A C++ implementation of the Binary GCD (Stern's) algorithm is shown in the Related Link below.
Here are the letter notes i used them myself xx RIGHT HAND A#-A-G-E-D-G-B-A-B-C-G-A-A#-A-G-E-D-G-B-A-B-C-F#-G-Eb-E-G-A-G-A-B-F#-B- G-F#-B-F#-B-G-A-B-F#-E-D-C-B-A-G-F#-D-B-A#-A-G-E-D-G-B-A-B-B-C-C-F#-F#- A-B-G-A-B-D-D-C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-A-A-A-G-E-E -D-B-A-GE-GE-BG-BG-DB-DB-F#D-F#B-CA-CA-BG-GE-GE-F#D-DB-DA-G-A-B-D-D- C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-B LEFT HAND AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-GDA-GDG-AEB-AEA- GDA-GDG-AEB-AEA-AEB-AEA-AEB-AEA-CGC-CDG-EBF#-EAE-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-FACF-GBDG-DF#AD-CEGC-CEGC-DF#AD-DF#AD-FACF-GBDG-DF#AD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-BDF#/
For piano or keyboard it's: (Right Hand) A#-A-G-E-D-G-B-A-B-C-G-A-A#-A-G-E-D-G-B-A-B-C-F#-G-Eb-E-G-A-G-A-B-F#-B- G-F#-B-F#-B-G-A-B-F#-E-D-C-B-A-G-F#-D-B-A#-A-G-E-D-G-B-A-B-B-C-C-F#-F#- A-B-G-A-B-D-D-C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-A-A-A-G-E-E -D-B-A-GE-GE-BG-BG-DB-DB-F#D-F#B-CA-CA-BG-GE-GE-F#D-DB-DA-G-A-B-D-D- C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-B (Left Hand) AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-GDA-GDG-AEB-AEA- GDA-GDG-AEB-AEA-AEB-AEA-AEB-AEA-CGC-CDG-EBF#-EAE-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-FACF-GBDG-DF#AD-CEGC-CEGC-DF#AD-DF#AD-FACF-GBDG-DF#AD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-BDF#B (Part 1) http://www.youtube.com/watch?v=3df2QqFRHkE&feature=related (Part 2) http://www.youtube.com/watchannotation_id=annotation_948659&v=Q5qg20FpBQM&feature=iv
Bella's Lullaby was written and composed by Carter Burwell some time before the movie- he met Catherine Hardwicke in Oregon, and the choice was eventually chosen for him to do the lullaby for Bella. Watch the movie for the scene where Rob Pattinson plays the music on a piano (no fake acting, mwahaha). The idea for "River Flows in You", by Yiruma, was considered for the part, but was not chosen. It is a beautiful piece, but it did not give off what the director and Stephanie Meyers wanted for the movie. Click on the links below to go direct to Bella's Lullaby piano sheet music:
This version is by Carter Burwell, It's the piece Edward plays in the actual movie! (Right Hand) X) A#-A-G-E-D-G-B-A-B-C-G-A-A#-A-G-E-D-G-B-A-B-C-F#-G-Eb-E-G-A-G-A-B-F#-B- G-F#-B-F#-B-G-A-B-F#-E-D-C-B-A-G-F#-D-B-A#-A-G-E-D-G-B-A-B-B-C-C-F#-F#- A-B-G-A-B-D-D-C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-A-A-A-G-E-E -D-B-A-GE-GE-BG-BG-DB-DB-F#D-F#B-CA-CA-BG-GE-GE-F#D-DB-DA-G-A-B-D-D- C-C-B-B-A-E-E-G-G-B-B-D-D-D-D-C-C-B-B-A-E-E-G-G-B-B-E-E-F#-D-D-C-C-B-B-A-G-G-C-C-E-E-A-A-D-D-C-C-B-B-A-G-G-G-C-C-E-E-A-B (Left Hand) X) AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-AEB-AEA-AEB-AEA-CAD-CGE-AEB-AEA-GDA-GDG-AEB-AEA- GDA-GDG-AEB-AEA-AEB-AEA-AEB-AEA-CGC-CDG-EBF#-EAE-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-FACF-GBDG-DF#AD-CEGC-CEGC-DF#AD-DF#AD-FACF-GBDG-DF#AD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD-GBD-GCD-GCD-GBD- GBD-BDF#B Check out these links to figure out how to play them together: (Part 1) http://www.youtube.com/watch?v=3df2QqFRHkE&feature=related (Part 2) http://www.youtube.com/watchannotation_id=annotation_948659&v=Q5qg20FpBQM&feature=iv
b+b+b+c+c+c+c =3b+4c
c + c + 2c + c + c = 6c
b + b + b + c + c + c + c = 3b + 4c
To calculate the least common multiple (lcm) of decimals (integers) and fractions you first need to calculate the greatest common divisor (gcd) of two integers: int gcd (int a, int b) { int c; while (a != 0) { c = a; a = b % a; b = c; } return b; } With this function in place, we can calculate the lcm of two integers: int lcm (int a, int b) { return a / gcd (a, b) * b; } And with this function in place we can calculate the lcm of two fractions (a/b and c/d): int lcm_fraction (int a, int b, int c, int d) { return lcm (a, c) / gcd (b, d); }
4c
c + c + c + c + c = 5 * c.
To find the GCD of three numbers, a, b and c, you need to find the GCD of a and b first, such that d = GCD(a, b). Then call GCD(d, c). Although you could simply call GCD(GCD(a, b), c), a more useful method is to use an array and iteratively call the GCD(a, b) function, such that a and b are the first two numbers in the first iteration, which becomes a in the next iteration, while b is the next number. The following program demonstarates this method. Note that the GCD of two numbers can either be calculated recursively or iteratively. This program includes both options, depending on whether RECURSIVE is defined or not. In a working program you'd use one or the other, but the iterative approach is usually faster because it requires just one function call and no additional stack space. The program will create 10 random arrays of integers of length 3 to 5 and process each in turn. Note that the more numbers in the array, the more likely the GCD will be 1. #include<iostream> #include<time.h> #define RECURSIVE // comment out to use iterative method #define ARRAY // comment out to use non-arrays #ifdef RECURSIVE // Returns the GCD of the two given integers (recursive method) unsigned int gcd(unsigned int a, unsigned int b) { if(!a) return(b); if(!b) return(a); if(a==b) return(a); if(~a&1) { if(b&1) return(gcd(a>>1,b)); else return(gcd(a>>1,b>>1)<<1); } if(~b&1) return(gcd(a,b>>1)); if(a>b) return(gcd((a-b)>>1,b)); return(gcd((b-a)>>1,a)); } #else // Returns the GCD of the two given integers (iterative method) unsigned int gcd(unsigned int a, unsigned int b) { if(!a) return(b); if(!b) return(a); int c; for(c=0; ((a|b)&1)==0; ++c) { a>>=1; b>>=1; } while((a&1)==0) a>>=1; do{ while((b&1)==0) b>>=1; if(a>b) { unsigned int t=a; a=b; b=t; } b-=a; }while(b); return(a<<c); } #endif RECURSIVE // Returns the greatest common divisor in the given array unsigned int gcd(const unsigned int n[], const unsigned int size) { if( size==0 ) return( 0 ); if( size==1 ) return( n[0] ); unsigned int hcf=gcd(n[0],n[1]); for( unsigned int index=2; index<size; ++index ) hcf=gcd(hcf,n[index]); return(hcf); } int main() { using std::cout; using std::endl; srand((unsigned) time(NULL)); for(unsigned int attempt=0; attempt<10; ++attempt) { unsigned int size=rand()%3+3; unsigned int* num = new unsigned int[size]; unsigned int index=0; while(index<size) num[index++]=rand()%100; unsigned int hcf=gcd(num,size); cout<<"GCD("; index=0; cout<<num[index]; while(++index<size) cout<<','<<num[index]; cout<<") = "<<hcf<<endl; delete[]num; } cout<<endl; }
There are no "primary and secondary keys" in c and c plus plus.