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:

#### 0.10.090476190.0809523810.0714285710.0619047620.0523809520.042857143

This system shows descending trend of length division sizes.

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

#### 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): 