Some of my peers are already familiar with my feelings about tritones. I think they’re great. They’re historically one of the most misunderstood musical intervals, if not the most misunderstood interval. Way back when, before the times of Bach even, tritones were considered the ‘Devil’s interval’ because of their dissonance. Composers avoided them, or faced heavy criticism.
But tritones are so cool, and frankly, kind of unavoidable. So instead of making music really boring, these early musicians came up with rules for how tritones ought to be applied and resolved in harmonic settings. I’m not going to go too in depth, but I’ve included a brief music theory lesson to help you understand why tritones are so cool, if you don’t know already, at the bottom of this post.
Tritones are a curious topic in music perception. I first got really passionate about them in a music perception context when I saw that Narmour’s Implication Realization Model for melodic expectancy considers tritones a “threshold interval” (p.296 in Schellenberg’s article) that is neither large or small. Small intervals are made up of five semitones or fewer, large intervals consist of seven or more semitones. Narmour’s theory, without going on another music theory tangent, basically sums up that melodic expectancy is derived from the relationship between the first two notes in a melody, the implicative interval, which then sets up an expectation of what the next note, the realized interval, in the melody will be. If the implicative interval is an octave, an the realized interval is most likely going to be a smaller interval, perhaps a tone or a semi tone, going in the opposite direction. This model has been applied to all kinds of different genres, including atonal music, and to music of different cultures. So our expectation of how melodies will continue seems relatively constant on a global scale, but how we perceive tritones… ehh, not so much.
The tritone paradox is an an example of why tritones are so fantastic. In a series of experiments conducted by Deutsch (1987, 1990, 1991), participants were presented with pairs of tritones and asked whether the notes were ascending or descending in pitch. These tones had a bell-shaped spectral pitch with octave related harmonics. This makes the pitch of the tones easily distinguishable, but the height a little more difficult to define. Whether the tone pairs were heard as ascending or descending varied among participants. Shifting the tritone pair up or down a range of three octaves didn’t change how they heard them either.
If you’re interested to see how the experiment works, try it:
(I have been trying to force hear it both ways, with no success)
The twelve tones of the octave can be represented by a circle, which I’ve shown below. Deutsch and colleagues discovered that there is a correlation between the persons speaking register and which direction they perceived the tones to be going in. This also effected how they divided the pitch class circle; where half the notes are always perceived as higher, and the other half as lower.
I think this is an interesting example of how speech and music perception are interrelated. This is significant on a cultural level as well, since the way we speak (using influx for example) is influenced by our environment. Try listening to this phenomenon with a friend from another country and see if you hear it the same!
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Music theorist love to measure the spaces between two notes, called intervals. We have three and a half (I say three and a half because the octave and the unison aren’t very different) ‘perfect’ intervals: the octave (P8) (and P1, the perfect unison), the perfect fifth (P5), and the perfect fourth (P4). When you invert perfect intervals, the result is always another perfect intervals. So when you have a P5 from C to G, if you invert it you get a P4 from G to C. That’s what makes them so “perfect”. Other intervals don’t do that; if you have a minor third and invert it, you get a major 6th. An easy trick to remember your inversions is every time you invert it, the new interval will be the opposite of the original interval: majors become minor, diminished become augmented, etc.. Unless it is perfect, then it’s perfectly fine the way it is. The sum of both intervals is always nine. So 6+3 = 9, 4+5 = 9, 2+7 = 9 etc.
So I know you’re probably thinking, what does any of this have to do with tritones? Which you’ve completely avoided mentioning other than briefly before your little music theory spiel? Patience, young Padawan. I’m getting there.
Tritones consist of three whole tones, and it always falls on the note between the P5 and P4. What I mean by that is if you’re in C major, the P4 is C-F, the P5 is C-G. The tritone is C-F#, the note in between these two intervals There are 12 semitones in an octave, and the tritone is special because it splits the octave in half. If you invert a tritone, the result is always another tritone (C-F#, F#-C). A tritone is either an augmented fourth (X4) or diminished fifth (o5), depending on who you are, some argue that one is more correct than the other. I am not one of these people.
What you will notice tho is that 4+5 is 9, and the opposite of diminished is augmented, so no matter how you look at it, they’re both actually the exactsame thing. How cool is that? So, so cool.
This is also interesting when we consider the tritones relationship to closely related keys of the first note (C), but I digress for now. All you need to know is that tonal music consists of a lot of interesting mathematical relationships, and that music theory is basically the coolest kind of math you could ever invest your time in. Just saying.
Deutsch, D. (1991). The tritone paradox: An influence of language on music perception. Music Perception: An Interdisciplinary Journal, 8(4), 335-347.
Deutsch, D., North, T., & Ray, L. (1990). The tritone paradox: Correlate with the listener’s vocal range for speech. Music Perception: An Interdisciplinary Journal, 7(4), 371-384.
Deutsch, D. (1987). The tritone paradox: Effects of spectral variables. Attention, Perception, & Psychophysics, 41(6), 563-575.
Schellenberg, E. G. (1997). Simplifying the implication-realization model of melodic expectancy. Music Perception: An Interdisciplinary Journal, 14(3), 295-318.