Cool Math for Hot Music 輪読会 Sec.5

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

View Slide

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

View Slide

େ·͔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

1. ಋೖɿؔ਺ͷू߹ʢ഑ஔू߹ʣ

View Slide

b
a
1
2
3
x
y
f1
```python
def f1(a1):
return x
```
• ؔ਺ f : a → b ͸ͨ͘͞Μ͋Δ

View Slide

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

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

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

େ·͔ͳྲྀΕ
2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

View Slide

2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎

View Slide

• ͍͖ͳΓ Fig. 5.1 ΛݟͪΌ͍·͢ʢԻఔ interval ͷଌఆ͕໨ඪʣ

View Slide

c (1) ू߹ c = { 0, 1 } Λ༻ҙ

View Slide

(1) ू߹ c = { 0, 1 } Λ༻ҙ
(2) c ͷ 2 ఺Λ a ্࣠ͷ 2 ఺ʹରԠͤ͞Δ
ؔ਺ fa
Λ༻ҙ
c
fa

View Slide

(1) ू߹ c = { 0, 1 } Λ༻ҙ
(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

(1) ू߹ c = { 0, 1 } Λ༻ҙ
(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

(1) ू߹ c = { 0, 1 } Λ༻ҙ
(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
(5) c ͷ 2 ఺Λ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
ͤ͞Δؔ਺ f ͕͋ͬͯ΋ྑ͍͸ͣ
f

View Slide

(1) ू߹ c = { 0, 1 } Λ༻ҙ
(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
pra
prb
(6) ͱ͜ΖͰࣹӨ pra, prb
͕ଘࡏ
(5) c ͷ 2 ఺Λ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ

View Slide

໌Β͔ʹ ( fa , fb ) = ( pra ∘ f , prb ∘ f )
c
pra
prb
f
fb
fa
໰ɿ͜ͷؔ਺ f ͱ ( fa , fb ) ͸࣮࣭ಉ͡ʁ

View Slide

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

(1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
߹ซͱඇަ࿨ͷҧ͍͸ޱ಄Ͱผ్આ໌༧ఆ

View Slide

(1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
(2) a = {0}, b = {1} ͱ͓͘
a
b

View Slide

(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

(1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
(4) c ΛҰؾʹ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
f
(2) a = {0}, b = {1} ͱ͓͘

View Slide

(4) c ΛҰؾʹ ab ฏ໘্ͷ 2 ఺΁௚઀ରԠ
(1) ू߹ {0, 1} = {0} ∪ {1} ͱ෼ׂ
(5) ͱ͜ΖͰ୯ࣹ ina, inb
͕ଘࡏ
a
b
a
0
b
1
c
1
0
inb
ina
(2) a = {0}, b = {1} ͱ͓͘

View Slide

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

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

େ·͔ͳྲྀΕ
2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

View Slide

3. ௚ੵΛू߹ 2 ݸ͔Βແݶݸ΁֦ு

View Slide

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

View Slide

a × b = { (x, y) | x ∈ a, y ∈ b } (§4 Def. 6)
• ݁ہʮ࠲ඪͷू߹ʯ͡ΌΜ……ʁ

View Slide

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

View Slide

ެཧతू߹࿦
ܯ࡯Ͱ͢ʂʂ
QVSFTFUTʹ
࠲ඪ͸ଘࡏ
͠·ͤΜʂʂ

View Slide

• ͋ͱͰਅ໘໨ʹ΍Γ·͢……ɻ
• ࠲ඪͰ͸ͳ͘ N-tuple ͱߟ͑Ε͹ɺ·͋……ɻ

View Slide

େ·͔ͳྲྀΕ
2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

View Slide

4. Ұൠͷ௚ੵʹ͓͚Δʮॱংʯ

View Slide

• Ұൠͷ௚ੵू߹ͰʮݩͷॱংʯΛͲͷΑ͏ʹܾΊΔ͔
• Example 15ɿC Major ͱ C minor ͸ʮͲ͕ͬͪઌʁʯ
(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

• ܾ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

• ܾΊํ 2ɿ௚ੵॱংʢDef. 20 product relationʣ
• ೚ҙͷ੒෼Ͳ͏͕͠ಉ͡ॱংؔ܎Ͱ͋Ε͹ɺͦΕΛ௚ੵू߹ʹ͓͚Δʮॱংʯͱ͢Δ
... (tx)aR(sx)a iff txrxsx for each x ∈ a
(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

• Mazzola ઌੜͷ American Set Theory ʹର͢Δ dis Γ۩߹ΛຯΘ͏ʢpp.67ʣ
• American Set Theory ͷ࿈த͸ࣙॻࣜॱং΋஌ΒΜౕΒ͡Όɻ
ނʹ "most packed to the left" ͱ͔ݴ͍ͳ͕Βӈ͔Βൺֱ͓ͯ͠Δͷ͡Όɻ
(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

• 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
https://www.mta.ca/pc-set/pc-set_new/pages/sources/sources.html

View Slide

େ·͔ͳྲྀΕ
2. ௚ੵʢ௚࿨ʣͱࣹӨɾؔ਺ͷؔ܎ §5.2, 5.3
3. ௚ੵΛू߹ 2 ݸ͔Β N ݸ΁֦ு §5.6
5. ႈू߹ͷه๏ 2a ͷҙຯ §5.4, 5.5ʢ͓·͚ʣ

View Slide

5. ႈू߹ͷه๏ 2a ͷҙຯ

View Slide

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

View Slide

• Example 13
• ผͷϞσϧԽͱͯ͠ɺa ͷ஋Λड͚औͬͯʮb ͷ஋Λड͚औͬͯ c Λฦؔ͢਺ʯΛฦؔ͢਺ g
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

• f(3, 5) == g(3)(5) == 0
• ͲͪΒ΋݁Ռ͸ಉ͡ => f Λ g ͷܗ΁ҰରҰม׵Ͱ͖Δؔ਺͕ଘࡏ͢ΔͷͰ͸ʁ
```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

• ౴ɿఆཧ 3ʢႈͷීวੑʣ
શ୯ࣹͱͳΔؔ਺ δ : Set(a × b, c) → Set(a, cb) ͕ଘࡏ
```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

• 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

View Slide

• 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
```python
def c_major(pitch):
if pitch in [p0, p4, p7]:
return 0
return 1
```
```python
def c_minor(pitch):
return 0
return 1
```

View Slide

• ू߹ͱͯ͠ͷ࿨ԻΛɺؔ਺ͱͯ͠ͷ࿨Ի΁ҰରҰม׵Ͱ͖Δؔ਺͕ଘࡏ͢ΔͷͰ͸ʁ
```python
ch = [p0, p4, p7]
def c_major(pitch):
return 0
return 1
```

View Slide

```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):
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

• ͭ·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 }

View Slide

Cool Math for Hot Music 輪読会 Sec.5

Cool Math for Hot Music 輪読会 Sec.5

Ryo Sakuma

More Decks by Ryo Sakuma

Other Decks in Science

Featured

Transcript