$30 off During Our Annual Pro Sale. View Details »

Cool Math for Hot Music 輪読会 Sec.5

Cool Math for Hot Music 輪読会 Sec.5

2021-05-24

Ryo Sakuma

May 17, 2021
Tweet

More Decks by Ryo Sakuma

Other Decks in Science

Transcript

  1. HASH 2021-05-XX
    5. Universal Properties
    Cool Math for Hot Music ྠಡձ

    View Slide

  2. ໨ඪ
    1. Իָͷ Example ʹ͓͚ΔओுΛཧղ
    2. Mazzola ઌੜͷ American Set Theory ʹର͢Δ
    dis Γ۩߹ΛຯΘ͏

    View Slide

  3. େ·͔ͳྲྀΕ
    1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ§5.0
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
    3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு §5.6
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ §5.6
    5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

    View Slide

  4. 1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ

    View Slide

  5. 1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ
    b
    a
    1
    2
    3
    x
    y
    f1
    ```python
    def f1(a1):
    return x
    ```
    • ؔ਺ f : a → b ͸ͨ͘͞Μ͋Δ

    View Slide

  6. 1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ
    b
    a
    1
    2
    3
    x
    y
    f2
    ```python
    def f2(a1):
    if a1 == 1:
    return x
    if a1 == 2:
    return x
    else:
    return y
    ```
    ؔ਺ͷ໼ҹͰ͸ͳ͘
    ཁૉରԠͷ໼ҹ
    • ؔ਺ f : a → b ͸ͨ͘͞Μ͋Δ

    View Slide

  7. 1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ
    b
    a
    1
    2
    3
    x
    y
    fi
    ཁૉରԠͷ໼ҹΛଋͶͨ
    1 ຊͷؔ਺ͷ໼ҹ
    f ͷ࣮૷͸ 23 = 8 ௨Γ
    • ؔ਺ f : a → b ͸ͨ͘͞Μ͋Δ
    { f1, f2, ..., f8 }

    View Slide

  8. 1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ
    b
    a
    1
    2
    3
    x
    y
    fi
    ཁૉରԠͷ໼ҹΛଋͶͨ
    1 ຊͷؔ਺ͷ໼ҹ
    { f1, f2, ..., f8 }
    := baʮ഑ஔू߹ʯ
    := Set(a, b) ...ݍ࿦ Hom(a, b)
    • ؔ਺ f : a → b ͸ͨ͘͞Μ͋Δ
    දه๏ͷҙຯɿ
    ू߹ͷཁૉ਺Λ |A| Ͱॻ͘ͱ
    |ba| = |b||a| = 23
    f ͷ࣮૷͸ 23 = 8 ௨Γ

    View Slide

  9. େ·͔ͳྲྀΕ
    1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ§5.0
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
    3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ §5.6
    5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

    View Slide

  10. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎

    View Slide

  11. • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎

    View Slide

  12. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    c (1) ू߹ c = { 0, 1 } Λ༻ҙ

    View Slide

  13. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    (1) ू߹ c = { 0, 1 } Λ༻ҙ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fa
    Λ༻ҙ
    c
    fa

    View Slide

  14. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    (1) ू߹ c = { 0, 1 } Λ༻ҙ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fa
    Λ༻ҙ
    c
    fb
    (3) c ͷ 2 ఺Λ b ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fb
    Λ༻ҙ

    View Slide

  15. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    (1) ू߹ c = { 0, 1 } Λ༻ҙ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fa
    Λ༻ҙ
    c
    (3) c ͷ 2 ఺Λ b ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fb
    Λ༻ҙ
    (4) ab ฏ໘্ͷ࠲ඪʢ௚ੵ a×b ͷཁૉʣ͕
    2 ఺ܾ·ΔʢԻఔͷଌఆʣ

    View Slide

  16. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    (1) ू߹ c = { 0, 1 } Λ༻ҙ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fa
    Λ༻ҙ
    c
    (3) c ͷ 2 ఺Λ b ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fb
    Λ༻ҙ
    (4) ab ฏ໘্ͷ࠲ඪʢ௚ੵ a×b ͷཁૉʣ͕
    2 ఺ܾ·ΔʢԻఔͷଌఆʣ
    (5) c ͷ 2 ఺Λ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
    ͤ͞Δؔ਺ f ͕͋ͬͯ΋ྑ͍͸ͣ
    f

    View Slide

  17. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    (1) ू߹ c = { 0, 1 } Λ༻ҙ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fa
    Λ༻ҙ
    c
    (3) c ͷ 2 ఺Λ b ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
    ؔ਺ fb
    Λ༻ҙ
    (4) ab ฏ໘্ͷ࠲ඪʢ௚ੵ a×b ͷཁૉʣ͕
    2 ఺ܾ·ΔʢԻఔͷଌఆʣ
    pra
    prb
    (6) ͱ͜ΖͰࣹӨ pra, prb
    ͕ଘࡏ
    (5) c ͷ 2 ఺Λ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
    ͤ͞Δؔ਺ f ͕͋ͬͯ΋ྑ͍͸ͣ

    View Slide

  18. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    ໌Β͔ʹ ( fa , fb ) = ( pra ∘ f , prb ∘ f )
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    c
    pra
    prb
    f
    fb
    fa
    ໰ɿ͜ͷؔ਺ f ͱ ( fa , fb ) ͸࣮࣭ಉ͡ʁ

    View Slide

  19. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    c
    pra
    prb
    f
    fb
    fa
    ౴ɿఆཧ 1ʢ௚ੵͷීวੑʣ
    ࣍ͷؔ਺͕ଘࡏ͠ɺ͔ͭશ୯ࣹ
    Set(c, a × b) → Set(c, a) × Set(c, b)
    : f ( pra ∘ f , prb ∘ f )
    ໌Β͔ʹ ( fa , fb ) = ( pra ∘ f , prb ∘ f )
    ໰ɿ͜ͷؔ਺ f ͱ ( fa , fb ) ͸࣮࣭ಉ͡ʁ

    View Slide

  20. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
    ߹ซͱඇަ࿨ͷҧ͍͸ޱ಄Ͱผ్આ໌༧ఆ

    View Slide

  21. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
    (2) a = {0}, b = {1} ͱ͓͘
    a
    b

    View Slide

  22. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
    a
    b
    fa
    fb
    (3) ҎԼ 2 ͭͷؔ਺Λ༻ҙʢԻఔͷଌఆʣ
    a ͷཁૉ 0 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fa
    b ͷཁૉ 1 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fb
    (2) a = {0}, b = {1} ͱ͓͘

    View Slide

  23. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
    (3) ҎԼ 2 ͭͷؔ਺Λ༻ҙʢԻఔͷଌఆʣ
    a ͷཁૉ 0 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fa
    b ͷཁૉ 1 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fb
    (4) c ΛҰؾʹ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
    ͤ͞Δؔ਺ f ͕͋ͬͯ΋ྑ͍͸ͣ
    f
    (2) a = {0}, b = {1} ͱ͓͘

    View Slide

  24. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    (4) c ΛҰؾʹ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
    ͤ͞Δؔ਺ f ͕͋ͬͯ΋ྑ͍͸ͣ
    (1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
    (3) ҎԼ 2 ͭͷؔ਺Λ༻ҙʢԻఔͷଌఆʣ
    a ͷཁૉ 0 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fa
    b ͷཁૉ 1 Λ࠲ඪ 1 ͭʹରԠͤ͞Δؔ਺ fb
    (5) ͱ͜ΖͰ୯ࣹ ina, inb
    ͕ଘࡏ
    a
    b
    a
    0
    b
    1
    c
    1
    0
    inb
    ina
    (2) a = {0}, b = {1} ͱ͓͘

    View Slide

  25. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    a
    b
    a
    0
    b
    1
    c
    1
    0
    inb
    ina
    fa
    fb
    f
    ໌Β͔ʹ ( fa , fb ) = ( f ∘ ina , f ∘ inb )
    ໰ɿ͜ͷؔ਺ f ͱ ( fa , fb ) ͸࣮࣭ಉ͡ʁ

    View Slide

  26. 2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    • ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ
    c
    a
    b
    a
    0
    b
    1
    c
    1
    0
    inb
    ina
    fa
    fb
    f
    ౴ɿఆཧ 2ʢ௚࿨ͷීวੑʣ
    ࣍ͷؔ਺͕ଘࡏ͠ɺ͔ͭશ୯ࣹ
    Set(a b, c) → Set(a, c) × Set(b, c)
    : f ( f ∘ ina , f ∘ inb )
    ༨ੵ coproduct ͸௚࿨ͷݍ࿦༻ޠ
    ໌Β͔ʹ ( fa , fb ) = ( f ∘ ina , f ∘ inb )
    ໰ɿ͜ͷؔ਺ f ͱ ( fa , fb ) ͸࣮࣭ಉ͡ʁ

    View Slide

  27. େ·͔ͳྲྀΕ
    1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ§5.0
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
    3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ §5.6
    5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

    View Slide

  28. 3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

    View Slide

  29. • ू߹ 2 ݸͷ௚ੵ a × b ͱ͸ʮॱংରͷू߹ʯ
    a × b = { (x, y) | x ∈ a, y ∈ b } (§4 Def. 6)
    3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

    View Slide

  30. • ू߹ 2 ݸͷ௚ੵ a × b ͱ͸ʮॱংରͷू߹ʯ
    a × b = { (x, y) | x ∈ a, y ∈ b } (§4 Def. 6)
    • ݁ہʮ࠲ඪͷू߹ʯ͡ΌΜ……ʁ
    3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

    View Slide

  31. • ू߹ 2 ݸͷ௚ੵ a × b ͱ͸ʮॱংରͷू߹ʯ
    a × b = { (x, y) | x ∈ a, y ∈ b } (§4 Def. 6)
    • ݁ہʮ࠲ඪͷू߹ʯ͡ΌΜ……ʁ
    • ू߹ N ݸͷ௚ੵ a1 × ... × aN
    ͱ͸ʮ࠲ඪͷू߹ʯ
    a1 × ... × aN = { (x1, ..., xN) | x1 ∈ a1, ..., xN ∈ aN }ʢDef. Hashʣ
    3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

    View Slide

  32. ެཧతू߹࿦
    ܯ࡯Ͱ͢ʂʂ
    QVSFTFUTʹ
    ࠲ඪ͸ଘࡏ
    ͠·ͤΜʂʂ

    View Slide

  33. • ͋ͱͰਅ໘໨ʹ΍Γ·͢……ɻ
    • ࠲ඪͰ͸ͳ͘ N-tuple ͱߟ͑Ε͹ɺ·͋……ɻ
    3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

    View Slide

  34. େ·͔ͳྲྀΕ
    1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ§5.0
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
    3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ §5.6
    5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

    View Slide

  35. 4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ

    View Slide

  36. • Ұൠͷ௚ੵू߹ͰʮݩͷॱংʯΛͲͷΑ͏ʹܾΊΔ͔
    • Example 15ɿC Major ͱ C minor ͸ʮͲ͕ͬͪઌʁʯ
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ
    (0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1) (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1)
    C ͔Β B ·Ͱ 12 Ի
    ࢖༻͢ΔԻʹ͸ 0
    ࢖༻͠ͳ͍Իʹ 1
    ΛׂΓ౰ͯͯΈΔ
    True/False ͷϑϥάͱ
    ٯͳؾ͕͢Δ͚ΕͲ
    ޙͰҙຯ͕෼͔Δ
    େલఏɿ֤੒෼Ͳ͏͠͸ൺֱՄೳʢઢܕॱংʣ

    View Slide

  37. • ܾΊํ 1ɿࣙॻࣜॱংʢDef. 19 lexicographic orderingʣ
    • ֤੒෼Ͳ͏͠Λࠨ͔Βॱʹݟ͍ͯͬͯɺॳΊͯ஋͕ҟͳΔҐஔ y Λ୳ͯ͠ɺ
    ... the smallest index y, where ty
    ≠ sy
    • Ґஔ y ͷ੒෼Ͳ͏͠ͷॱংΛɺͦͷ··௚ੵू߹ʹ͓͚Δʮॱংʯͱ͢Δ
    ... (tx)a < (sx)a iff (snip), has ty 4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ
    (0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1) (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1)
    t4 s4
    >
    >
    C Major (tx)a C minor (sx)a
    ઌʂ

    View Slide

  38. • ܾΊํ 2ɿ௚ੵॱংʢDef. 20 product relationʣ
    • ೚ҙͷ੒෼Ͳ͏͕͠ಉ͡ॱংؔ܎Ͱ͋Ε͹ɺͦΕΛ௚ੵू߹ʹ͓͚Δʮॱংʯͱ͢Δ
    ... (tx)aR(sx)a iff txrxsx for each x ∈ a
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ
    (0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1) (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1)
    t4 s4
    >
    ?
    C Major (tx)a C minor (sx)a
    t5 s5
    <
    ௚ੵॱং͸Ұൠʹઢܕॱংͱ͸ͳΒͳ͍

    View Slide

  39. • Mazzola ઌੜͷ American Set Theory ʹର͢Δ dis Γ۩߹ΛຯΘ͏ʢpp.67ʣ
    • American Set Theory ͷ࿈த͸ࣙॻࣜॱং΋஌ΒΜౕΒ͡Όɻ
    ނʹ "most packed to the left" ͱ͔ݴ͍ͳ͕Βӈ͔Βൺֱ͓ͯ͠Δͷ͡Όɻ
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ
    (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1) > (0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1)
    s8
    t8
    >
    C Major
    C minor
    ࣙॻࣜॱংͱ͸ٯͷ݁Ռ

    View Slide

  40. • American Set Theory ͸ࠞཚ͍ͯ͠Δʂʂʂ
    • Forte, Allen. (1973). The Structure of Atonal Music
    => packed from the left
    • Rahn, John. (1980). Basic Atonal Theory
    => packed from the right
    • Straus, Joseph N. (1990). Introduction to Post-Tonal Theory
    => packed from the right
    • Mazzola ઌੜ͕Ҿ༻͍ͯ͠Δ [180] ͸ Straus 1990
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ
    https://www.mta.ca/pc-set/pc-set_new/pages/sources/sources.html

    View Slide

  41. େ·͔ͳྲྀΕ
    1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ§5.0
    2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
    3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
    4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ §5.6
    5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

    View Slide

  42. 5. ႈू߹ͷه๏ 2a ͷҙຯ

    View Slide

  43. • Example 13
    • c = { 0, 1 } ͱͯ͠ɺԻ͕৐͍ͬͯΔηϧ͕ 0ɺ৐͍ͬͯͳ͍ηϧ͕ 1
    • ࡞ۂͱ͸ؔ਺ f : a × b → { 0, 1 } ͰϞσϧԽͰ͖Δ
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    def f(a1, b1):
    if (a1, b1) == (3, 5):
    return 0
    # do stuff
    return 1
    ```
    a
    b

    View Slide

  44. • Example 13
    • ผͷϞσϧԽͱͯ͠ɺa ͷ஋Λड͚औͬͯʮb ͷ஋Λड͚औͬͯ c Λฦؔ͢਺ʯΛฦؔ͢਺ g
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    a
    b
    ```python
    def g(a1):
    def b_to_c(b1):
    if (a1, b1) == (3, 5):
    return 0
    # do stuff
    return 1
    return b_to_c
    ```

    View Slide

  45. • f(3, 5) == g(3)(5) == 0
    • ͲͪΒ΋݁Ռ͸ಉ͡ => f Λ g ͷܗ΁ҰରҰม׵Ͱ͖Δؔ਺͕ଘࡏ͢ΔͷͰ͸ʁ
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    def g(a1):
    def b_to_c(b1):
    if (a1, b1) == (3, 5):
    return 0
    # do stuff
    return 1
    return b_to_c
    ```
    ```python
    def f(a1, b1):
    if (a1, b1) == (3, 5):
    return 0
    # do stuff
    return 1
    ```

    View Slide

  46. • ౴ɿఆཧ 3ʢႈͷීวੑʣ
    શ୯ࣹͱͳΔؔ਺ δ : Set(a × b, c) → Set(a, cb) ͕ଘࡏ
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    def delta(f1):
    def take_a(a1):
    def take_b(b1):
    return f1(a1, b1)
    return take_b
    return take_a
    ```
    ```python
    def f(a1, b1):
    if (a1, b1) == (3, 5):
    return 0
    # do stuff
    return 1
    ```

    View Slide

  47. • Example 14ɿ࿨ԻΛ਺ֶతʹදݱ͢Δ
    • 12 Իͷू߹ P = { p0, p1, ..., p12 }
    • ࿨Ի ch ͱ͸ ch ⊂ P Ͱ͋Γ ch ∈ 2P Ͱ΋͋Δʢe.g. C Major { c, e, g } = { p0, p4, p7 }ʣ
    5. ႈू߹ͷه๏ 2a ͷҙຯ

    View Slide

  48. • Example 14ɿ࿨ԻΛ਺ֶతʹදݱ͢Δ
    • 12 Իͷू߹ P = { p0, p1, ..., p12 }
    • ࿨Ի ch ͱ͸ ch ⊂ P Ͱ͋Γ ch ∈ 2P Ͱ΋͋Δʢe.g. C Major { c, e, g } = { p0, p4, p7 }ʣ
    • ผͷݟํͱͯ͠ɺP → { 0, 1 } ͷʮؔ਺ʯͦΕࣗମΛ࿨ԻͱಉҰࢹ
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    def c_major(pitch):
    if pitch in [p0, p4, p7]:
    return 0
    return 1
    ```
    ```python
    def c_minor(pitch):
    if pitch in [p0, p3, p7]:
    return 0
    return 1
    ```

    View Slide

  49. • ू߹ͱͯ͠ͷ࿨ԻΛɺؔ਺ͱͯ͠ͷ࿨Ի΁ҰରҰม׵Ͱ͖Δؔ਺͕ଘࡏ͢ΔͷͰ͸ʁ
    5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    ch = [p0, p4, p7]
    def c_major(pitch):
    if pitch in [p0, p4, p7]:
    return 0
    return 1
    ```

    View Slide

  50. 5. ႈू߹ͷه๏ 2a ͷҙຯ
    ```python
    def chi(ch):
    def chord_as_func(pitch):
    if pitch in ch:
    return 0
    return 1
    return chord_as_func
    ```
    ```python
    ch = [p0, p4, p7]
    def c_major(pitch):
    if pitch in [p0, p4, p7]:
    return 0
    return 1
    ```
    • ౴ɿఆཧ 4ʢ෦෼ର৅෼ྨࢠ subobject classifierʣ
    શ୯ࣹͱͳΔؔ਺ χ : 2a → Set(a, 2) ͕ଘࡏ
    ͨͩ͠ 2 := { 0, 1 } Ͱɺ͜ΕΛ෦෼ର৅෼ྨࢠʢݍ࿦༻ޠʣͱݺͿ

    View Slide

  51. • ͭ·Γ 2a ͱ͸ɺa ͷ෦෼ू߹Λ 2 := { 0, 1 } ΁ૹΔؔ਺ͷू߹ʢ഑ஔू߹ʣ
    • P.43 Lemma 1 ΍ P.52 Example 2 Λࢀর
    • 0 := ∅
    • 1 := 0+ = 0 ∪ { 0 } = ∅ ∪ { ∅ } = { ∅ } = { 0 }
    • 2 := 1+ = 1 ∪ { 1 } = { ∅ } ∪ { { ∅ } } = { ∅, { ∅ } } = { 0, { 0 } } = { 0, 1 }
    5. ႈू߹ͷه๏ 2a ͷҙຯ

    View Slide