We've formed PERMUTATION or SYMMETRIC GROUPS of LETTERS and of COLORS. Now we shall form some of musical tones. Actually, this is a long hallowed ART in "cathedral towns" of England, where aficionados RING ALL THE CHANGES (PERMUTATIONS) ON A GIVEN NUMBERS OF BELLS, under two conditions:
- only immediate neighbors in an order can be interchanged;
- at the end of the last set of changes, you must return to the ORIGINAL ORDERING.
If you look back at the previous PERMUTATIONS ON LETTERS OR COLORS, exactly those conditions are satisfied!
20th century mathematicians have named these "campanological groups", since "campanolgy" is "the art of bell ringing". But none of these bell-ringers are aware of the fact that "all the changes" constitutes that MATHEMATICAL STRUCTURE, THE GROUP!
In Dorothy Sayers' detective novel, The Nine Tailors ( 9 bells), some of the changes are rung one New Year's Eve, after which a man is found dead in the belfry, killed by the bell- ringing. However, not all the 9! = 362880 permutations were rung, since that would that would take too long. There's a claim that once all changes on 8 bells were rung (8! = 40320 permutations), taking 2 1/2 days! So, rinring all changes on 9 bells might take 9 times that long, or 22 1/2 days!
To illustrate, let's take the the four bell-tones that ring out "The Big Ben Chime". In the first order they descend in solfeggio as ""mi, re, do, sol". Then, as with the four letters, ABCD, in a previous file, or their colorings in its successor, there are exactly 24 PERMUTATIONS, hence, 24 CHANGES ON THE BELLS.
(Personal Note: In the summer of 1957, I initiated and co-organized THE FIRST NATIONAL SCIENCE FOUNDATION SEMINAR FOR HIGH SCHOOL TEACHERS at Inter American University, San Germ&aacut;m, Puerto Rico, where I taught math. As part of this SEMINAR, I hired the University organist to ring all these tones on the Campus Carillons: 96 ding-dongs -- which people outside my class thought very strange.
Let's represent those tones as the notes above, and including, "middle C", on the piano, descending to G below "middle C": EDCG. Using the ALGORITHM for LETTERS or COLORS -- since (as noted above) it the CAMPANOLOGICAL CONDITIONS, WE HAVE THE 24 CHANGES ON THE "BIG BEN" DING-DONGS:
EDCG GECD CEDG GCDE DCEG GDEC EDGC EGCD CEGD CGDE DCGE DGEC EGDC ECGD CGED CDGE DGCE DEGC GEDC ECDG GCED CDEG GDCE DECG