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

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

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

1. ಋೖɿָཧͷྺ࢙

• (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

• 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

2. Lewin ͷཧ࿦ͱ܈࡞༻

• ̇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

• ̇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

• 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

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)

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൒

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

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

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

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

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

3. يಓʹΑΔྨผͱ࿨Իͷ෼ྨ

• ܈࡞༻ 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

• 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

• ܈ 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෼ྨ

4. ޙষʹඞཁͳ஌ࣝ

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