AID: Difference between revisions

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For an intervallic system with n divisions, [http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29 AID] is considered as [http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html 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 :

If the first division is A1 and the last, An , with common difference of d , we have :

:: A1 = A1

:: A2 = A1+d

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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))'''

:: '''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:

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 .

# 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 :

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

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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 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 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.

* If choosing A1 equal to division size in equal degree EDO, d is zero.

[[file:AIDO-custom-size-298-402.jpg]]

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[[Category:AID]]

[[Category:~~Shaahin_Mohajeri~~]]

[[Category:Shaahin Mohajeri]]

[[Category:~~todo~~:cleanup]]

[[Category:Todo:cleanup]]

@@ Line 3: / Line 3: @@
 For an intervallic system with <u>n</u> divisions, [http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29 AID] is considered as [http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html arithmetic sequence] with divisions of system as terms of sequence.
-If the first division is <u>A1</u> and the last , <u>An</u> , with common difference of <u>d</u> , we have :
+If the first division is <u>A1</u> and the last, <u>An</u> , with common difference of <u>d</u> , we have :
 :: A1 = A1
 :: A2 = A1+d
@@ Line 16: / Line 16: @@
 As we can consider <u>Sn</u> of system to be 1200 cent or anything else (octavic or non-octavic system ) then <u>d</u> is most important to make an AID with n divisions with A1. So, the common difference between divisions is :
-:: '''d =( <u>2(Sn - nA1))/((n(n-1))</u>'''
+:: '''d =( <u>2(Sn - nA1))/(n(n-1))</u>'''
-By considering Sn=1200 , A1=70 , n=12 , d will be 5.454545455 and our 12-tone scale is equal to:
+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 <u>n</u>=8 and <u>A1</u>=150 , then we have 8-EDO .
+# we choose d=0 so, A1 = Sn/n .. Consider <u>n</u>=8 and <u>A1</u>=150, then we have 8-EDO .
 # for a constant <u>n</u> and different <u>A1</u>, if <u>d</u> and (Sn/A1) are integers, we have a subset of EDO or EDI (Equal divisions of Interval).
-Consider <u>Sn = 1400</u> , <u>n</u>=8 and <u>A1</u>=70 , then we have a subset of a 140-ED (1400.) with Degrees as 7 17 30 46 65 87 112 140 :
+Consider <u>Sn = 1400</u> , <u>n</u>=8 and <u>A1</u>=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
@@ Line 35: / Line 35: @@
 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 <u>A1</u> greater than division size in equal degree EDO , <u>d</u> is negative and <u>AID</u> is descending.
+* If choosing <u>A1</u> greater than division size in equal degree EDO, <u>d</u> is negative and <u>AID</u> is descending.
-* If choosing <u>A1</u> smaller than division size in equal degree EDO , <u>d</u> is positive and <u>AID</u> is ascending.
+* If choosing <u>A1</u> smaller than division size in equal degree EDO, <u>d</u> is positive and <u>AID</u> is ascending.
-* If choosing <u>A1</u> equal to division size in equal degree EDO , <u>d</u> is zero.
+* If choosing <u>A1</u> equal to division size in equal degree EDO, <u>d</u> is zero.
 [[file:AIDO-custom-size-298-402.jpg]]
@@ Line 58: / Line 58: @@
 [[Category:AID]]
-[[Category:Shaahin_Mohajeri]]
+[[Category:Shaahin Mohajeri]]
-[[Category:todo:cleanup]]
+[[Category:Todo:cleanup]]