Previously, we said that the notes written on the staff were approximations of the pitches of the harmonics. But what kind of sense does that make?

## Intervals and ratios

Let’s look at the picture we had indicating harmonics, their frequency ratios and the equivalent notes on the staff:

This picture is claiming that when you play note 1 (a.k.a. C2) on the piano, the strings for that note vibrate at some frequency (as it happens, roughly 65.4 vibrations per second), and when you play note 2 (C3) on the piano, the strings vibrate twice as fast (about 130.8 times per second). Similarly, note 3 is three times as fast as note 1, and so on.

I’m going to assert here that regarding octaves, there is no approximation going on. The vibration frequency of note 2 (C3) really is exactly twice that of note 1 (C2). Similarly the frequency of note 4 (C4) is twice that of note 2, and the frequency of note 8 (C5) is twice that of note 4. In addition to the Cs (notes 1, 2, 4, and 8) the picture shows another example of an octave: note 6 (G4) has exactly twice the vibration frequency of note 3 (G3).

## Ratios and intervals

Since an octave corresponds to a frequency ratio of 2-to-1, one would start to suspect that any musical interval between two notes has a certain frequency ratio. For example, there would be some ratio corresponding to the interval of a major third (it wouldn’t be an integer multiplier, but rather one plus some fraction). So if you start with one note, multiply its frequency by some number, and make that result the frequency of your second note, the second note would sound a major third higher than the first note.

When we discussed intervals and their full names, we noted that every interval has a certain number of half steps. And long ago we noted that each successive white or black key on the piano keyboard is one half step higher than the one before, as you move to the right.

OK, bear with me here. If every interval has a corresponding frequency ratio, there must be some frequency ratio corresponding to the interval of a half step. Suppose I start with the note C2. I multiply its frequency by some mystical number (which we’ll call X for now) and that gives me the frequency of a note that is one half step higher, which would be C#2. Multiplying that new frequency by X again gives me the frequency of a note that is yet another half step higher, which would be D2. By multiplying by X some number of times, I can compute the frequency of a note that is some number of half steps higher than the original.

It turns out we can compute what X must be! Since there are twelve half steps in an octave, and going up an octave doubles the frequency of a note, then X must be something that when you multiply something by it twelve times, the thing doubles. I’ll just tell you that X is the twelfth root of 2, or in a plain old decimal fraction, about 1.059463. For those who like percentages: when you multiply the frequency by X you are increasing it by 5.9463 percent, or roughly 6 percent.

## Two methods for computing intervals

Now in the discussion of harmonics, we said that the interval of an octave plus a fifth was what you get when you multiply the frequency by 3. In the harmonics picture you can see that we went up an octave (as shown by note 2) by doubling the frequency, so the ratio of the frequencies of note 3 to note 2 is 3-to-2. This is to say that note 3’s frequency is 1 and a half times as much as note 2. So by the harmonics picture, it looks like in order to raise a pitch by a perfect fifth, you’d multiply its frequency by 1.5.

But also we have this magic multiplier X (roughly 1.06) that increases the frequency by a half step, and we know that a perfect fifth is seven half steps in size. So if I multiply a note’s frequency by X seven times, I should move the pitch up by a perfect fifth. If we’re lucky, the harmonics method (multiplying by 1.5) would give the same result as multiplying by X seven times. Now I’ve already done the math, so let’s see what we get:

- Multiplier to go up by a perfect fifth according to harmonics: 1.5
- Multiplier from multiplying by X seven times: 1.4983

Uh-oh, it doesn’t match! It is close, but not the same. Let’s try this with some other intervals. On the picture of harmonics we can pick out ratios for a perfect fourth, which is the interval from note 3 up to note 4, and this ought to be five half steps. Also, we can identify a major third between note 4 and note 5, and a minor third between note 5 and note 6. I’m sure you’re ahead of me here; you already know they’re not going to match up. Let’s see what happens:

Interval | Ratio | Harm. mult. | Half steps | Mult. from X | Error |

Perfect fifth | 3 : 2 | 1.5 | 7 | 1.4983 | 0.113% |

Perfect fourth | 4 : 3 | 1.3333 | 5 | 1.3348 | 0.113% |

Major third | 5 : 4 | 1.25 | 4 | 1.2599 | 0.794% |

Minor third | 6 : 5 | 1.1666 | 3 | 1.1892 | 0.899% |

So the intervals going up by X’s are not the same as the intervals in the harmonic series, and that was the reason for the caveat in the previous discussion that the notes on the staff only approximated the pitches in the harmonic series. In our modern system of tuning, we use the X-method for half steps. Each successive white or black key on the piano keyboard (assuming the piano is in tune) actually does produce a note with X times the frequency of the one before.

## Is this wrong?

In the discussion of harmonics we hinted at the possibility of a relationship between the harmonic series and the way things sound naturally to the human ear. This being the case, you would imagine that the intervals in the harmonic series would sound “right” to the ear, and you would be absolutely correct.

But wait, wouldn’t the intervals according to X sound wrong, then? OK, so the fifths and fourths are only off by about a tenth of a percent, but the thirds are off by almost a whole percent. And a half step is about six percent, and the pitch difference in two notes a half step apart is very obvious. Wouldn’t you hear a difference of almost one percent in frequency? And again, you would be absolutely correct, you can hear this difference in frequency. It’s not a large difference, but you can hear it.

But wait, hold it, what’s up here? Which one is right? The harmonic series, or X? The answer is that our modern tuning system uses the principle of X, and these intervals do actually sound slightly “wrong” in our tuning system.

We sort of mathematically showed that using the equal half steps of the X principle, you can go around the circle of fifths and end up back where you started.

You can imagine that if you tried to use the perfect fifth defined by the harmonic ratio 3-to-2 to do this, you would not quite end up where you started. To do this you’d keep multiplying the frequency by 1.5. True enough; if you started at the C at the top of the circle and stepped all the way around multiplying by 1.5, after twelve steps you’d end up with a frequency about 1.36 percent higher than a true C.

In fact, in the old days, people knew they liked the sound of the harmonic series intervals, and they tried to use those intervals when they tuned. But there was a trade-off here. The cost of making many intervals sound really good was that other intervals sounded really bad. If you had a piano tuned this way, songs in some keys would sound very nicely in tune, and songs played in other keys would sound out of tune. And this is really what happened!

In time, however, the system of tuning we use today won out, and essentially our intervals are out of tune with the harmonic series but because the badness is distributed evenly, none of the keys sound too terribly out of tune. (This is also because our ears are used to hearing this sort of tuning.) If you were to ask your piano tuner, he’d be well aware of this out-of-tune-ness. As he starts out by tuning the central octave or so of the piano, he’s actually listening for the right amount of “error” in each interval to make them all come out evenly.