Cool Math for Hot Music 輪読会 Sec.5

Slide 1

Slide 1 text

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

Slide 2

Slide 2 text

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

Slide 3

Slide 3 text

େ·͔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

Slide 4

Slide 4 text

1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ

Slide 5

Slide 5 text

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

Slide 6

Slide 6 text

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

Slide 7

Slide 7 text

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 }

Slide 8

Slide 8 text

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

Slide 9

Slide 9 text

େ·͔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

Slide 10

Slide 10 text

2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎

Slide 11

Slide 11 text

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

Slide 12

Slide 12 text

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

Slide 13

Slide 13 text

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

Slide 14

Slide 14 text

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

Slide 15

Slide 15 text

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

Slide 16

Slide 16 text

Slide 17

Slide 17 text

Slide 18

Slide 18 text

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

Slide 19

Slide 19 text

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

Slide 20

Slide 20 text

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

Slide 21

Slide 21 text

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

Slide 22

Slide 22 text

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

Slide 23

Slide 23 text

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

Slide 24

Slide 24 text

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

Slide 25

Slide 25 text

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

Slide 26

Slide 26 text

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

Slide 27

Slide 27 text

Slide 28

Slide 28 text

3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

Slide 29

Slide 29 text

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

Slide 30

Slide 30 text

• ू߹ 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΁֦ு

Slide 31

Slide 31 text

• ू߹ 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΁֦ு

Slide 32

Slide 32 text

ެཧతू߹࿦ ܯ࡯Ͱ͢ʂʂ QVSFTFUTʹ ࠲ඪ͸ଘࡏ ͠·ͤΜʂʂ

Slide 33

Slide 33 text

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

Slide 34

Slide 34 text

Slide 35

Slide 35 text

4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ

Slide 36

Slide 36 text

• 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

Slide 37

Slide 37 text

• ܾ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 > C Major (tx)a C minor (sx)a ઌʂ

Slide 38

Slide 38 text

• ܾ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

Slide 39

Slide 39 text

• 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

Slide 40

Slide 40 text

• 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

Slide 41

Slide 41 text

Slide 42

Slide 42 text

5. ႈू߹ͷه๏ 2a ͷҙຯ

Slide 43

Slide 43 text

• 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

Slide 44

Slide 44 text

• 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 ```

Slide 45

Slide 45 text

• 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 ```

Slide 46

Slide 46 text

• ౴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 ```

Slide 47

Slide 47 text

• 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

Slide 48

Slide 48 text

• 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 ```

Slide 49

Slide 49 text

• ू߹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 ```

Slide 50

Slide 50 text

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

Slide 51

Slide 51 text

• ͭ·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