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Curiosity really is a dangerous thing! I've always wondered why the black keys are arranged the way they are on the piano. This curiosity goes all the way back to when I had piano lessons from age six to twelve. An article in the Washington Post gave me a clue on how to discover the reason. This set me on a path during which friends with a music theory background would tell me, for example, that the phenomenon that I had just observed on my spreadsheet was the Pythagorean Comma. Also some of the tricks I used in massaging the data yielded interesting results not found using the standard approaches. I ended up writing 30 pages. The table below is from the paper.
From this table, all twelve of the major scales can be
extracted. For example, for C Major, from the bottom row,
read C D E, all in red, and from the row above read F G A B, also in
red. For G Major read G A B in green background from the next to bottom line,
and C D EG♭ ,
also in green background,
from the bottom line. The notes of the major keys A through G as well
as C♯, D♯,
G♯ and A♯ can be easily generated in the same fashion as for C and G
Major. To explain why F♯ is defective and how the defect can be
remedied by tempering, read the paper. Click here.
for how it works. Oh yeah, I showed why
the black keys are where they are on the piano. It's in the first five pages.
The paper can now be seen at the on-line musical journal Eunomios. Click here to read
A local copy of the paper can obtained by clicking here.
The paper can also be found on the internet by using the search string :
"musical theory" four-thirds three-halves
on the Google, Yahoo or Bing search engine.
at the www.eunomios.org site. Please email me comments, criticisms or sugestions to the email address shown above. I really would like to hear from you.
Stuff I have worked on:
Past Cooperative Efforts with University Faculty (while working at the Topographic Engineering Center)
John R. Benton and V.S. Subrahmanian, Automated Siting of Air Defense Missile Batteries, Proceeding, Army Science Conference, Orlando FL, June 1994, pp. 141-148.
John R. Benton and V.S. Subrahmanian, Using Hybrid Knowledge Bases for Missile Siting Problems, June, 1993, Technical Report, University of Maryland Computer Science Dept. Also in Proceedings, Tenth Conference on Artificial Intelligence for Applications, IEEE Computer Society, San Antonio, TX, (1994).