John Benton
email: Click here to get my Curriculum Vitae Tel: (703)556-7783

What I Just Did:

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.

Super Rows
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
C♯' D♯' F' G' A' B' D♭' E♭'
F♯ G♯ A♯ C D E G A B

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.

An Archaeological Dig into the Mathematical Foundations of Western Music

at the 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:

  • Interactive Maryland Platform for Agents Collaborating Together (IMPACT)
  • Heterogeneous Reasoning and Mediator System (HERMES)
  • Hacking in Lisp on route planners
  • Past Cooperative Efforts with University Faculty (while working at the Topographic Engineering Center)

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