When I was studying at the University of Iowa, I took an independent study with Larry Fritts during which we engaged in a wide range of conversations about music and music-related math. I began experimenting with different tuning systems, some of which were based in some way on the overtone series. For example, I created a 12-tone equal temperament scale using a 3:2 “octave” rather than the 2:1 octave. In other words, I split the pure P5, rather than the P8, into 12 equal steps. I then translated a Bach invention into my new system to fascinating results. One of the more interesting facets of the result was the fact that while the “perfect consonant” of the P8 was translated into another “perfect consonant” of the P5, many other consonances became dissonance through the translation and vice versa. The dissonance of a tritone, for example, became an interval that rests between the m3 and the M3, both of which are consonances. The consonant P5 becomes a large M3, while the consonance of a M3 becomes an even larger M2.
(I also did the same translation with a 5:2 “octave”—that is, a pure M10. This translation produced even more fascinating results in terms of dissonances & consonances.)
While pondering the overtone series and the extent to which our equal tuning system fails to correspond to the “pure” intervals found in the series, I asked myself this very unimportant question: “So what is the ‘pure’ tritone?”
We know that the “pure” perfect fifth (3:2) is produced by the combination of the second and third partials and that it is approximately 2 cents larger than the equal tempered perfect fifth. (100 cents = semitone). We know that the “pure” major third (5:4) is produced by the combination of the fourth and fifth partials and that it is approximately 14 cents smaller than the equal tempered major third.
Questions about “pure” intervals begin with the minor third. In terms of the overtone series, which minor third provides us with the “pure” minor third? Is it the 6:5 minor third (316¢), the first minor third appearing in the overtone series? Is it the 7:6 minor third (267¢)? Or is it the first minor third produced with the fundamental (partial 1) as its lower tone, the 19:16 minor third (298¢)?
What I find interesting about the tritone as found in the overtone series is that there are no fewer than five differently sized tritones embedded within the first 16 partials. (To put this in perspective, there are no P5s that differ in size from the 3:2 perfect fifth.) The only other interval that appears in more different sizes within the first 16 partials is the M2, another dissonant interval, which appears in six different sizes (8:7, 9:8, 10:9, 11:10, 12:11, 15:13). It is often pointed out that we hear the consonantness of our consonant intervals because of their low position in the overtone series. While it is true that the first five partials produce the major triad, it is usually overlooked that so many dissonances appear very low in the series as well.
The five differently sized tritones are:
- 7:5 (582.512¢)
- 10:7 (617.488¢)
- 11:8 (551.318¢)
- 13:9 (636.618¢)
- 16:11 (648.682¢)
You may have noticed that the 14:10 interval produces a tritone. However, this is simply a duplication of the 7:5 interval in terms of size.
Hear now what they sound like, in order of smallest size to largest, with the equal tempered tritone included:
- 11:8 Tritone (551¢)
- 7:5 Tritone (583¢)
- Equal Tempered Tritone (600¢)
- 10:7 Tritone (617¢)
- 13:9 Tritone (637¢)
- 16:11 Tritone (649¢)
So is the first tritone appearing the overtone series the “pure” tritone, the 7:5 tritone (582.512¢)? Or is it the first tritone appearing with the fundamental as the lower note, the 11:8 tritone (551.318¢)? This tritone is actually nearly half way between the equal tempered tritone and the equal tempered perfect fourth. Since it rests closer to the P4 than the others, does it have less of a pull towards resolution? If so, does that make it more or less pure? Perhaps the “pure” tritone is the 10:7 tritone (617.488¢), which is the inversion of the 7:5 tritone. Or maybe it’s the 13:9 tritone (636.6177¢)? Surely it’s not the 16:11 tritone (648.682¢), the one closest to the perfect fifth, right?
Hearing these tritones in succession, they begin to sound very different from each other. After all, the distance between the smallest tritone (11:8) and the largest (16:11) is 2 cents shy of a semitone. Whichever you decide is the purest of them all, perhaps you will never hear the tritone (or tritones) the same way again.
…Or maybe you will.