# “Tritone” intervals in n-tone equal temperament

In 12-tone equal temperament, the tritone interval is made of three whole tones (six semitones) and is considered, if I am not mistaken totally, as the most dissonant interval.

I have two questions:

• Is there an explanation why tritone interval seems so dissonant in 12-tone equal temperament?
• Is there a generalization of this “tritone” dissonance in the n-tone equal temperament where n is a natural integer other than 12? How can we construct that? Is it unique?

One possible answer to the latter question is that “tritone” intervals can be defined as an half of an octave. Obviously, this only works when n is even. Is it possible to generalize the “tritone” quality even to scales with an odd number of equal tones?

• I'm pretty rusty with this area of theory. That being said I think some consider the minor second (a half step) the most dissonant harmonic interval. Jun 13 '15 at 16:03
• It's pretty dissonant in other temperaments as well - in Medieval times it was avoided in music - especially vocal.
– Tim
Jun 13 '15 at 16:53
• I think it would be very hard to back up your statement that the tritone is the single most dissonant interval. Who says so? The minor second and the major seventh are equally dissonant. Jun 13 '15 at 20:35
• @MattL. And the augmented octave even more so. Jun 15 '15 at 8:53
• Semantically, "tritone" is three whole tones. Would you have a definition of "tone" in a n-temperament scale? Then take three of these. If you are looking for the most dissonent (it's arguable if the tritone is the most dissonant interval in 12-temperament. There is a tritone in a dominant 7 chord), then find the "weidest" frequency ratio. Jun 15 '15 at 8:59

As some of the other answers have eluded to, there are two basic problems with your question:

• The first is the question of how you generalize a "tritone" in a non-12-TET based system. One possibility is to interpret it literally as three whole tones (which then begs the question as to how you define a whole tone in a non 12-tone system). Another possibility would be to define it as a half an octave, which still leaves tritones undefined for odd n. Because tritones are inherently defined around a diatonic system consisting of whole tones and half tones, it is not clear what the proper generalization to other systems should be. For example, perhaps with an odd n, you could define the two intervals on either side of n/2 as a minor tritone and a major tritone (bothe of which would likely be dissonant). But to my knowledge, this isn't standard terminology.

• The second issue is with your assertion that the tritone is the most dissonant. This is not necessarily true, as we'll see below, and depends on how you define dissonance.

It's not really worth trying to argue around the semantics of an undefined tritone (besides, what about other dissonant intervals, or intervals in non-ET systems?). As a result, I will generalize your question to the following:

Is there an explanation why any interval seems so dissonant in any tuning system?

When formulated in this manner, the question merges into the following question which already has an excellent answer: Is there a way to measure the consonance or dissonance of a chord? (although the title says "chord", much of the answer is concerned with the consonance/dissonance of intervals).

To summarize the answer from that page, two pure frequencies will produce a "roughness" (caused by the interference of the two sound waves) which gradually increases as the frequencies get closer together, and is maximal somewhere around a minor 2nd. Frequencies much closer than that start to be perceived as out of tune versions of the same frequency, and the perceived roughness rapidly decreases until it reaches zero when they are in unison.

However, most instruments do not produce a single pure frequency, but create a harmonic series of frequencies. In order to compute the dissonance of two notes, you have to sum all the "roughness" between all the pairs of frequencies in the two instruments spectra. In this approach, the tritone is primarily dissonant because it is a m2 away from the second overtone of the fundamental (a P5), and inversely, it's second overtone (a P5) is a m2 away from the fundamental.

When you plot this out for all frequencies, you get what is called a Plomp-Levelt curve which looks like the following, though it will probably differ depending on how you weight the formulas, and what the individual spectra of the instruments look like. You'll notice in this graph that, while the tritone isn't the most dissonant interval (at least according to this mathematical model of dissonance) it is the most dissonant interval between a m3 and a m7. Furthermore, the local maximum is actually pitched slightly higher than a tritone (somewhere between a tritone and a fifth). You'll also note that the axis across the top is equally divided into 12 parts from unison to octave. For any other system, you would merely divide the axis into a different number of subdivisions. Or, for other non-equal tuning systems, you would just find the corresponding subdivisions.

Because the above curve that models the degree of consonance for various intervals is determined, in part, by the location of an instrument's overtones, this suggests that instruments with inharmonic overtones (such as xylophones, bells, or synthesizers) might be able produce curves with consonances at different locations, including locations more suited for alternate tuning systems. Sethares has performed some research in this area, and in the linked article, provides a series of timbres theoretically suitable for 10-TET -- including a consonant tritone! Unfortunately, I don't see an audio sample provided.

You ask specifically about a uniquely "most dissonant" note, but such a thing does not exist, either at the tritone, or at any other interval. Such a question assumes the existence of a single, objective, quantitative scale by which dissonance can be measured. There is no such scale. There are many factors that come into play: acoustics/physics, biomechanics, neurophysics, and -- importantly -- ethnocultural experiences. The Plomp-Levelt curve provided above is merely one model that attempts to explain the phenomenon of dissonance, which depends on specific instruments' spectra, and it's inharmonicity, as well as making assumptions about how the ear rounds nearby pitches. As shown above, these parameters can be tweaked to produce different curves. As an example of a different model, here is one based on neural synchronization, which, according to the abstract,

"explore[s] the theory of synchronization properties of ensembles of coupled neural oscillators to demonstrate why simple frequency ratios may have achieved a special status and why they are important for auditory perception."

Since there is no single objective quantitative definition of dissonance, there cannot be an objectively "most dissonant" interval.

1. Two other importantnotes in a given key (which are the basis for main chords) are the 4th and 5th notes of a major scale, being 5 and 7 semitones from the root respectively. Giving the subdominant and the dominant. That puts a tritone right between them, neither one or the other, but too close to sound consonant. Yes, mathematically it could be construed that it ought to sound good, but musically it doesn't - unless one has acquired the taste, so to speak.

2. Don't know!!

• No, mathematically there is no reason to think it would sound consonant. Assuming equal temperament these are both approximations, but the perfect 4th is a frequency ration of 3:4, and the perfect 5th is 2:3, which are both round fractions. Exactly half an octave is a frequency ratio of 1: root(2), which is a horrible irrational number, so no wonder it sounds nasty. (Check: 4th plus 5th = 4/3 * 3/2 = 2; root(2)*root(2) = 2.) Jun 13 '15 at 16:50
• @BrianChandler - so, horrible fraction = nasty sound?!
– Tim
Jun 13 '15 at 16:56
• @Tim: yes, you could indeed say so. Jun 13 '15 at 23:25
• @Tim it's not that simple. Dominant 7th chords contain tritones. Do they sound "nasty" to you? They don't to me.
– user19146
Jun 16 '15 at 13:52
• @BrianChandler or maybe the irrational square root of two sounds okay because it approximates the ratio 25:18 or 45:32. Do you think semitones are dissonant? Does the answer depend on whether the ratio is 25:24 or 16:15 or the twelfth root of two? Nov 19 '18 at 4:34

To answer 2: yes, for any n, if n is even, then half an octave is n/2 divisions. (This is rather boringly obvious, so perhaps I don't understand the question.) I don't understand what you mean by "unique".

• Why half an octave is always the most dissonant interval for all `n`? Your property is asserted without clues. Jun 13 '15 at 17:18
• By ``uniqueness'', I mean the existence of a single most dissonant interval in the `n` equal temperament for all `n`, like in the case `n = 12`. From what I know, there is no reason for the fact there cannot be two different most dissonant intervals for some values of `n`. Jun 13 '15 at 17:30
• In most edo-tunings, almost all intervals sound horrible. The tritone usually won't stand out much... For instance in 16-edo, diminished seventh chords are pretty much the only proper chords you have, so they take on an almost consonant role. Jun 13 '15 at 23:27
• I did not mean to claim that half an octave is "most dissonant"; I do not think there is really a way to determine exact rankings of dissonance. I only point out that half an octave means a frequency ratio of 1: root(2), which is irrational, and therefore dissonant because it cannot be a ratio of integers. The number of divisions makes no difference as long as n is even. Jun 14 '15 at 5:38
• I don't see the point of this question: assuming that "tritone" means "half octave", then obviously in n-equal-temperament where n is odd, n/2 is not an integer, so there is no exact half octave. The closest approximation is (n+1)/2 or (n-1)/2, each of which is an equally bad approximation; and in fact the half octave is maximally difficult to approximate. Jun 14 '15 at 12:09

Whether it is dissonant or not is a matter of cultural practice. However, splitting an octave in half (harmonically) means that the product of the ratio of these intervals is Sqrt(2). It has been known (at least from the time of Pythagoras) that this number is irrational; it cannot be written as a ratio of whole numbers. Thus, any "just" set of intervals cannot include this division (nor can any finite combination thereof). For example, the "large" just whole tone has a ratio of 9/8 (and a "small" whole tone of 10/9; just intonation cannot even create a single whole tone. Combining three of these gives one of the following (9/8)^3=729/512, (9/8)^2*10/9=45/32, 9/8*(10/9)^2=25/18, and (10/9)^3=1000/729. None of these exactly equal the Sqrt(2). From the theory of continued fractions, the best approximations (those which need a bigger denominator to be better) are 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169... None of these can be made easily from other intervals.

This does reflect musical notation in that F# and Gb are different notes and reflects the normal usage where diminished fifths shrink and augmented fourths expand. B-F resolves to C-E while F-B resolves to E-C.

Generally, one may have to adjust notes on-the-fly (as in which of the two wholetones should be used) to keep close to just intervals.

• The discussion of just ratios isn't relevant. The question is about equal temperament with an arbitrary number of divisions to the octave. That F♯ and G♭ are different pitches is irrelevant, and also, for 12-tone equal temperament, it is not correct. Feb 12 '21 at 3:59