Fundamentals of Music Processing (Chapter 5)

Transcript

1. Fundamentals of Music Processing
   Chapter 5: Chord Recognition
   

   খྛ ᕣՏ
   εϥΠυʹؚ·ΕΔਤ͸Fundamentals of Music ProcessingΑΓҾ༻
   

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2. Chapter 5: Chord Recognition
   Chord(࿨Ի)
   • 3ͭҎ্ͷҟͳΔԻූ͔Βߏ੒͞ΕΔԻͷ͜ͱ
   Harmony(࿨੠)
   • ෳ਺ͷ࿨Ի͔ΒͳΔܥྻɺ࿨Իਐߦ
   FM7 G7 Em7 Am
   Harmony
   

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3. Chapter 5: Chord Recognition
   Chord Recognition(࿨Իೝࣝ)
   • Ի੠͔Β࿨ԻਐߦΛೝࣝ͢Δٕज़
   ԻָϑΝΠϧ͔ΒίʔυේΛࣗಈͰ࡞੒ग़དྷΔ
   ࿨Իೝ͕ࣝ͏·͍͘͘ͱ…
   

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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
   

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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
   

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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]
   

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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
   

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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
   

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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
   

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10. Chapter 5.3: HMM-Based Chord Recognition
    

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11. Chapter 5.3: HMM-Based Chord Recognition
    

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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])
    

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13. Chapter 5.3: HMM-Based Chord Recognition
    

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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཰
    

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15. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒֬཰
    ભҠ֬཰ ॳظঢ়ଶͷ֬཰
    ੜ੒͞Εͨ༗޲άϥϑ
    

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16. Chapter 5.3: HMM-Based Chord Recognition
    ViterbiΞϧΰϦζϜʹΑͬͯ

    ࠷΋໬΋Β͍͠ܦ࿏ʹ͍ͭͯܭࢉΛߦ͏
    

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17. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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18. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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19. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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20. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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21. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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22. Chapter 5.3: HMM-Based Chord Recognition
    ੜ੒͞Εͨ༗޲άϥϑ
    ViterbiΞϧΰϦζϜͰ͸֤ҐஔͰɺ

    ֤࿨ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ

    ͦͷ௚લʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
    

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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)
    

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