Koga Kobayashi
December 12, 2019
50

# Fundamentals of Music Processing (Chapter 5)

## Koga Kobayashi

December 12, 2019

## Transcript

1. ### Fundamentals of Music Processing Chapter 5: Chord Recognition   খྛ

ᕣՏ εϥΠυʹؚ·ΕΔਤ͸Fundamentals of Music ProcessingΑΓҾ༻
2. ### Chapter 5: Chord Recognition Chord(࿨Ի) • 3ͭҎ্ͷҟͳΔԻූ͔Βߏ੒͞ΕΔԻͷ͜ͱ Harmony(࿨੠) • ෳ਺ͷ࿨Ի͔ΒͳΔܥྻɺ࿨Իਐߦ

FM7 G7 Em7 Am Harmony

4. ### Chapter 5.3: HMM-Based Chord Recognition Chapter 5.1~5.2 • ಛ௃ྔ͔Β࿨ԻΛ͋Δఔ౓ਪఆग़དྷΔ ͔͠͠ɺ͜Ε͸࿨ԻҰͭҰ͔ͭ͠ݟ͍ͯͳ͍

Α͘ग़Δ࿨Ի΍ܨ͕Γͷڧ͍࿨Իʹண໨͍ͨ͠ ྫ: I–IV–V–Iਐߦ • F΍G͸සग़͠ɺ͍͖ͳΓFmʹߦ͘͜ͱ͸΄΅ແ͍ HMM(ӅΕϚϧίϑϞσϧ)Λར༻ͯ͠࿨ԻਪఆΛߦ͏
5. ### Chapter 5.3: HMM-Based Chord Recognition ࿨Ի: , ঢ়ଶ: (for )

ͱͨ͠ͱ͖ɺϚϧίϑੑΛԾఆ͢Δͱ Ͱ͋Δ֬཰͸ A := {α1 , α2 , ⋯, αI } s i ∈ [1 : I] sn+1 = αj P[sn+1 = αj |sn = αi , sn−1 = αk , ⋯] = P[sn+1 = αj |P[sn+1 = αj |sn ] ͜͜Ͱ ΛҎԼͷΑ͏ʹఆٛ͢Δɻ(for ) aij i, j ∈ [1 : I] ͜Ε͸ঢ়ଶ͕ ͔Β ʹભҠ͢Δ֬཰ͱߟ͑Δ͜ͱ͕ग़དྷΔ αi αj aij := P[sn+1 = αj |sn = αi ] ∈ [0,1] ·ͣɺϚϧίϑ࿈࠯Λར༻ͨ͠࿨Ի༧ଌʹ͍ͭͯઆ໌͢Δ
6. ### Chapter 5.3: HMM-Based Chord Recognition ۩ମྫ; , ͷͱ͖ I =

3 A := {α1 = C, α2 = G, α3 = F} ·ͨ࠷ॳͷঢ়ଶ͕ Ͱ͋Δ֬཰ΛҎԼͷΑ͏ʹఆٛ͢Δ αi ci := P[s1 = αi ] ∈ [0,1]
7. ### Chapter 5.3: HMM-Based Chord Recognition , , ͷͱ͖ ঢ়ଶܥྻ: ʹ͍ͭͯߟ͑Δ

I = 3 A := {α1 = C, α2 = G, α3 = F} C = (c1 , c2 , c3 )T = (0.6,0.2,0.3)T S = (C, C, C, G, G, F, F, C, C) ࠷ॳͷঢ়ଶ͕ Ͱ͋Δ֬཰ΛҎԼͷΑ͏ʹఆٛ͠ɺ αi ci := P[s1 = αi ] ∈ [0,1] ۩ମྫ
8. ### Chapter 5.3: HMM-Based Chord Recognition ભҠ֬཰͕ҎԼͷͱ͖ ͷΑ͏ͳ࿨Իਐߦ͕ى͖Δ֬཰͸ S S =

(C, C, C, G, G, F, F, C, C) = c1 ⋅ a11 ⋅ a11 ⋅ a12 ⋅ a22 ⋅ a23 ⋅ a33 ⋅ a31 ⋅ a11 ≈ 1.29 ⋅ 10−4
9. ### Chapter 5.3: HMM-Based Chord Recognition ઌఔঢ়ଶܥྻ Λ༻͍ͯɺ࿨Իਐߦͷ֬཰Λܭࢉ͕ͨ͠  ࣮ੈքͰ͸औΓ͏Δ͢΂ͯͷ ʹ͍ͭͯܭࢉ͸ෆՄೳ  S

S ྫ: 10छྨͷ࿨Իɺ20͔ͭΒߏ੒͞ΕΔۂͷ৔߹  ύλʔϯͷ֬཰Λܭࢉ͢Δඞཁ͕͋Δ 1020 ͦ͜Ͱ಺෦ͷঢ়ଶͰ͸ͳ͘ɺಛ௃ϕΫτϧΛ༻͍ͯ  ࿨ԻਐߦΛٻΊΔํ๏ͱͯ͠HMMΛར༻͢Δɻ

12. ### Chapter 5.3: HMM-Based Chord Recognition Input: Ի੠σʔλ͔Β؍ଌͨ͠ܥྻ ͔Β  ϞσϧΛར༻͠ɺ؍ଌܥྻ Λ࡞੒ɻ

O = (o1 , ⋯, oN ) B = (β1 , ⋯, βN ) ؍ଌܥྻ ͸  ؍ଌγϯϘϧ    ͔Βߏ੒͞ΕΔ B = (β1 , ⋯, βN ) ℬ = {β1 , ⋯, βk } (for k ∈ [1 : K])

14. ### Chapter 5.3: HMM-Based Chord Recognition Viterbi: ؍ଌܥྻ ͱ  ॳظঢ়ଶͷ֬཰

ੜ੒֬཰ͱભҠ֬཰͔Β༗޲άϥϑΛ࡞੒ B = (β1 , ⋯, βN ) C = (c1 , c2 , c3 )T = (0.6,0.2,0.3)T ੜ੒֬཰ ભҠ֬཰

23. ### Chapter 5.3: HMM-Based Chord Recognition ࠷ޙʹɺͦͷϙΠϯλΛḷΓ࠷΋໬΋Β͍͠࿨ԻਐߦΛ  ٻΊΔ A := {α1

= C, α2 = G, α3 = F} ͷͱ͖ ̂ S = (C, C, C, G, G, F)