слайды - MachineLearning.ru

Êðàòêîñðî÷íîå ïðåäñêàçàíèå ìóçûêàëüíûõ
ïðîèçâåäåíèé ñ èñïîëüçîâàíèåì
ïîñëåäîâàòåëüíîñòåé àêêîðäîâ
Ìèõàèë Ìàòðîñîâ
1,2,3
,
íàó÷íûé ðóêîâîäèòåëü Â. Â. Ñòðèæîâ
1,2,3
,
1
êîíñóëüòàíò Àíòîí Ìàòðîñîâ
1 Ìîñêîâñêèé Ôèçèêî-Òåõíè÷åñêèé Èíñòèòóò
2 Ñêîëêîâñêèé Èíñòèòóò Íàóêè è Òåõíîëîãèé
3 Âû÷èñëèòåëüíûé Öåíòð Ðîññèéñêîé Àêàäåìèè Íàóê
Âû÷èñëèòåëüíûé Öåíòð Ðîññèéñêîé Àêàäåìèè Íàóê
Èþíü 2015, Ìîñêâà, ÐÔ
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
1 / 28
Öåëü èññëåäîâàíèÿ
Ïðåäñêàçàòü ñëåäóþùèé ýëåìåíò â ïîñëåäîâàòåëüíîñòè
àêêîðäîâ â òåîðåòèêî-ãðóïïîâîì ïðåäñòàâëåíèè, íå ó÷èòûâàÿ
òåìïîðàëüíóþ ñîñòàâëÿþùóþ (àðïåäæèî, äëèòåëüíîñòè,
ïàóçû).
Íîâèçíà:
Ìåòîä:
áîëåå òî÷íîå (áîëüøå èíôîðìàöèè î êàæäîì
àêêîðäå) ïðåäñòàâëåíèå ìóçûêàëüíîé ïîñëåäîâàòåëüíîñòè.
Pitch
êîìïîçèöèÿ áàéåñîâñêèõ êëàññèôèêàòîðîâ.
Time
, not
A sequence of simple tones
Òèçåð: 92.5% Õýììèíã-ñõîäñòâî (58% ïî-àêêîðäíî)
ìåæäó ïðåäñêàçàíèåì è îðèãèíàëüíîé ìåëîäèåé èç
òåñòîâîé âûáîðêè.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
2 / 28
Îêòàâà
Îêòàâà ñîñòîèò èç 12 ïîëóòîíîâ. Ìóçûêàíò ìîæåò ñûãðàòü âñå
íîòû â ïðåäåëàõ îäíîé îêòàâû, ïðè ýòîì çâó÷àíèå ìåëîäèå
èçìåíèòñÿ íå çíà÷èòåëüíî. Ïî-ýòîìó ìû ïðåäñòàâëÿåì àêêîðäû
â ïðåäåëàõ òîëüêî îäíîé îêòàâû.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
3 / 28
Àêêîðäû
Êàæäûé
àêêîðä
ñîäåðæèò îò 1 äî 12 îäíîâðåìåííî çâó÷àùèõ
ïîëóòîíîâ. Àêêîðä ìîæåò áûòü òðàíñïîíèðîâàí íà íåñêîëüêî
ïîëóòîíîâ ââåðõ èëè âíèç. Åãî òàê æå ìîæíî èçîáðàçèòü íà
êðóãå, ïðè ýòîì âðàùåíèå ýêâèâàëåíòíî èçìåíåíèþ âûñîòû
òîíà.
Triadic chord examples (key of C)
A
G#b A Bb#
A
C DC#
b
D#
Eb E
D#
Eb E
F
D#
Eb E
A
G#b A Bb#
A
B
D
D
D
F
F#
Gb G
A
G#b A Bb#
A
B
C DC#
b
F#
Gb G
B
diminished
minor
C DC#
b
F
major
F#
Gb G
Seventh Chords (key of C)
F#
Gb G
A
G#b A Bb#
A
A
G#b A Bb#
A
F
F
A
G#b A Bb#
A
C DC#
b
D#
Eb E
D#
Eb E
D#
Eb E
F#
Gb G
Ïðåäñêàçàíèå ìóçûêè
B
D
D
D
Ì. Ï. Ìàòðîñîâ
B
C DC#
b
F#
Gb G
B
diminished
minor
C DC#
b
F
major
4 / 28
Ñîçâó÷èå è áàçîâûé òîí
Íèæå èçáðàæåíû åùå íåñêîëüêî àêêîðäîâ. Ñóùåñòâóåò 351
âîçìîæíîå ñîçâó÷èå.
Êàæäîå èç íèõ ìîæåò áûòü ñûãðàíî â 12 ðàçëè÷íûõ êëþ÷àõ,
êðîìå íåñêîëüêèõ ñèììåòðè÷íûõ ñëó÷àåâ (äîâîëüíî ðåäêèõ).
Èòîãî
212 − 1 = 4095 âîçìîæíûõ àêêîðäîâ.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
5 / 28
Ïðåäñòàâëåíèå àêêîðäîâ
Ìåëîäèÿ ïðåäñòàâëÿåòñÿ â âèäå ïîñëåäîâàòåëüíîñòè àêêîðäîâ
(öåëûõ îò 1 äî
212 − 1), i
çäåñü è äàëåå îáîçíà÷àåò âðåìÿ.
c = {ci } , ci ∈ C,
melody sequence
C = {1, 2, 3, . . . , 4095}
ïðîñòðàíñòâî
àêêîðäîâ
Êàæäûé àêêîðä èìååò ñâîþ ôîðìó (ñòðîé
(s, z):
c = s × z,
chord strum key
c = {(s, z)i },
melody
pairs
B
Z12
ñòðîåâ
, 351
ãðóïïà âû÷åòîâ ïî ìîäóëþ 12.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
D#
Eb E
ìíîæåñòâî óíèêàëüíûõ
ýëåìåíòîâ.
D
Ïðîèçâåäåíèå îçíà÷àåò òðàíñïîçèöèþ.
C DC#
b
A
G#b A Bb#
A
ïîñëåäîâàòåëüíîñòü ïàð
S
è áàçîâûé òîí
Òàêèì îáðàçîì, âñÿ ìåëîäèÿ ïðåäñòàâëÿåòñÿ êàê
F#
Gb G
∈ Z12 ).
F
(z
.
s ∈ S)
6 / 28
Ïîñëåäîâàòåëüíîñòü ýëåìåíòîâ
Ìåëîäèÿ ìîæåò áûòü òðàíñïîíèðîâàíà, òàê ÷òî íîòû íóæíî
ñîèçìåðÿòü ñ ïðåäøåñòâóþùèìè:
ri = zi − zi−1
Ïàðà (s, r) íàçûâàåòñÿ
mod 12, r1 = z1 .
ýëåìåíòîì
è îáîçíà÷àåòñÿ
x ∈ E.
C = S × Z12 = S × R12 = E,
ìíîæåñòâî óíèêàëüíûõ
,
,
,
Z12 âû÷åòû ïî ìîäóëþ 12, N = 12,
R12 ñìåæíûå ðàçíîñòè Z12 , N = 12,
E ïðîñòðàíñòâî
ýëåìåíòîâ
B
C
A
G#b A Bb#
A
,
, â êîòîðîì ìû ïðåäñêàçûâàåì.
po , r
s
an n
Tr sitio
D#
Eb E
àêêîðäîâ N = 4095
ñòðîåâ N = 351
F#
Gb G
ïðîñòðàíñòâî
F
C
S
Ïðîñòðàíñòâî ýòî ìíîæåñòâî ñ
îïåðàöèåé òðàíñïîíèðîâàíèÿ (èçìåíåíèÿ
âûñîòû òîíà).
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
7 / 28
N-ãðàììû
N -ãðàìì
N ýëåìåíòîâ x ∈ E
x = {xi }. N -ãðàìì ðàçìåðà 1 ïîäïîñëåäîâàòåëüíîñòü èç
äàííîé ïîñëåäîâàòåëüíîñòè
ïðîñòî îäèí ýëåìåíò
x ∈ E.
ýòî
Íàïðèìåð:
xN
i = {xi , xi+1 , . . . , xi+N−1 }.
 äàííîé ðàáîòå
N -ãðàììû
èñïîëüçóþòñÿ êàê õàðàêòåðèçóþùèå
âåêòîðà, îïèñûâàþùèå òåêóùóþ òî÷êó â ìóçûêàëüíîé
ïîñëåäîâàòåëüíîñòè.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
8 / 28
Íàáîð äàííûõ Midi50k
50 000 ñëó÷àéíûõ midi-ôàéëîâ
áûëè ñîáðàíû â èíòåðíåòå.
Êàæäûé midi-ôàéë ïðåîáðàçîâàí â ïîñëåäîâàòåëüíîñòü
àêêîðäîâ
c = {ci }, ci ∈ C
ïî ñëåäóþùåìó àëãîðèòìó:
îòêðûòü midi-ôàéë êàê piano roll,
îòáðîñèòü ïåðêóññèþ,
êâàíòîâàòü ñ ÷àñòîòîé
2 · tempo ,
îòáðîñèòü íîìåð îêòàâû (pitch
= pitch mod 12).
600 àêêîðäîâ
30 ìèëëèîíîâ
Ñðåäíèé midi-ôàéë ñîäåðæèì
äàåò ñóììàðíî
àêêîðäîâ, ÷òî
àêêîðäîâ.
Ìåëîäèÿ ïîñëåäîâàòåëüíîñòü ýëåìåíòîâ (èíäåêñ ýòî âðåìÿ):
x = {xi }, xi ∈ E.
X íàáîð ìåëîäèé: X = {xj }.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
9 / 28
Ðàçáèåíèå âûáîðêè
Äëÿ îöåíèâàíèÿ àëãîðèòìà ïîëíûé íàáîð Midi50k
X0
äåëèëñÿ
íà äâå ÷àñòè ðàçíîãî ðàçìåðà. Êàæäûé ðàç äåëåíèå
ïðîèçâîäèëîñü ñëó÷àéíûì îáðàçîì èç íàáîðà âûáèðàëîñü
ïîäìíîæåñòâî çàäàííîãî ðàçìåðà
M.
Xtraining ⊂ X0 ,
|Xtraining | = M.
Äëÿ ïðîâåðêè àëãîðèòìà èñïîëüçîâàëàñü îñòàâøàÿñÿ ÷àñòü
íàáîðà:
Xtesting = X0 \ Xtraining .
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
10 / 28
Ïîñòàíîâêà çàäà÷è â òåðìèíàõ ìèíèìèçàöèè îøèáêè
Äëÿ òðåíèðîâî÷íîãî íàáîðà äàííûõ
ïàðàìåòðîâ àëãîðèòìà
w,
X
íàéòè âåêòîð
ìèíèìèçèðóþùèé ôóíêöèþ îøèáêè:
ŵ = arg min S(w, X).
w∈R2K
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
11 / 28
Ôóíêöèÿ ïðåäñêàçàíèÿ
Ïðåäñêàçàíèå äåëàåòñÿ ïî âçâåøåííîé ñóììå êëàññèôèêàòîðîâ:
xi+1 = f (X, {x1 , . . . , xi },
w
={uk ,vk },
k=1,...K
K
X
) = arg max
(Aek uk + Bek vk ),
e∈E
k=1
(1)



Aek ∝ N {xi−k+2 , . . . , xi , e}
{z
}
|
k -gram

in

X ,
|{z}
Training set


Bek ∝ N {xi−k+2 , . . . , xi , e}
|
{z
}
k -gram
in

{x1 , . . . , xi }  ,
|
{z
}
Part of melody before i+1
ãäå "∝"îçíà÷àåò, ÷òî
A∗k è B∗k íîðìèðîâàíû íà L1, xi ∈ E,
N(g in dataset) êîëè÷åñòâî k -ãðàììîâ g ∈ Ek â íàáîðå,
w = {uk , vk |k = 1, . . . K } ∈ R2K âåêòîð ïàðàìåòðîâ, K ñëîæíîñòü ìîäåëè (ìàêñèìàëüíàÿ äëèíà N -ãðàììîâ).
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
12 / 28
Ôóíêöèÿ îøèáêè
Ïóñòûå êðóãè èñòèííûå ïîëóòîíà, òî÷êè ïðåäñêàçàííûå,
îøèáêè ïîäñâå÷åíû, ãîðèçîíòàëüíàÿ îñü âðåìÿ.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
13 / 28
Ôóíêöèÿ îøèáêè
Ôóíêöèÿ
f
ýòî êëàññèôèêàòîð, ïðåäñêàçûâàþùèé ñëåäþùèé
ýëåìåíò. Òîãäà ôóíêöèÿ îøèáêè (i îçíà÷àåò âðåìåííîé
èíòåðâàë):
S(w, X) =
x −1
X NX
x∈X i=1


xi+1 6= f (X, {x1 , . . . , xi } , w).
|
{z
}
Prev. part of the melody
Brackets stand for 1 if the statement inside is true and 0 if false.
Êâàäðàòíûå ñêîáêè äàþò 1, åñëè âûðàæåíèå âíóòðè èñòèííî, è
0, åñëè ëîæíî.
X
íàáîð ìåëîäèé,
w ∈ R2K
Ì. Ï. Ìàòðîñîâ
âåêòîð ïàðàìåòðîâ.
Ïðåäñêàçàíèå ìóçûêè
14 / 28
Ñãëàæåííàÿ ôóíêöèÿ îøèáêè
Âââåäåì òàêæå ñãëàæåííóþ ôóíêöèþ îøèáêè, ÷òîáû
èçáàâèòüñÿ îò âîçìîæíûõ ïðîáëåì ñî ñòàáèëüíîñòüþ ïðîöåññà
ìèíèìèçàöèè:
Ssmooth (w, X) =
x −1
X NX
x∈X i=1
B xi+1 , fˆ(X, {x1 , . . . , xi } , w),
{z
}
|
Prev. part of the melody
B(x, fˆ) = tanh(−s
ãäå
M1
è
M2
M1 − M2
· e),
|M1 + M2 |
ïåðâûé è âòîðîé íàèáîëüøèå êîìïîíåíòû
ðàñïðåäåëåíèÿ âåðîÿòíîñòåé
ïàðàìåòð,


e=1
fˆ èç f
â (1),
s
åñëè ïðåäñêàçàíèå âåðíî è
Ì. Ï. Ìàòðîñîâ
ìàñøòàáíûé
−1
èíà÷å.
Ïðåäñêàçàíèå ìóçûêè
15 / 28
Ñãëàæåííàÿ ôóíêöèÿ îøèáêè
Ìåíüøèé ïàðàìåòð ìàñøòàáà
s
ôóíêöèè îøèáêè
Ssmooth
äåëàåò
ñòóïåíüêè ìåíåå çàìåòíûìè, òåì ñàìûì ïðîöåññ îïòèìèçàöèè
ïðîùå, íî ìåíåå òî÷íûì.
0.5
0
S(x)
−0.5
−1
−1.5
s = 100
s = 10
s=1
−2
−2
−1
0
1
2
3
x
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
16 / 28
Ñãëàæåííàÿ ôóíêöèÿ îøèáêè
Ñãëàæåííàÿ
Ssmooth (w, X)
è èñõîäíàÿ
S(w, X)
ôóíêöèè îøèáêè
â áîëüøèíñòâå ñëó÷àåâ èìåþò áëèçêèå ìèíèìóìû.
0.325
0.32
0.315
0.31
0.305
0.3
0.295
Smooth
Raw
0.29
−0 8
−0 6
−0 4
−0 2
Ì. Ï. Ìàòðîñîâ
0
02
04
06
Ïðåäñêàçàíèå ìóçûêè
08
1
17 / 28
Ñòîõàñòè÷åñêèé ãðàäèåíòíûé ñïóñê
Ïðåäñêàçàíèå è ôóíêöèÿ îøèáêè âû÷èñëÿåòñÿ äî 100 ÷àñîâ.
Ëó÷øå äåëàòü
ìàëåíüêèå øàæêè
äëÿ íåáîëüøèõ ÷àñòåé
òðåíèðîâî÷íîãî ìíîæåñòâà ñëó÷àéíîå ïîäìíîæåñòâî èç
ìåëîäèé
100
. Äëÿ ñðàâíåíèÿ, â îáû÷íîì íàáîðå ìîæåò áûòü äî
10 000. Òàê îïòèìèçàöèþ ìîæíî ïðîâåñòè áûñòðåå.
−4
2
i
L2 distance ρ(w −w
optimal
)
x 10
1
0
0
10
20
30
40
50
Iteration number
Ì. Ï. Ìàòðîñîâ
60
70
Ïðåäñêàçàíèå ìóçûêè
80
90
18 / 28
Ìóçûêà îáëàäàåò ñâîéñòâàìè åñòåñòâåííîãî ÿçûêà
Ðàñïðåäåëåíèå
N -ãðàììîâ
ïî ÷àñòîòå äëÿ ðàçëè÷íûõ
ñîáûòèé ýòî ÷èñëî ïîÿâëåíèé
ðàâåí
−0.6,
N -ãðàììà
N.
×èñëî
â Midi50k. Íàêëîí
ðàñïðåäåëåíèå ïîõîæå íà ðàñïðåäåëåíèå ñëîâ â
åñòåñòâåííîì ÿçûêå (çàêîí Öèïôà).
7
10
6
1−grams
Number of events
10
5
10
4
10
3
10
16−grams
2
10
1
10
0
10 0
10
1
10
2
10
3
10
4
10
5
10
Most popular N−grams
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
19 / 28
Âåðîÿòíîñòü ýëåìåíòîâ
E
Êàðòà ðàñïðåäåëåíèÿ ÷àñòîò ýëåìåíòîâ
ýòî ÷èñëî ïîÿâëåíèé
N -ãðàììà
E.
×èñëî ñîáûòèé â Midi50k. Ïîðÿäîê ñòðîåâ (ïî
ãîðèçîíòàëè) ïðîèçâîëüíûé ðåçóëüòàò ïðåäñòàâëåíèÿ
àêêîðäà â âèäå ïàðû
(s, r ).
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
20 / 28
Ïðîñòðàíñòâî of 2-ãðàììîâ
Ðàñïðåäåëåíèå êîìïîçèöèé ïî æàíðàì ñïðîåöèðîâàííîå èç
ïðîñòðàíñòâà áèãðàììîâ àêêîðäîâ íà äâóìåðíóþ ïëîñêîñòü
ìåòîäîì PCA.
Cluster Assignments
0.6
blues
classical
hip hop
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.4
−0.3
−0.2
−0.1
Ì. Ï. Ìàòðîñîâ
0
0.1
0.2
0.3
0.4
Ïðåäñêàçàíèå ìóçûêè
0.5
21 / 28
Êà÷åñòâî ïðåäñêàçàíèÿ
×èñëî ïàðàìåòðîâ ðàâíî 16, òðåíèðîâî÷íûé íàáîð âåñü
Midi50k. Ñðåäíÿÿ âåëè÷èíà îøèáêè ðàâíà
óñïåøíî ïðåäñêàçàííûõ ýëåìåíòîâ
x ∈ E).
0.42
(îçíà÷àåò 58%
Ñóùåñòâóþò
ìåëîäèè, ïðåäñêàçàííûå íà 100%, òàê æå êàê è ìåëîäèè
ïðåäñêàçàííûå ïëîõî (<5%).
4000
Number of melodies
3500
3000
2500
2000
1500
1000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Average prediction error S(w). Lower is better.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
0.9
1
22 / 28
Ðàçìåð òðåíèðîâî÷íûõ äàííûõ
S(w, X) =
x −1
X NX
x∈X i=1


xi+1 6= f (X, {x1 , . . . , xi } , w).
{z
}
|
Prev. part of the melody
Average error S(w), lower is better
0.7
Train
Test
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
1
10
2
10
3
10
4
10
Number of melodies in dataset
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
23 / 28
Ñëîæíîñòü ìîäåëè
Ôóíêöèÿ îøèáêè
S(w, X)
è ÷èñëî ïàðàìåòðâ
K (w ∈ R2K ),
ïðîâåðî÷íàÿ âûáîðêà:
Average prediction error, S(w)
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0
2
4
6
8
10
12
14
16
Number of model parameters
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
24 / 28
Ñðàâíåíèå ìåòîäîâ
Quality
Main idea
Mozer[1]
Conklin[2]
Proposed
Bayes classiers
Neural network
Music patterns
Chords
40%
Pitches
90%
Durations
Datasize
[1]
93%
20
95%
58.0%
92.5%
75%
4500
50 000
Neural network music composition by prediction
Multiple viewpoint systems for music prediction
M. Mozer,
Connection Science, 1994.
[2]
D. Conklin,
I. Witten, Journal of New Music Research, 1995, rev. 2002.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
25 / 28
Ðåçóëüòàòû âû÷èñëèòåëüíîãî ýêñïåðèìåíòà
Îïòèìàëüíàÿ ñëîæíîñòü ìîäåëè (ìàêñèìàëüíàÿ äëèíà
N -ãðàììîâ)
ðàâíà 8, íî ÷åì áîëüøå òåì ëó÷øå.
×èñëî ìåëîäèé â òðåíèðîâî÷íîì íàáîðå æåëàòåëüíî
áîëüøå 1000.
Êà÷åñòâî ïðåäñêàçàíèÿ 58% (ïî-àêêîðäíî, 0.024% åñëè
íàóãàä), ðàññòîÿíèå Õýìèíãà ðàâíî 0.075 (92.5%
ñîâïàäàþùèõ ïîëóòîíîâ, 50% äëÿ ñëó÷àéíîãî óãàäûâàíèÿ).
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
26 / 28
Çàùèùàåìûå óòâåðæäåíèÿ
Ïðåäñòàâëåíèå ïîñëåäîâàòåëüíîñòè àêêîðäîâ êàê
ïîëåäîâàòåëüíîñòè ýëåìåíòîâ ñïåöèàëüíî
ñêîíñòðóèðîâàííîé ãðóïïû âïîëíå îáîñíîâàíî.
Ðàçðàáîòàí àëãîðèòì ïðåäñêàçàíèÿ îäíîãî ñëåäóþùåãî
ýëåìåíòà ìóçûêàëüíîé ïîñëåäîâàòåëüíîñòè.
Àëãîðèòì ïðîòåñòèðîâàí íà íàáîðå Midi50k è ïðîâåäåíî
åãî ñðàâíåíèå ñ äðóãèìè ðàáîòàìè (Mozer, Conklin) ñ òî÷êè
çðåíèÿ êà÷åñòâà ïðåäñêàçàíèÿ.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
27 / 28
Ïóáëèêàöèè
M. Matrosov, V. Strijov, A. Matrosov. Short-term forecasting
of musical compositions using chord sequences. Conference of
International Federation of Operational Research Societies, July 2014, Barcelona, Spain.
M. Matrosov, V. Strijov. Short-term forecasting of musical
compositions using chord sequences. 57-th Scientic
Conference of MIPT. November 2014, Dolgoprudny, Russia.
Seminar Music and Science, Moscow State Conservatory
named for P. I. Tchaikovsky. January 20, 2015, Moscow,
Russia.
Ì. Ï. Ìàòðîñîâ
Ïðåäñêàçàíèå ìóçûêè
28 / 28