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2/6: Didymus and Ptolemy

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Didymus and Ptolemy

I know that I am mortal by nature, and ephemeral; but when I trace at my pleasure the windings to and fro of the heavenly bodies I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia   ― Ptolemy

Greek music theory was almost lost to antiquity after the Roman conquest.  The theorist Didymus (living in occupied Greece around 60 A.D.) helped preserve and expand upon the theories of The Harmonists. It was Didymus who pointed out the difference between the 3-limit D and the five-limit D1 (in relation to the starting note C), a difference which had the ratio of 81/80 and became known as the Didymic comma. As we will see, these two D’s have very different musical meanings, and you, as a 53-eq or just intonational composer, must choose precisely what you mean to say harmonically with each.

The second-century Alexandrian theorist Ptolemy (northern Egypt) both preserved and rebutted the ancient Pythagorean and Harmonist theories, claiming they were both improperly reasoned. He boiled down the traditional scales to the simplest possible ratios, which for a time brought the fifth harmonic back into common usage. He also explored the tonal possibilities of the seventh and eleventh harmonics, allowing both into his scales! Interestingly, the 53-eq system does not come terribly close to 11th harmonic tunings (it’s 8 cents, or 1/12th of a half step off). However, the 11th harmonic sits almost perfectly in 24-eq or quarter-tone tuning (only one cent off). But for the lower harmonics, 53-eq is a great system for integrating Pythagorean and Ptolemic (3-limit and 5-limit) theories, since, in 53-eq, both systems are almost perfectly in tune.

If you would like to learn more about the 53-equal tuning that embraces all of ancient music, you can buy the entire book, The Grand Unified Theory of Music, in pdf form for $25 with hundreds of embedded musical examples of scales and chords from all over the world.

A free introduction to what The Grand Unified Theory of Music offers is on this website and includes both text and a few musical examples from each webpage. If you would like to learn more about this chapter and the full contents of this entire e-book, you can buy The Grand Unified Theory of Music for $25, with hundreds of embedded musical examples of scales and chords from all over the world — and ideas for how to set up your computer system —


You’ll get a personalized password you can use to see the entire e-book. Inside the full book, you will also get a link to the complete pdf file of this e-book, which you can read on your Kindle or similar device. The links to the hundreds of mp3 sound files – the same ones you can hear on the website — will also be included. This is “Version 1.0” of The Grand Unified Theory of Music. Because it is an e-book, additions, corrections and improvements in the sound may be added at any time. The Grand Unified Theory of Music is Copyright © 2018 by Christopher Mohr. All rights reserved.

One person per password. Sharing this password with others is a violation of copyright. Do not allow others to use your password or link to the pdf file!

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