2006年 09月 17日
The Fourth Subject of Contrapunctus XIV of Die Kunst der Fuge
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Die Kunst der Fuge (The Art of Fugue) BWV 1080 is an unfinished
work by J.S. Bach published after his death in 1750, which contains
fourteen fugues and four canons. among them the last fugue is the
final unfinished Contrapunctus XIV. as I previously mentioned in the
third subject of the fugue his name "B-A-C-H" had been embedded,
while the fourth subject had no shape. we are going to seek the
glimpse of the never written subject.
Contrapunctus XIV breaks off abruptly in the middle of the third
section at the 239th measure: lets compose the Mathematical Mozu Song without the help of
PMT(Prouhet-Thue-Morse sequence).
Definition A: Mathematical Mozu Song is an infinite sequence
of 4-notes {0,1,*,.}. '0'<->'1' and '*'<->'.' are complements.
~S and @S stand for the complement of S and the palindrome
of S respectively. the following procedure constructs the song.
(1)S=0
(2)S=S~S (append the complement of S)
(3)S=S@S (append the palindrome of S)
(4)exchange '11' in S with '*' (eliminate central square '11')
(5)continue (2)
here is the expansion steps.
#0: 0
#1: 01
#2: 0*0
#3: 0*01.1
#4: 0*01.*.1 0*0
#5: 0*01.*.10*01.10*.*01.1
#6: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
#7: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
1.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.1
#8: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
1.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.*
.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.10
*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
I once proved cube-freeness of PMT by using *scales*. but it is
very hard for us to find a scale in this case because in this formation
complementary operation and reverse operation alternate. I tried
but failed. so I consult with Thue, and finally find a pretty morphism.
now redefine the Mathematical Mozu Song.
Definition B: Mathematical Mozu Song is an infinite string over
alphabet Z={0,1,*,.}. (endo)morphism M is a mapping:
{0,1,*,.}*->{0,1,*,.}* such that
'0'->'0*0', '1'->'1.1', '*'->'1.*.1', '.'->'0*.*0'.
Song is a sequence of letters generated by M^i(0)=M(M^(i-1)(0)),
for i=0 to oo. lets see how it is looked like:
#0: 0
#1: 0*0
#2: 0*0 1.*.1 0*0
#3: 0*0 1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
#4: 0*0 1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
1.1 0*.*0 1.1 0*0 1.*.1 0*.*0 1.*.1 0*0 1.1 0*.*0 1.*.1
0*.*0 1.1 0*0 1.*.1 0*.*0 1.*.1 0*0 1.1 0*.*0 1.1 0*0
1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
the composition is highly symmetric and balanced.
vocabularies are {0, 1, 0*0, 1.1, 0*.*0, 1.*.1},
all words are palindrome and each word has its
complement. and of course all sentences(strings)
are palindrome. a perfect symmetry. Beautiful! now
that there is no doubt that MMS is a square-free
infinite sequence. perhaps I can find a template for
the proof in Thue's paper or other place. ^^; is it
too exaggerated to say that the MMS is the missing fourth subject
that Bach had abandoned? now I will call the song the "infinitely
palindromic square free sequence". do you know other IPSF?
M...
PS: what happens if I left-shift the song by one? it is sure that
the song is still square-free, but the palindrome property would
be harmed... consider a palindrome string abcdefgfedcba. if I cut
off the first character, then I must cut the tail, too. isnt it? like
bcdefgfedcb, but how can we cut the tail out of an infinite string?
can you imagine an infinite symmetry? I believe that if you have
a chance to hear the song playing, then you would really touch
the Eternity. you would see the birth and the death simultaneously,
even if you hear just the opening bars of a piece of the music
because you have an organ to perceive a perpetual algorithm.
links: The Art of Fugue
http://en.wikipedia.org/wiki/The_Art_of_Fugue
http://tech.groups.yahoo.com/group/theory-edge/message/12387
work by J.S. Bach published after his death in 1750, which contains
fourteen fugues and four canons. among them the last fugue is the
final unfinished Contrapunctus XIV. as I previously mentioned in the
third subject of the fugue his name "B-A-C-H" had been embedded,
while the fourth subject had no shape. we are going to seek the
glimpse of the never written subject.
Contrapunctus XIV breaks off abruptly in the middle of the third
section at the 239th measure:
PMT(Prouhet-Thue-Morse sequence).
Definition A: Mathematical Mozu Song is an infinite sequence
of 4-notes {0,1,*,.}. '0'<->'1' and '*'<->'.' are complements.
~S and @S stand for the complement of S and the palindrome
of S respectively. the following procedure constructs the song.
(1)S=0
(2)S=S~S (append the complement of S)
(3)S=S@S (append the palindrome of S)
(4)exchange '11' in S with '*' (eliminate central square '11')
(5)continue (2)
here is the expansion steps.
#0: 0
#1: 01
#2: 0*0
#3: 0*01.1
#4: 0*01.*.1 0*0
#5: 0*01.*.10*01.10*.*01.1
#6: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
#7: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
1.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.1
#8: 0*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
1.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.*
.10*.*01.10*01.*.10*.*01.*.10*01.10*.*01.10
*01.*.10*01.10*.*01.*.10*.*01.10*01.*.10*0
I once proved cube-freeness of PMT by using *scales*. but it is
very hard for us to find a scale in this case because in this formation
complementary operation and reverse operation alternate. I tried
but failed. so I consult with Thue, and finally find a pretty morphism.
now redefine the Mathematical Mozu Song.
Definition B: Mathematical Mozu Song is an infinite string over
alphabet Z={0,1,*,.}. (endo)morphism M is a mapping:
{0,1,*,.}*->{0,1,*,.}* such that
'0'->'0*0', '1'->'1.1', '*'->'1.*.1', '.'->'0*.*0'.
Song is a sequence of letters generated by M^i(0)=M(M^(i-1)(0)),
for i=0 to oo. lets see how it is looked like:
#0: 0
#1: 0*0
#2: 0*0 1.*.1 0*0
#3: 0*0 1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
#4: 0*0 1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
1.1 0*.*0 1.1 0*0 1.*.1 0*.*0 1.*.1 0*0 1.1 0*.*0 1.*.1
0*.*0 1.1 0*0 1.*.1 0*.*0 1.*.1 0*0 1.1 0*.*0 1.1 0*0
1.*.1 0*0 1.1 0*.*0 1.*.1 0*.*0 1.1 0*0 1.*.1 0*0
the composition is highly symmetric and balanced.
vocabularies are {0, 1, 0*0, 1.1, 0*.*0, 1.*.1},
all words are palindrome and each word has its
complement. and of course all sentences(strings)
are palindrome. a perfect symmetry. Beautiful! now
that there is no doubt that MMS is a square-free
infinite sequence. perhaps I can find a template for
the proof in Thue's paper or other place. ^^; is it
too exaggerated to say that the MMS is the missing fourth subject
that Bach had abandoned? now I will call the song the "infinitely
palindromic square free sequence". do you know other IPSF?
M...
PS: what happens if I left-shift the song by one? it is sure that
the song is still square-free, but the palindrome property would
be harmed... consider a palindrome string abcdefgfedcba. if I cut
off the first character, then I must cut the tail, too. isnt it? like
bcdefgfedcb, but how can we cut the tail out of an infinite string?
can you imagine an infinite symmetry? I believe that if you have
a chance to hear the song playing, then you would really touch
the Eternity. you would see the birth and the death simultaneously,
even if you hear just the opening bars of a piece of the music
because you have an organ to perceive a perpetual algorithm.
links: The Art of Fugue
http://en.wikipedia.org/wiki/The_Art_of_Fugue
http://tech.groups.yahoo.com/group/theory-edge/message/12387
by exod-US
| 2006-09-17 19:36
| 我が命運の尽きる日まで