Arithmetic divisions of lenght     Back to tuning models
 

For a length of L (in meter) with n divisions , ADL is considered as arithmetic sequence with divisions of length as terms of sequence.     

         If the first division is l1 and the last , ln , with common difference of d , we have :              

         l1 = l1 

         l2= l1+d         

         l3= l1+2d       

         l4 = l1+3d      

         ………

         ln = l1+(n-1)d

         So sum of the divisions is L :

 L = n[2 * l1+(n-1)d]

                2        

The common difference between divisions is :       

 d =  2(L -n*l1)

                                                                                            n(n-1)

For any length ratio asa/b,wehaveL=1- (b/a) :

 -For string length , L =1
- For octave length , L = 0.5

Consider an octave lenght with7divisions which l1=0.1m andd=-0.00952381m . We have 7-EDL as:

 

0.000
182.404
365.825
548.229
726.865
898.153
1057.627
 1200.000

 

            Length division sizes are (in meter):

0.1
0.09047619
0.080952381
0.071428571
0.061904762
0.052380952
0.042857143

          This system shows descending trend of length division sizes.                                                                          

You can click here to download an excel spreadsheet about this system.

 

An irregular temperament based on 12-ADL

            

 Consider an octave lenght with 12 length divisions which l1= 0.0547 meter and d= - 0.002369697 meter .The result is an irregular temperament:     

   

0.000
97.387
195.980
295.656
396.257
497.590
599.416
701.444
803.326
904.649
1004.926
1103.593
1200.000

 

                    You can click here to download  VISUAL TEMPERAMENT ANALYZER about this system.

 

             Here, you can see interval matrix and its graphical presentation of this 12-ADL irregular temperament : 

    

 
 

 

Here is rational approximation of this 12-ADL irregular temperament (By scala):