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KÖVESS MINKET A FACEBOOKON IS: https://www.facebook.com/Edesviz KERESD A WEBOLDALUNKON: http://webaruhaz.edesviz.hu/kalandok-a-leleknek.html ÉDESVÍZ+ magazin: http://edesvizkiado.hu/ Ér...
A bok that explore all the implication of harmony and shows a theoryFull description
A bok that explore all the implication of harmony and shows a theoryDescripción completa
Harmony theory by Sir John StainerDescripción completa
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Descripción completa
Harmony theory by Sir John Stainer
A bok that explore all the implication of harmony and shows a theoryFull description
KÖVESS MINKET A FACEBOOKON IS: https://www.facebook.com/Edesviz KERESD A WEBOLDALUNKON: http://webaruhaz.edesviz.hu/kalandok-a-leleknek.html ÉDESVÍZ+ magazin: http://edesvizkiado.hu…Full description
Solution ManualFull description
We consider then U
r x log r x
U
1
p x log p x
log r x
r x
log p x
1
If p x is varied at a particular argument xi
q x log q x p x
dx
1
log q x
q x
dx
si , the variation in r x is
r x
q xi
si
and U
q xi
si log r xi dx i
log p si
0
and similarly when q is varied. Hence the conditions for a minimum are q xi
si log r xi dx i
log p si
p xi
si log r xi dx i
log q si
If we multiply the first by p si and the second by q si and integrate with respect to s i we obtain
or solving for
and
H 3
H 1
H 3
H 2
and replacing in the equations H 1
q xi
si log r xi dx i
H 3 log p si
H 2
p xi
si log r xi dx i
H 3 log q si
Now suppose p xi and q xi are normal p xi q xi
Ai j
n 2
2
n 2
Bi j
n 2
2
n 2
1 2
exp
1 2
exp
∑ Ai j xi x j
∑ Bi j xi x j
Then r xi will also be normal with quadratic form C i j . If the inverses of these forms are a i j , b i j , c i j then ci j
ai j
bi j
We wish to show that these functions satisfy the minimizing conditions if and only if a i j give the minimum H 3 under the constraints. First we have n
log r xi q xi
si log r xi dx i
1 C i j 2 2 n 1 C i j log 2 2 log
1 2
∑ C i j xi x j 1 2
∑C i j sis j
1 2
∑C i j bi j
This should equal H 3 n H 1 2
which requires Ai j
H 1 H 3
log
C i j . In this case A i j
1 Ai j 2
1 2
∑ Ai j si s j
H 1 Bi j and both equations reduce to identities. H 2
