Why are tritones not consonant, confusion with the definition of a perfect fifth

A fifth is a ratio of 1.5, and 1.5 is the middle between 1 and 2.

A tritone is exactly in the center of my 12 notes.

Stacking two tritones leads to an octave.

Adding 1 + 0.5 + 0.5 leads me to the next octave as well.

So why are tritones and fifths not the same? I mean, obviously they are not, but what is wrong about my understanding of the above definitions?

EDIT: is it because 1.5 * 1.5 = 2.25, so its a little bit more than a tritone?

• Try comparing x*x with x+x and looking at a logarithm curve. Also try applying fifth two times, where do you end up?
– Emil
Jan 11 at 19:45
• Your understanding is wrong because it's based on the assumption that ratios add up in a linear way, but frequencies are not linear, and ratios are relative proportions, not absolute. The distance of each frequency increases logarithmically: as much as it doubles at each octave (440, 880, 1760), the difference between a semitone at the beginning of an octave and its end is not the same (it's almost double), meaning that the "middle" of an octave (the tritone) is not halfway between those two frequencies. Jan 11 at 20:07
• A fifth is not halfway. It's 7 half steps. A tritone is six half steps which is only available in the 4 through 7 of a diatonic major or 1 though 4 of the minor. Jan 12 at 5:19
• @musicamente - For clarification, your "nearly double" statement is true if we measure frequencies with hertz(/Hz) but not with cents (cents are similarly logarithmic). Jan 12 at 13:59

Since intervals are ratios, they are combined by multiplication, not addition. For example, an octave is 2:1 (i.e., 2.0), two octaves is 4:1, three octaves is 8:1, etc. So to combine three single octaves, the calculation is 2 * 2 * 2 rather than 2 + 2 + 2.

Similarly with perfect fifths, the calculation is 1.5 * 1.5 = 2.25 = 9/4, which is a ninth (as expected).

One way to relate fifths and octaves is to observe that an octave can be constructed by "adding" a perfect fifth and a perfect fourth, which is 3/2 * 4/3 = 12/6 = 2.

The half-way point of an octave is SQRT(2) ~= 1.414. This is consistent with expectation, since a tritone is smaller than a perfect fifth. And, of course, SQRT(2) * SQRT(2) = 2.

• Nitpick: isn't this answer comparing an equal temperament tritone with a just intonation perfect fifth? Jan 12 at 4:08
• @JosephSible-ReinstateMonica Strictly speaking, yes, but the point is to illustrate the math rather than get into the distinctions between the two tuning systems. Jan 12 at 4:10
• "To construct an octave, we use a perfect fifth plus a perfect fourth": that is one of several possibilities. We could use any interval and its inversion, which follows from the definition of inversion. Jan 13 at 5:19
• “To construct an octave” is a weird phrasing. It's normally taken as axiomatic that an octave is a double of frequency, and then the inversion of an interval with frequency ratio x (whether rational or irrational) is defined as having the frequency ratio 2/x. Jan 14 at 0:56

Expressions like "halfway" are rather confusing when speaking about intervals. As Aaron pointed out, value of 1.5 corresponds to frequency ratio, so it should be rather viewed in the logarithmic scale than linear.

A more intuitive unit are cents, which are additive, see: https://en.wikipedia.org/wiki/Cent_(music). A half-tone has 100 cents, whole tone has 200 cents, and so on. A perfect fifth, which is 7 half-tones has 700 cents, and an octave has 1200 cents. Half of 1200 cents is 600 cents, and that corresponds to tritone.

• Exactly! 1.5 is only arithmetically halfway between 1 and 2. Rather, geometrically it's the square root of two that lies halfway between 1 and 2, since that is the ratio factor that applied twice results in a doubling from 1 to 2. And guess which interval corresponds best to the square root of 2? +1 Jan 13 at 3:54

octaves and perfect fifths are perfect consonants. (The fifth is the first overtone that we can hear very strongly when playing a tone.)

Sixths and thirds are consonants too - but imperfect. (In the middle age aera they were also called dissonants - like the major and minor seventh and the seconds.)

The tritone must have come from the devil! ;) Why? As we hear the perfect fifth above the prime the tritone is only a semitone apart from the fifth, that's why we hear a dissonance between them: the tritone and the perfect fifth are colliding together like minor seconds are.

• It's also, famously, an irrational number, i.e. cannot be reduced to any fraction, as proved by an ancient Greek ... who was promptly murdered by angry Pythagoreans. A diabolical number indeed, both in music and mathematics! plus.maths.org/content/maths-minute-square-root-2-irrational Jan 12 at 16:37
• @user_1818839: The interval can be approximated by a rational number (7/5, 10/7, 11/8, 13/9, 16/11, 17/12, etc.), though. And in Equal Temperament, all intervals that aren't whole octaves are irrational. So it's not immediately obvious why A4 would be "worse" than other intervals. Jan 12 at 22:01
• @dan04 From a math perspective it is. "Nice" intervals have good rational approximations; sqrt(2) is about as bad as it can get (it has no good rational approximations). It's calculable, but a little technical - check out continued fractions if you're interested. Jan 13 at 3:32
• @user_1818839 the tritone is only an irrational number when you're using an irrational tuning system such as equal temperament, in which case the perfect fifth is an irrational number. The Pythagorean tritone is 729/512, which is rational. Jan 13 at 5:25
• @Spitemaster from the perspective of practical application, the irrational system of equal temperament is an approximation to the rational intervals, not the other way around. That is, 2^(7/12) is an approximation of 1.5. This is why equal temperament is often said to be "out of tune." Jan 13 at 5:28