Interval Training


That’s it, hold the F sharp! Six more beats, then eight beats of G, then a whole rest, then start again from the beginning! Breath is for the weak!

The problem with learning about cool things is that usually there’s way more than I can fit into one little blog post. Sometimes I just deal with it and move on, but sometimes I end up deciding I really need two (or more) posts to cover a topic. This happened to to me last week while I was researching the history of pitch.

Today I’m going to focus on two of the things I came across: octave equivalency and equal temperament. That sounds like a lot of equality, but I hope you’ll agree with equanimity that I’m not equivocating when I say that these two ideas are definitely not equivalent.

Octave Equivalency

The name pretty much gives this one away. Basically, it refers to the fact that we hear notes that are an octave apart as essentially similar, even though we can still distinguish between the different pitches in a series of octaves.

Mathematically, octaves are related by double or half frequencies—so a note at 200 Hz will have one note an octave lower at 100 Hz and another an octave higher at 400 Hz, then at 50 Hz and 800 Hz, and so on. This is why I said in my previous post that logarithms are a good way to think of octaves.

Take the example of scientific pitch, which isn’t commonly used but sets middle C to a frequency of 256 Hz. That means that every other C increases or decreases from 256 by powers of two, so we have:

C (Hz) = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192…

We can also think of this as a logarithm in base two, so you could describe which C you’re talking about by finding what power 2 is taken to in order to get its frequency. That would mean C0 = 1 Hz, C1 = 2 Hz, and so on up to C8 = 256 Hz.

Music Octave Waves.png

Forgive the drawing of sound waves as transverse waves. They’re longitudinal, but it’s just easier, and I’m lazy.

Standard concert pitch (A = 440 Hz) doesn’t have the handy powers-of-two frequencies for any particular note, but octaves are still defined by doubling the frequency each time. That means that notes in an octave series share harmonics, or the higher frequencies that can also come out of a particular string, pipe, or however you’re getting your note. The lowest frequency in the series has all the higher ones as harmonics, and the next highest one has all the ones above it, and so on. Each higher note has a little less in common with the ones below, but they’re all in the same series.

Music Octave Harmonics.png

Etcetera, etcetera.

Humans hear these notes as similar because of the common harmonics, through a process that’s regulated in the thalamus of the brain. (Other intervals like perfect fifths also sound “right” to us because of other harmonics, the ones that aren’t in the octave series.) Monkeys have a similar reaction, according to nonverbal tests scientists have developed to test octave equivalency.

However (and this is so cool), there’s some evidence that some birds don’t! They’re much better at recognizing different pitches than we are, since that’s how they send and receive signals with other birds, but they may not hear octaves the same way we do. (Research does show that whether or not they perceive octaves as such, some songbirds sing in “keys” or harmonic series, as we tend to do ourselves.)

The idea of birds not recognizing octaves seems odd, since musical pitch is so much more important to birds than to us. However (based partly on my limited understanding of neurology), I’d guess that if we do perceive octaves differently, that difference in importance must be the key. After all, if we know one thing about our own brains, it’s that we can only handle so much information at once. Finding or assigning patterns helps us stave off information overload by reducing the complexity of stuff to sort through.

With that in mind, it makes sense that we’d categorize pitches into octave groups. After all, I can’t imagine that there are many situations where we’d get much useful information from knowing that we just heard the A above middle C instead of the A below it. Songbirds, on the other hand, will get all kinds of new information based on pitch, so the harmonic similarities of two notes may be outweighed by the usefulness of recognizing a sharp distinction between them. For us it’s incidental, but for them it’s survival.

Equal Temperament

Another fun aspect of music theory is that there are different definitions of the intervals between notes. This is a dramatic battle that highlights the difference between equality…and justice.

Just kidding. That kind of debate is beyond the scale of this blog (sorry not sorry). We are going to look at the difference between equal temperament and just intonation, though!

Today, most instruments use equal temperament, which means that the octave is divided into equal parts (usually twelve in western music, though the same idea applies with other numbers). In twelve-tone equal temperament, these intervals are a half-step each and can be combined to make a succession of half and whole steps for major or minor modes. Here you can see an octave divided into twelve equal parts, with notes labeled starting with C (for your convenience) and the intervals of the major scale marked.

Music Equal Temp.png

This system is easy to visualize and has the great advantage of letting you transpose music easily to another key on the same instrument. Let’s see how that looks going up a whole step from a C major to a D major scale:

Music Equal Scales.png

Note that even though flats and sharps are the same note and both are shown here, a D scale has only sharps. Basically, it’s whatever arrangement makes sure there’s only one of each letter per scale.

Starting two half-steps up with D, the same pattern of whole and half steps uses some different frequencies (color-coded here) from the C scale. However, while the pitches change, the intervals stay the same.

Why would we want to use anything else? Well, according to Pythagoras (legend has it, at least) and certainly plenty of others since then, the human ear prefers intervals that are defined by ratios of small numbers. (For example, the first whole step in a major scale might be defined by 9/8, compared to the root note at 1 and the end of the octave at 2.)

This is called just intonation. There are multiple forms, like Pythagorean tuning and meantone intonation, but I’ll just illustrate one. Here’s how the intervals in an octave are spaced, again arbitrarily starting with C. You can C how unequal they are:

Music Just Temp.png

If these intervals sound better, why aren’t we using them? Well, the aesthetic advantage of just intonation is balanced by a pretty serious practical disadvantage. Let’s take a look. This is a C major scale compared to the equally-tempered C scale:

Music C Scales.png

You can see the differences, though the notes are very similar for the second, fourth, and fifth (D, F, and G). Now let’s look at a D major scale:

Music D Scales.png

Still really similar on the fourth (G), but visible differences otherwise. The real problem, though, is obvious when you compare both scales for just intonation:

Music Just Scales.png

Yikes! Those are clearly not the same scale. The fourth and sixth (F/G and A/B) seem fine, but nothing else. (For reference, those differences are on the order of ten musical “cents,” or about 1/120 of an octave. This doesn’t sound like much, but toward the higher end of our hearing range, a whole octave might be, say, 4800 Hz. That would mean 10 cents = 40 Hz, and the human ear can hear differences of only a few Hz. You can hear at least a subtle difference, more with a trained ear.)

This is the real disadvantage of just intonation: the starting note isn’t arbitary anymore. You can’t just transpose music up or down a step or so and play the same intervals like you can with equal temperament. Instead, if you’re playing a fixed-pitch instrument (pretty much anything except unfretted string instruments like the violin family) you have to tune the whole thing up or down if you want to play in a different key.

Understandably, this isn’t a step most musicians are willing to take. It’s much simpler to make instruments equally tempered, so switching keys is easy. (Certainly easier than stopping to retune the whole band for a key change!) Basically, the collective will of musicians has consigned us all to suffer under unjust tuning, which purists scoff at as a poor substitute for true harmony. They’re not just being snobby, either—remember, those ten cents can make a real difference!

Luckily, we can still experience just intonation with the most versatile instrument of all—the human voice, which can change pitch by whatever increment the singer wants. In fact, most a cappella groups end up using just intonation, as do many string ensembles that don’t have to worry about their inflexible fixed-pitch bandmates. It’s not going to revolutionize the music industry, but it’s not dying out anytime soon either.

You’ve probably guessed that (as usual) this is just the tip of the musical iceberg when it comes to pitch perception and harmonics. Look into it yourself and you’ll find plenty to satisfy your curiosity!

For more on just intonation, check out the Wikipedia page, which has audio samples to test your ear. For octave equivalency, check out a few discussions, like this one explaining some of the complicating factors and this timeline of octave equivalency research. Happy researching!


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