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
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.important to make an AID with n divisions with A1.
so, the common difference between divisions is :
d = 2(Sn - nA1)
n(n-1)
c 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.
171.4285714 is point of intersection in these 3 trends:
Baran scale
Baran scale is a hybrid 12-tone scale which has two segments as 7-aid(700.) and 5-aid(500.) with a1=80 for both:
0
80
166.7
260
360
466.7
580
700
780
870
970
1080
1200
This scale is a subset of a 360-EDO with these degrees: 24 50 78 108 140 174 210 234 261 291 324 360
Here you see interval matrix of baran scale and its graphical deviation from 12-EDO:
Reverse baran scale is a hybrid 12-tone scale which has two segments as 5-aid(500.) and 7-aid(700.) with a1=80 for both:
0
120
230
330
420
500
620
733.333
840
940
1033.333
1120
1200
This scale is a subset of 360-EDO with these degrees: 36 69 99 126 150 186 220 252 282 310 336 360
Here you see interval matrix of reverse baran scale and its graphical deviation from 12-EDO:
