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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 :1- we choose d=0 so , A1 = Sn/n .. Consider n=8 and A1=150 , then we have 8-EDO .
2- for a constant n and different A1, if d and (Sn/A1) are Integer number , we have a susbet 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 :
0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000
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 sytem 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.**

external image AIDO-custom-size-298-402.jpg

171.4285714 is point of intersection in these 3 trends:



external image AIDO2.jpg
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**