# AID

## Arithmetic irrational divisions

For an intervallic system with __n__ divisions, AID is considered as arithmetic sequence with divisions of system as terms of sequence.

If the first division is __A1__ and the last , __An__ , with common difference of __d__ , we have :

- A1 = A1
- A2 = A1+d
- A3 = A1+2d
- A4 = A1+3d
- ...
- An = A1+(n-1)d

So sum of the divisions is __Sn__ :

**Sn =(**__n[2A1+(n-1)d])/2__

As we can consider __Sn__ of system to be 1200 cent or anything else (octavic or non-octavic system ) then __d__ is most important to make an AID with n divisions with A1. So, the common difference between divisions is :

**d =(**__2(Sn - nA1))/((n(n-1))__

By considering Sn=1200 , A1=70 , n=12 , d will be 5.454545455 and our 12-tone scale is equal to:

**0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**

Scales based on AID can be subsets of EDO if:

- we choose d=0 so , A1 = Sn/n .. Consider
__n__=8 and__A1__=150 , then we have 8-EDO . - for a constant
__n__and different__A1__, if__d__and (Sn/A1) are integers, we have a subset of EDO or EDI (Equal divisions of Interval).

Consider __Sn = 1400__ , __n__=8 and __A1__=70 , then we have a subset of a 140-ED (1400.) with Degrees as 7 17 30 46 65 87 112 140 :

- 0.0 70.0 170.0 300.0 460.0 650.0 870.0 1120.0 1400.0

And now for Sn=1400 and n=8,

- If A1=175.0 then we have 8-AID(1400.)
- If A1=56 then we have 700-AID(1400.) with Degrees as 28 73 135 214 310 423 553 700
- If A1=87.5 then we have 112-AID(1400.) with Degrees as 7 16 27 40 55 72 91 112

AID system shows different ascending, descending or linear trend of change in divisions sizes due to relation between n and A1 in AID and EDO with equal degree:

- If choosing
__A1__greater than division size in equal degree EDO ,__d__is negative and__AID__is descending. - If choosing
__A1__smaller than division size in equal degree EDO ,__d__is positive and__AID__is ascending. - If choosing
__A1__equal to division size in equal degree EDO ,__d__is zero.

171.4285714 is point of intersection in these 3 trends:

We can have different kinds of AID:

- AIDO = Arithmetic irrational divisions of octave
- AIDINO = Arithmetic irrational divisions of irrational non-octave
- AIDRNO = Arithmetic irrational divisions of rational non-octave
- AIDRI = Arithmetic irrational divisions of rational interval
- AIDII = Arithmetic irrational divisions of irrational interval

Example: Baran scale