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Fundamentals of Music Processing (Chapter 5)
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Koga Kobayashi
December 12, 2019
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Fundamentals of Music Processing (Chapter 5)
Koga Kobayashi
December 12, 2019
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Transcript
Fundamentals of Music Processing Chapter 5: Chord Recognition খྛ
ᕣՏ εϥΠυʹؚ·ΕΔਤFundamentals of Music ProcessingΑΓҾ༻
Chapter 5: Chord Recognition Chord(Ի) • 3ͭҎ্ͷҟͳΔԻූ͔Βߏ͞ΕΔԻͷ͜ͱ Harmony() • ෳͷԻ͔ΒͳΔܥྻɺԻਐߦ
FM7 G7 Em7 Am Harmony
Chapter 5: Chord Recognition Chord Recognition(Իೝࣝ) • Ի͔ΒԻਐߦΛೝࣝ͢Δٕज़ ԻָϑΝΠϧ͔ΒίʔυේΛࣗಈͰ࡞ग़དྷΔ Իೝ͕ࣝ͏·͍͘͘ͱ…
Chapter 5.3: HMM-Based Chord Recognition Chapter 5.1~5.2 • ಛྔ͔ΒԻΛ͋Δఔਪఆग़དྷΔ ͔͠͠ɺ͜ΕԻҰͭҰ͔ͭ͠ݟ͍ͯͳ͍
Α͘ग़ΔԻܨ͕Γͷڧ͍Իʹண͍ͨ͠ ྫ: I–IV–V–Iਐߦ • FGසग़͠ɺ͍͖ͳΓFmʹߦ͘͜ͱ΄΅ແ͍ HMM(ӅΕϚϧίϑϞσϧ)Λར༻ͯ͠ԻਪఆΛߦ͏
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] ͜Εঢ়ଶ͕ ͔Β ʹભҠ͢Δ֬ͱߟ͑Δ͜ͱ͕ग़དྷΔ αi αj aij := P[sn+1 = αj |sn = αi ] ∈ [0,1] ·ͣɺϚϧίϑ࿈Λར༻ͨ͠Ի༧ଌʹ͍ͭͯઆ໌͢Δ
Chapter 5.3: HMM-Based Chord Recognition ۩ମྫ; , ͷͱ͖ I =
3 A := {α1 = C, α2 = G, α3 = F} ·ͨ࠷ॳͷঢ়ଶ͕ Ͱ͋Δ֬ΛҎԼͷΑ͏ʹఆٛ͢Δ αi ci := P[s1 = αi ] ∈ [0,1]
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 Ͱ͋Δ֬ΛҎԼͷΑ͏ʹఆٛ͠ɺ αi ci := P[s1 = αi ] ∈ [0,1] ۩ମྫ
Chapter 5.3: HMM-Based Chord Recognition ભҠ͕֬ҎԼͷͱ͖ ͷΑ͏ͳԻਐߦ͕ى͖Δ֬ S S =
(C, C, C, G, G, F, F, C, C) = c1 ⋅ a11 ⋅ a11 ⋅ a12 ⋅ a22 ⋅ a23 ⋅ a33 ⋅ a31 ⋅ a11 ≈ 1.29 ⋅ 10−4
Chapter 5.3: HMM-Based Chord Recognition ઌఔঢ়ଶܥྻ Λ༻͍ͯɺԻਐߦͷ֬Λܭࢉ͕ͨ͠ ࣮ੈքͰऔΓ͏Δͯ͢ͷ ʹ͍ͭͯܭࢉෆՄೳ S
S ྫ: 10छྨͷԻɺ20͔ͭΒߏ͞ΕΔۂͷ߹ ύλʔϯͷ֬Λܭࢉ͢Δඞཁ͕͋Δ 1020 ͦ͜Ͱ෦ͷঢ়ଶͰͳ͘ɺಛϕΫτϧΛ༻͍ͯ ԻਐߦΛٻΊΔํ๏ͱͯ͠HMMΛར༻͢Δɻ
Chapter 5.3: HMM-Based Chord Recognition
Chapter 5.3: HMM-Based Chord Recognition
Chapter 5.3: HMM-Based Chord Recognition Input: Իσʔλ͔Β؍ଌͨ͠ܥྻ ͔Β ϞσϧΛར༻͠ɺ؍ଌܥྻ Λ࡞ɻ
O = (o1 , ⋯, oN ) B = (β1 , ⋯, βN ) ؍ଌܥྻ ؍ଌγϯϘϧ ͔Βߏ͞ΕΔ B = (β1 , ⋯, βN ) ℬ = {β1 , ⋯, βk } (for k ∈ [1 : K])
Chapter 5.3: HMM-Based Chord Recognition
Chapter 5.3: HMM-Based Chord Recognition Viterbi: ؍ଌܥྻ ͱ ॳظঢ়ଶͷ֬
ੜ֬ͱભҠ͔֬Β༗άϥϑΛ࡞ B = (β1 , ⋯, βN ) C = (c1 , c2 , c3 )T = (0.6,0.2,0.3)T ੜ֬ ભҠ֬
Chapter 5.3: HMM-Based Chord Recognition ੜ֬ ભҠ֬ ॳظঢ়ଶͷ֬ ੜ͞Εͨ༗άϥϑ
Chapter 5.3: HMM-Based Chord Recognition ViterbiΞϧΰϦζϜʹΑͬͯ ࠷Β͍͠ܦ࿏ʹ͍ͭͯܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ੜ͞Εͨ༗άϥϑ ViterbiΞϧΰϦζϜͰ֤ҐஔͰɺ ֤ԻʹͨͲΓண͘·Ͱͷ࠷దίετͱ ͦͷલʹࢸΔ·ͰͷϙΠϯλΛ֮͑ͳ͕ΒܭࢉΛߦ͏
Chapter 5.3: HMM-Based Chord Recognition ࠷ޙʹɺͦͷϙΠϯλΛḷΓ࠷Β͍͠ԻਐߦΛ ٻΊΔ A := {α1
= C, α2 = G, α3 = F} ͷͱ͖ ̂ S = (C, C, C, G, G, F)