Introduction to Set Theory  

The Absolute-Do (abdo) System in Twelve-Tone Theory

copyright © 1997 by Larry Solomon  

Confusion can result because of the different systems in use for determining the index number (or transposition number). For example in the absolute-do (abdo) system C is always zero (0), E is always 4, G is always 7, etc. But in reldo (relative-do) any pc may be set to zero; e.g., G can become zero. Both systems are currently in use, so one needs to have some familiarity with both of them.

The row G A# B F E C# C A G# D D# F# is expressed in abdo as: 7AB541098236. To form a matrix with this row as Po, first write the numbers in the top row.

7  A  B  5  4  1  0  9  8  2  3  6

To form the inversion of this (Io), first double the first number (7x2=14). Set S=equal to this result (14) and subtract the row numbers from S, beginning with the second number. Write the results vertically down from the first row number (7). If the resulting subtraction is greater than 11, subtract 12; e.g., 14-1=13 for the sixth note, so subtract 12 (13-12=1), giving 1 as the sixth note of the inversion.

7  A  B  5  4  1  0  9  8  2  3  6
4
3
9
A
1
2
5
6
0
B
8

Finally write the transpositions of P starting with the first number of each row. E.g., the second row starts with 4, which results from subtracting 3 from the top row numbers (or adding 9, mod12). The third row, which starts with 3 is one less than the second row, or may be found by subtracting 4 from the top row, etc.

7 A B 5 4 1 0 9 8 2 3 6
4 7 8 2 1 A 9 6 5 B 0 3
3 6 7 1 0 9 8 5 4 A B 2
9 0 1 7 6 3 2 B A 4 5 8
A 1 2 8 7 4 3 0 B 5 6 9
1 4 5 B A 7 6 3 2 8 9 0
2 5 6 0 B 8 7 4 3 9 A 1
5 8 9 3 2 B A 7 6 0 1 4
6 9 A 4 3 0 B 8 7 1 2 5
0 3 4 A 9 6 5 2 1 7 8 B
B 2 3 9 8 5 4 1 0 6 7 A
8 B A 6 5 2 1 A 9 3 4 7

Notice that with this row in abdo notation, Po starts with 7 (G) and ends with 6. These two numbers become the indexes for the inversions and retrograde forms respectively; i.e., Po and Io start with 7, and Ro and RIo start with 6. P1 and I1 both start with 8, and R1 and RI1 both start with 7, etc. Thus, the identity of row forms is quite different than in reldo. Ro and RIo start with the last pc of Po in abdo, but Ro and RIo (as well as Po and Io) always start with zero (0) in reldo.

  

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