The neddiH dlroW of Intervals

© 1990 Paul Cooijmans B., B.

In 1990, during my study of composition at the Brabants Conservatorium of Tilburg, Netherlands, I found a mathematical method to model certain aspects of music. I had the bad luck no one grasped the import of my work. The abstraction level and simple mathematics involved were beyond even the most enlightened figures in contemporary Netherlandic music. This lack of understanding was one of my reasons to join the high-I.Q. society Mensa.

After qualifying I wrote an article about my method in Mensa's journal. Indeed I got a reaction from a member who had studied mathematics and had music as a hobby, and understood my work. She advised me to publish it in a music theory magazine. I sent it to the only appropriate one at the time, but the editor returned it to me, saying he could not understand a word of it, even after multiple readings. I learnt: write beyond the editor's comprehension and you will not get published. I realized I was too smart for my peers and ahead of my time. For protection, I had the treatise I'd written on the matter registered, and decided the Netherlandic music scene would have to do without me for the moment.

Meanwhile, I joined other I.Q. clubs with higher requirements, and achieved world fame as a test creator, founder of societies and author of articles. And I tried again, and wrote an explanation in layman's terms for these societies' members:

In Western music, twelve tones are used. When all possible intervals (that is, combinations of two tones) between those are considered, only seven are distinct, that is, can not be made to coincide with any of the other six by permutation of their two tones.

A local musical situation (set of tones occurring together) can be characterized by its intervallic contents. For example, four tones have six intervals among them. A stunning discovery I have made is that most of the (theoretically possible) intervallic contents can not occur in a set of tones. In other words, that most interval combinations will never sound.

For better understanding one may study the analogous phenomenon for the digits 1 to 9, and verify for oneself that indeed many interval combinations can not occur in any digit combination.

Next, I have come to see that relations between two sets of tones can also be characterized by intervallic content; e.g., two sets of respectively three and five tones have fifteen intervals between them. In terms of intervals, there is no essential difference between these "relation" interval combinations and the "set of tones" interval combinations. The latter sound in the set, the first between sets. As relation interval combinations, all theoretical possibilities may occur.

An interval combination can be written as a row of seven numbers, each the occurrence frequency of one of the seven interval types. This row can be taken as a seven-dimensional coordinate set; music thus corresponds to seven-dimensional spatial events.

Research has shown that properties of these spatial events bear relation to aspects of the corresponding music. A few examples: For musical situations corresponding to a straight line, the length of the line is indicative of the discordance perceived. For situations corresponding to a triangle, the area is indicative of the discordance. "Discordance" by the way should not be confused with "dissonance"; the first refers to a relation between two situations, the latter to the tension within a situation.

A more detailed explanation of this is in this treatise (English online version). A Netherlandic version is available as a P.D.F. file and online. A computer program that executes these calculations is the Qoymans Intervallic Converter.