July 29, 2016
Last time I talked about a theory I developed to understand the complex harmonic structure of intervals in a simple way, and I made the following offhand remark:
"If you program computers, you can see how it would be very easy to write a program which selects levels and places to produce music in Just Intonation based on given fundamental frequencies."
After posting that I realized that it may not actually be so obvious how to do that, so I went back to the paper I had written in 1999 and copied the following equations which show how the (level, place) notation works with any pair of numbers. In the paper instead of the term "level" I was using the term "order", hence the letter o is used for the level. The term "primary" used below means any rational interval having a power of 2 in the denominator. (A rational interval having a denominator which is not a power of 2 is called "incidental".) The basic idea of the system is to represent all intervals in a so-called "primary" form, within a couple of cents accuracy (obviously, the close to the target the better). The reason to do this is so that all intervals share a common fundamental. Doing that makes complex harmonic relationships much easier to manage, and allows for quick comparative analysis of complex interval structures.
Just in case it seems confusing that x is in the denominator, let me explain. The preferred notation for harmonic interval ratios is horizontal form using a colon x:y and in all cases this means x is less than y. This way, intervals are written in ascending order from left to right. However, ratios in general are normally written in vertical form with the larger number on top, that is the form preferred for individual tones.
Having these equations to work with, hopefully now any programmer can now see how easy it would be to write a computer program to work with this kind of data to make music in Just Intonation.
Regards,
Aaron
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