7. 2. Lewin ͷཧ࿦ͱ܈࡞༻
   

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8. • ̇Example 46 લ൒
   • Lewin ͷ "Transformation Theory" ʹ͓͚Δ࠷ॳͷ࢓ࣄ͸ɺԻఔͷ֓೦ΛҰൠԽͨ͠
   "Generalized Interval System (GIS)" ͷఏএ
   • ௨ৗͷʮԻఔʯΛ࢖ͬͨ GIS ͷྫɿ
   2. Lewin ͷཧ࿦ͱ܈࡞༻
   C
   C#
   D
   Ĕ
   E
   F
   F#
   G
   Ă
   A
   B̆
   B
   Իָతର৅ͷू߹ S = { C, C#, ..., B }
   int(C, E) + int(E, G) = int(C, G) ... ݁߹ଇ͕੒Γཱͭ
   int(C, E) = 4 ͸ C Λ 4 ൒Ի্͛Δʮ࡞༻ʯͱΈͳͤΔ
   int(s, t) ͸ԋࢉ + (mod 12) Ͱ܈Λͳ͢
   ܈Λ IVLS ͱ͓͘ͱɺؔ਺ int : S × S → IVLS
   GIS ͸ (S, IVLS, int)
   int(C, E) = 4
   int(E, G) = 3
   ԻఔΛଌΔؔ਺ int(C, E) = 4, int(E, G) = 3, int(C, G) = 7
   

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9. • ̇Example 46 લ൒
   • Իָతର৅ͷू߹ S ΍ɺԻఔΛଌΔؔ਺ int Λ͏·͘ม͑ͯ͋͛Ε͹ɺଞʹ΋৭ʑͳաఔɾมԽ
   ΛදݱͰ͖Δʢ࿨ԻɺϦζϜɺetc.ʣ
   • ͏·͘ม͑ͯ 㱻 ݁߹ଇ΍աఔͷҰҙੑͳͲʢޙड़ʣ
   2. Lewin ͷཧ࿦ͱ܈࡞༻
   C
   C#
   D
   Ĕ
   E
   F
   F#
   G
   Ă
   A
   B̆
   B
   Իָతର৅ͷू߹ S = { C, C#, ..., B }
   int(C, E) + int(E, G) = int(C, G) ... ݁߹ଇ͕੒Γཱͭ
   int(C, E) = 4 ͸ C Λ 4 ൒Ի্͛Δʮ࡞༻ʯͱΈͳͤΔ
   int(s, t) ͸ԋࢉ + (mod 12) Ͱ܈Λͳ͢
   ܈Λ IVLS ͱ͓͘ͱɺؔ਺ int : S × S → IVLS
   GIS ͸ (S, IVLS, int)
   int(C, E) = 4
   int(E, G) = 3
   ԻఔΛଌΔؔ਺ int(C, E) = 4, int(E, G) = 3, int(C, G) = 7
   

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10. • int(C, E) = 4 ͸ C Λ 4 ൒Ի্͛Δʮ࡞༻ʯͱΈͳͤΔ
    • ݴ͍׵͑Ε͹ (int(C, E), C) ͔Β E ΁ͷରԠ
    • ରԠؔ܎ʢ࡞༻ʣΛ܈શମʹΘͨͬͯूΊͨ΋ͷ → ܈࡞༻
    2. Lewin ͷཧ࿦ͱ܈࡞༻
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    Իָతର৅ͷू߹ S = { C, C#, ..., B }
    ԻఔΛଌΔؔ਺ int(C, E) = 4, int(E, G) = 3, int(C, G) = 7
    int(C, E) + int(E, G) = int(C, G) ... ݁߹ଇ͕੒Γཱͭ
    int(C, E) = 4 ͸ C Λ 4 ൒Ի্͛Δʮ࡞༻ʯͱΈͳͤΔ
    int(s, t) ͸ԋࢉ + (mod 12) Ͱ܈Λͳ͢
    ܈Λ IVLS ͱ͓͘ͱɺؔ਺ int : S × S → IVLS
    GIS ͸ (S, IVLS, int)
    int(C, E) = 4
    int(E, G) = 3
    

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11. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • Def. 52ʢ܈࡞༻ʣ
    • ܈ (G, *) ͱɺ೚ҙͷू߹ X ʹ͍ͭͯߟ͑Δ
    • ܈࡞༻ͱ͸࣍ͷ 2 ৚݅Λຬͨؔ͢਺ f : G × X → X
    • (i) ∀x f(e)(x) = x
    • (ii) ∀x f(g * h)(x) = f(g)(f(h)(x))
    • ͜͜Ͱ؆ศͷͨΊ f(g)(x) = gɾx ͱ͍͏ه๏Λಋೖ͢Ε͹ɺ
    • (i) eɾx = x
    • (ii) (g * h)ɾx = gɾ(hɾx)
    

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12. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • ̇Example 46 ޙ൒ʢWe want to show ʙʣ
    • IVLSopp ͱ͍͏ͷ͸܈ IVLS ʹର͢Δٯ܈
    • ٯ܈ (Gopp, *') ͱ͸ɺ܈ (G, *) ʹରͯ͠ g1 *' g2 = g2 * g1
    Λຬͨ͢܈
    • ܈ G ͷ (g * h)ɾx = gɾ(hɾx) Λຬͨ͢࡞༻͸ʮࠨ܈࡞༻ʯͱ͍͏
    • ͜ͷͱ͖ٯ܈͸ xɾ(g * h) = (xɾg)ɾh Λຬͨ͢͜ͱ͕஌ΒΕ͍ͯΔʢӈ܈࡞༻ʣ
    • Lewin ͷΦϦδφϧ൛͸ӈ܈࡞༻Λ࢖ͬͨఆٛͳͷͰɺࠨ܈࡞༻ʢ܈࿦ͷ׳ྫʣͷఆ͔ٛΒߟ͑
    Δͱٯ܈ͩΑͶɺΛԆʑͱॻ͍͍ͯΔͷ͕̇Example 46 ͷޙ൒
    

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13. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • Def. 52ʢيಓʣ
    • ܈࡞༻ G × X → X ʹ͓͍ͯɺX ্Ͱఆٛ͞ΕΔҎԼͷؔ܎ ʙ ͸ʮಉ஋ؔ܎ʯʹͳΔ
    • x ʙ y 㱻 ∃g ∈ G : gɾx = y
    • ͋Δ x ∈ X ʹ͍ͭͯɺ্هͷಉ஋ؔ܎ʹΑΔಉ஋ྨ [ x ] ΛʮيಓʯͱΑͼɺGɾx ͱॻ͘
    • (∀x) Gɾx = X Λຬͨ͢ͱ͖ɺ܈࡞༻͸ʮભҠతʯͰ͋Δͱ͍͏ʢيಓ͕ͨͩҰͭʣ
    • ભҠత͔ͭ (∀x ∀y ∈ X) ∃!g : gɾx = y Λຬͨ͢ͱ͖ɺ܈࡞༻͸ʮ୯७ભҠతʯͰ͋Δͱ͍͏
    

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14. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • يಓͷྫ 1
    • ର৅ X = { C, ..., B }ɺ܈ G = { e, I0 } (= { ແ, C-F#࣠Ͱࠨӈ൓స })
    • يಓ GɾC = { C }, GɾC# = { C#, B }, GɾF = { F, G }, ...
    • يಓ͕ෳ਺͋ΔͷͰɺ܈࡞༻͸ભҠతͰͳ͍
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    I0
    

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15. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • يಓͷྫ 2
    • ର৅ X = { C, ..., B }ɺ܈ G = { e, T1 , ..., T11 } (= { ແ, 1 ൒Ի্͛Δ, ..., 11 ൒Ի্͛Δ })
    • يಓ GɾC = { C, ..., B } = X, GɾC# = { C, ..., B } = X, GɾF = { C, ... , B } = X, ...
    • يಓ͸།Ұɺ͔ͭ 2 ఺ؒͷม׵͕ҰҙͳͷͰɺ܈࡞༻͸ભҠత͔ͭ୯७ભҠత
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B T2
    T2
    

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16. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • يಓͷྫ 3
    • ର৅ X = { C, ..., B }ɺ܈ G = { e, T1 , ..., T11, I0, T1*I0, ..., T11*I0 }
    • يಓ GɾC = { C, ..., B } = X, GɾC# = { C, ..., B } = X, ... ΑͬͯભҠత
    • C → C# ͷ࡞༻ͱͯ͠ T1
    ͱ T1*I0
    ͷՄೳੑ͕ 2 ͭ͋ΔͷͰɺ୯७ભҠతͰ͸ͳ͍
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    T1
    T1 I0
    T1*I0
    

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17. 2. Lewin ͷཧ࿦ͱ܈࡞༻
    • Ti*I0 = Ii
    ͱ͓͘
    • άϥϑͷ § 15 ʹग़͖ͯͨ Klumpenhouwer Network ͷ T ͱ I ͸
    ͜ͷ܈࡞༻Λද͍ͯ͠Δʢ࡞༻ʹண໨ͨ͠άϥϑʣ
    • ͳ͓ɺKlumpenhouwer ͸ Lewin ͷఋࢠ
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    I7
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    I5
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    T2
    Fig. 15.2
    

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18. 3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ
    

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19. • ܈࡞༻ G × X → X ʹ͓͍ͯɺX ্Ͱఆٛ͞ΕΔҎԼͷؔ܎ ʙ ͸ʮಉ஋ؔ܎ʯʹͳΔ
    • x ʙ y 㱻 ∃g ∈ G : gɾx = y
    • ͋Δ x ∈ X ʹ͍ͭͯɺ্هͷಉ஋ؔ܎ʹΑΔಉ஋ྨ [ x ] ΛʮيಓʯͱΑͼɺGɾx ͱॻ͘
    • ҎԼͷਤʹ͓͚ΔيಓɿGɾC = { C }, GɾC# = { C#, B }, GɾF = { F, G }, ...
    3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    I0
    

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20. • يಓ Gɾx ͸ର৅ͷू߹ X Λྨผ͢Δ → 2 ͭͷيಓ͸૬౳͔ഉ൓ͷͲͪΒ͔
    • ূ໌ʣ2 ͭͷيಓ Gɾx, Gɾy ͷڞ௨ݩ z ∈ X Λߟ͑Δ
    • يಓͷఆٛΑΓ z = g1
    ɾx = g2
    ɾy ͳͷͰɺx = (g1-1*g2)ɾy
    • ͢ͳΘͪ೚ҙͷ x ͸ y ͷيಓʹؚ·ΕΔͷͰɺ2 ͭͷيಓ͸૬౳͔ഉ൓ͷͲͪΒ͔˙
    3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ
    C
    C#
    D
    Ĕ
    E
    F
    F#
    G
    Ă
    A
    B̆
    B
    I0
    

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21. • ܈ G = { e, T1 , ..., T11, I0, I1, ..., I11 }
    • ू߹ X = { C, ..., B } ʹ͍ͭͯɺ2X ্ʹ͓͚Δ܈࡞༻Λߟ͑Δʢ࿨Ի ch ∈ 2X ʣ
    • ࿨Իશମ 2X Λ܈ G ʹΑΔيಓͰྨผʢʹ෼ྨʣ͢Δ
    • ྫ͑͹ɺC+5M7 ͱ F#mM7 ͸ I5
    ʹΑͬͯಉ͡يಓ → ಉ͡άϧʔϓͱͯ͠෼ྨՄೳ
    3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ
    

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22. 4. ޙষʹඞཁͳ஌ࣝ
    

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23. • ਖ਼ن෦෼܈ɺ঎܈ a.k.a. ৒༨܈ɿ§ 20 Ͱग़ͯ͘Δ
    • ܈ͷ௚ੵɿ§ 21 Ͱग़ͯ͘Δ
    • ࢿྉ੍࡞͕ྗਚ͖ͨͷͰɺ͕࣌ؒ༨ͬͨΒखॻ͖ղઆ
    4. ޙষʹඞཁͳ஌ࣝ
    

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