My book is an in-depth music theory text, outlining a musical notation based on prime-factorization of frequency ratios, and the design of both my lattice diagrams and planetary graphs to represent those prime factors. The second part of the book is a very extensive examination of different historical just-intonation tuning systems, as well as an exploration of myriad different tunings used by contemporary microtonal composers. It's been about 14 years in the works.

I should state at the outset that I find it impossible to believe that "atonality" can exist, since there are always proportional relationships between simultaneously-heard notes, whether simple and clearly perceived, or complex and vague.

My work began originally as an attempt to create what would be for me a more lucid microtonal notation, avoiding the proliferation of large numbers in ratios, and enabling me to display the pitches on a staff with as much standard music-notation convention as possible.

As I've explored the subject more deeply, and debated it with colleagues on the Tuning Forum, it has grown into something far deeper and more meaningful to me. I have by now become caught up in the numerological belief that there is something special inherent in the series of prime numbers, as a final reduction or simplification of our understanding of musical harmony or tonality.

The introductory chapters present the background on human perception of pitch and our increasing sophistication in notating musical pitches.

Next are mathematical explanations of ratios, equal-temperaments, and prime-factoring. Then I describe a graph of sonance, Matrix Addition for the calculation of intervals, and my Planetary Graphs.

The rest of the book shows how my lattice diagrams are designed by making use of this prime factorization, with the diagrams becoming more and more complex as I give an overview thru both increasing prime limits and a comprehensive historical chronology.

Partch claimed that

"In terms of consonance man's use of musical materials has followed the scale of musical intervals [which] begins with absolute consonance (1 to 1) and progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase."

Although I find an element of truth in this, I wish to show by my manner of presentation that suprisingly large prime numbers were a part of many older theories.

The historical aspect gives all of the different just-intonation tuning systems I have found in doing my research, presented from one viewpoint--one which, I have found, makes it easier to understand how different tuning theories have evolved and interacted with each other.

It goes back to ancient Greek and Indian systems, analyzes the work of several medieval European theorists, including in-depth surveys of Boethius's On Music, the Musica Enchiriadis, and Marchetto of Padua's Lucidarium, and progresses all the way up to present-day composers such as La Monte Young, Ben Johnston, Ezra Sims, and many younger composers, with special emphasis given to Harry Partch and Arnold Schoenberg. I also present what I believe are the first microtonal analyses of gifted 'popular' composer/performers such as Robert Johnson and Jimi Hendrix.

As can be inferred by my inclusion of Schoenberg, I have also given some reference to some of the more popular equal-temperaments, especially (unavoidably)12-equal, and how some theorists have "justified" or explained the use of these temperaments by emphasis on their good or bad representation of particular ratios.

David Beardsley wrote, in Onelist Tuning Digest # 206, message 9:

So what does your book offer that one can't already find in the writings of Partch, Helmholtz or Doty? Or even a trip to the library wouldn't dig up?

ORDERING INFORMATION

JustMusic is available!

for $40 US plus postage ($3 in the US).

My post office has given me the following sample rates for postage. Air mail takes 7 to 9 days, parcel takes 4 to 6 weeks.

PARCEL: $9.66 US anywhere in the world
AIR: $9.15 US to South America, $14.40 US to Europe, $20.05 US to Australia.

Send me an email notifying me of your interest before sending money.

Thank you.

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I'm still working on the 4th edition of it (October 2000), so it's not really ready to be printed. A text of the 3rd edition is available for download here in a zip file (291 K), in Microsoft Word format (212 pages). It does not include any musical examples or lattice diagrams.

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If you don't understand my theory or the terms I've used, start here or try some definitions.

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to my home page return to the Sonic Arts home page