TWELVE-TONE SCALE?
Nigel Nettheim
This item first appeared in Mathematics Magazine,
Vol 42 No. 4, September 1969, pp. 216-218.
Published by the Mathematical Association of America.
Problem 714. [January, 1969] Proposed by Samuel Wolf, Linthicum Heights, Maryland.
Find a solution in base 12 for V I O L I N
V I O L I N
V I O L A
C E L L O
Q U A R T E T
Solution by Nigel F. Nettheim, U. S. Department of Commerce, Washington, D. C.
I have regarded this as a problem in computer programming. There are 8 solutions in base 12 and they are given below. Computer time on a GE625 was 46 seconds.
11 4 3 6 4 1 8 3 7 5 3 2 9 0 4 10 0 2 9 5 4 10 5 3
11 4 3 6 4 1 8 3 7 5 3 2 9 0 4 10 0 2 9 5 4 10 5 3
11 4 3 6 5 8 3 7 5 0 9 0 4 10 3 9 5 4 10 2
9 8 6 6 3 9 4 5 5 7 5 8 10 10 4 6 7 10 10 4
2 0 5 7 10 8 10 1 6 0 10 11 4 11 1 7 3 6 11 8 11 1 8 2 11 0 7 0
5 9 8 3 9 6 6 8 0 3 8 9 11 7 8 5 7 10 6 7 10 8 7 11
5 9 8 3 9 6 6 8 0 3 8 9 11 7 8 5 7 10 6 7 10 8 7 11
5 9 8 3 11 6 8 0 3 4 11 7 8 5 9 6 7 10 8 4
10 2 3 3 8 5 11 3 3 0 6 3 5 5 8 5 9 8 8 10
1 0 11 4 7 2 7 1 2 4 7 10 11 10 2 0 9 4 1 3 1 1 2 4 3 0 9 0
There are 40 solutions in base 13, for example:
11 3 6 7 3 0
11 3 6 7 3 0
11 3 6 7 9
4 8 7 7 6
1 10 9 12 2 8 2
There are many solutions in higher bases.