Ok, it's a trick question. The answer is it depends on which octaves the notes are in, and specifically which note is higher. That's the concept of interval inversions that I will talk about here. I suggest reviewing my lessons on The Musical Alphabet and Intervals.

For reference, here is the table of interval names.

A good first guess answer to the above question would be minor third, since there are three half-steps moving upward from A to C (from A to A#, from A# to B, and from B to C).

But what if we take the A and raise it up an octave. Then we're counting half-steps starting from C instead of from A. Like so:

1. C to C#
2. C# to D
3. D to D#
4. D# to E
5. E to F
6. F to F#
7. F# to G
8. G to G#
9. G# to A

We can see that is nine half-steps. Or, if we're clever (which we are), we can remember that an octave is 12 half-steps, so just subtract three from 12, and get nine that way. Either way, we see that the inversion of a minor third is a major sixth.

This process is called inverting an interval. That is, it's when you take the lower note in an interval and increase it by an octave.

One important thing to notice is that if you invert an interval twice, you wind up with the same interval you started with. Using the above example, this becomes obvious. Start with the minor 3rd from A to C. First, raise the A an octave, giving a major 6th interval from C to the A above. Then raise the C an octave, and we're back to the minor 3rd from A to C.

So what?

When dealing with harmonies, an interval and its inversion are like cousins, really close-knit cousins. Because they are really built from the same notes, but moved to different octaves, one can be substituted for each other. In isolation the we're looking at them now, that's not such a big deal, but use them to build chords, and it becomes a hugiferously important tool for making interesting harmonies. Next lesson, I'll cover chord inversions, and you'll see more clearly what I mean.

Special case: the tritone

If you're good at math, you may notice that 12 - 6 = 6. Interesting, eh? That means that the interval of six half-steps is it's own inversion. Checking the table, this is the awkwardly named diminished fifth / augmented fourth (those are actually two alternate names). It also has a third name, the tritone. It's exactly half an octave, and actually not a particularly pleasant sounding interval. So much for cosmic balance.

Reference

Here is a listing of intervals and their inversions.

The random part

Turns out my instincts were correct, to a certain extent: our system of music is indeed arbitrary, but there is also some hidden magic. If you traveled back in time and listened to an orchestra from 400 years ago, it would sound flat, but the tunes would still be familiar. That's because it's not the notes themselves that are special, but the relationship between them.

Not too long ago (relatively), the International Organization for Standardization decided that an audio tone at a frequency of 440Hz would be called A. That's the A above the C in the center of a piano keyboard, and from this frequency we get all the rest. The choice of 440 had no basis in cosmic laws or the circumference of the Earth or anything like that. In fact, in previous times different countries used different frequencies, which is why a standard had to be set. And 440 was as good as any other frequency. So there's the start.

The magic part

Now where do we get the rest of the notes? That's where Pythagoras comes in.

Pythagoras was a fascinating character. He's probably most known for his theorem about the proportions of the sides of a right triangle, but in his day he was prominent in the philosophical, religious, and musical areas as well. He founded a group, first in Greece, then in the south of Italy, a group that was somewhere between a religious cult, university study group, and a garage band. He and his students, and after he died his followers, came up with many significant ideas in mathematics and astronomy, as well as some kooky religious ideas. But what we are concerned with here are their musical contributions.

First the word frequency, what does it mean, anyway? It's just how often something happens, right? As in, "I drink with a frequency of five beers per day." When we're talking about music, it refers to how fast a string or a column of air or a drum head vibrates. A guitar string playing that middle A moves up and down 440 times per second. 440 Hertz (named after a guy named Mr. Hertz), written as 440Hz. Sounds pretty fast, and it is, to look at, but the human ear can hear frequencies up to around 20,000Hz. Try that with beer.

Ok, next thing is the idea of harmonic frequencies. That's where the real magic comes in. Turns out, when you pluck a string that's tuned to 440Hz, that's not the only frequency of sound that comes out. You get the 440Hz, but you also get a slightly quieter 880Hz tone (which is 440 times two), as well as another one, even quieter, at 1320Hz (440 times three), and so on. These "extra" notes are called harmonics, and they occur everywhere in nature, when you play an instrument, when you bang two sticks together, when birds sing. They are natural, they are cosmic, and we hear them so much we figure they must sound nice. (Pythagoras loved them so much he had a theory that the planets "sang" in harmonic frequencies as they orbited around the Earth, but we'll ignore that kookiness.)

Here are the harmonic frequencies produced when plucking an A. (Don't worry, you don't have to memorize them. I'll tell you why they're important in a second.)

The next thing Pythagoras discovered is that if you cut the string in half the frequency doubles, from 440 to 880. This is the same thing you're doing when you hold down a guitar string at the 12 fret. The new harmonic frequencies are:

 Now look back at the shaded columns in the first chart. Notice something? The numbers in the second chart are the same as every-other column in the first chart! That really is magic. There must be something special about the relationship between these two notes. In fact, they sound so similar we give them the same name: A. That's right, they're both As. The second one is an octave higher than the first. So now we know why the scale is as long as it is: it's just a doubling of the beginning frequency. Now we can work on filling in the middle notes, the ones between A at 440Hz and A at 880Hz.

Try and stay awake for this part

What do we know so far? Harmonic frequencies sound good together, and doubling a frequency gives a similar sounding note. Ok, well cutting something in half is the same as doubling, but backwards, so let's try this: take the number from column three of table 1, the 1320Hz, which we know sounds good with the A, and cut it in half, so it fits between 440 and 880. We get 660Hz, another nice looking number, if I do say so myself. So we tune up our little test strings, one to 440Hz, another to 660Hz, and we pluck them together. What do you know, they sound sweet! Let's give it a name. I'm gonna go with E, just because. Now we have three notes in our scale: A, E, and another A.

Well, if we can do that with the third harmonic, we can do it with the fifth, which is 2200Hz. Dividing it in two gives us 1100Hz, which is still too high, so divide it in two again, giving us 550Hz. I like that number, too. Let's call it D. (Don't worry about where these names come from, for now. They're basically arbitrary.) Now we've got A, D, E, and another A. We're on our way to scale-hood!

Do you see what we're doing here? We're taking the harmonics of the starting frequency and dividing by two until they fit into the octave. We can do the same thing starting on D, or E, or any of the new frequencies we come up with. I'm doing this with the help of a calculator and modern technology which allows us to measure how many times a piece of string vibrates in a second. (Let's pause for a second to consider just how cool that is... Ok, back on topic.) Pythagoras and crew did the same thing by cutting and measuring strings. I think he wins.

I could keep going with the math, but that would be boring and tedious. Just trust me that if we were to keep going we'd end up with something pretty close to the notes of the major scale we still use today, and if you start in different places, you get pretty close to all twelve notes we use today. "Hold on a second, Eddy!" I hear you saying. "What do you mean, 'pretty close?' If these ratios are magical, why aren't they exactly what we use?" Ok, you caught me. That's where even tempered tuning comes in.

Even tempered tuning

Turns out the numbers don't work out exactly as perfectly as the Pythagoreans thought. The problem is, we end up with slightly different numbers when we start on, for example, an E, instead of an A. That wouldn't be a big deal if you and all the members of your band could agree to always play every song in the same key. But then if one day you got sick of A and wanted to try a song in the key of E, you'd have to retune all your instruments slightly. That's no fun, so somebody came up with "even tempered tuning."

When Pythagoras followed the above exercise, he came up with twelve notes all together. He could have stopped earlier or kept going to get a different number of notes in the scale, like they have in India or Africa, but he didn't. Twelve is what he came up with, and that's what we still use here in what we for some reason call the "western world." Anyway, as I said the notes were not quite evenly spaced. Maybe that's why the planets don't actually sing. We can make them evenly spaced, but that messes with Pythagoras's beautiful ratios, and the planets, and yada-yada-yada, so it took thousands of years for someone to get over the fear of being struck by lightening and actually space them evenly. They came up with a ratio of around 1.059 between each note, which when multiplied by itself 12 times gives you 2, which is the octave we're looking for. Some dude named Bach wrote a piece of music called The Well-tempered Clavichord using this new-fangled tuning technique, and the rest, as they say, is rock-and-roll history. It caught on, and we've been using even tempered tuning ever since.

Rock on

So next time you find yourself thinking, "Hmm, I wonder why the scale has seven notes," you'll know you have a Greek named Pythagoras and his cult to thank. Maybe that won't turn your guitar solos into blistering works of pure genius, but it's a step towards understanding more about why music is the way it is. I hope you find the topic as interesting as I do, and my explanations made sense. There is a lot more detail to be discovered, so if you're still interested check out one of the references in the list that follows.

• Daniel Levitin: This is Your Brain on Music
• Trudi Hammel Garland and Charity Vaughan Kahn: Math and Music: Harmonious Connections
• Math and Musical Scales

Category: Music theory

Level: Beginner

In the last lesson, we talked about the musical alphabet, and how notes are named. We discussed that the distance between two adjacent notes is called a half-step, and that two half steps is called a whole-step. Very exciting, I'm sure.

The generic term for the distance between two notes - any two notes - is an interval. Turns out there are names for lots of intervals, not just the half-step and whole-step. And some of these intervals have more than one name. Remember how I said that much of music theory is giving fancy names to things you may already understand? Well, here is a perfect example. Intervals are something we get instinctively. It's when they get names like "diminished seventh" that people run for the hills. Don't. In this lesson, you can listen to them, hear them in context, and see where they are used in real life, and not just read about them abstractly.

The names are just names. You're probably not going to be sitting around with your friends talking about Eric Clapton's startling use of the augmented fifth, but it is amazingly useful to be able to recognize intervals by ear. This helps when learning a song, composing a song, soloing, and so on. The intervals have to be called something, so these names are as good as any other.

Here's a list of the first twelve intervals, along with some real life examples of their use. You can also click on the symbol to hear each interval. Bookmark this page so you can refer back to this table.

Number of half-steps	1
Names	half-step, semitone, minor second
Real life example	Joy to the World
Listen

Number of half-steps	2
Names	whole-step, major second
Real life example	Mary Had a Little Lamb
Listen

Number of half-steps	3
Names	minor third
Real life example	Greensleeves
Listen

Number of half-steps	4
Names	major third
Real life example	Blister in the Sun, Beethoven's Fifth Symphony
Listen

Number of half-steps	5
Names	perfect fourth
Real life example	Amazing Grace
Listen

Number of half-steps	6
Names	augmented fourth, diminished fifth
Real life example	Maria from Westside Story, The Simpsons theme
Listen

Number of half-steps	7
Names	perfect fifth
Real life example	Chariots of Fire theme
Listen

Number of half-steps	8
Names	augmented fifth, minor sixth
Real life example	Third and fourth notes of The Entertainer
Listen

Number of half-steps	9
Names	major sixth, diminished seventh
Real life example	NBC chimes
Listen

Number of half-steps	10
Names	minor seventh
Real life example	West Side Story: "Somewhere." Specific lyrics: There's a Place for Us
Listen

Number of half-steps	11
Names	major seventh
Real life example	(This is a really ugly interval, and doesn't show up very often.)
Listen

Number of half-steps	12
Names	octave
Real life example	Somewhere Over the Rainbow
Listen

Here's a little quiz to test how well you've learned the definitions. Feel free to refer back to the above list and to the list of note names in the Musical Alphabet lesson.

How did you do? Once you think you have a grasp of the names, you can try your ear at the Ear Trainer (audio interval quiz) on this site. Start nice and easy, with just a few intervals included in the test, then slowly add some more until you get good at naming them. It's a fun little game, and will improve dramatically your ability to recognize and reproduce melodies.

Interesting facts

• The tritone interval (six half-steps) was considered evil and was banned from use in churches during the Middle Ages.
• The Greek mathematician/philosopher/cult leader Pythagorus considered the intervals of a perfect fourth and a perfect fifth to be sacred and cosmic. That's why they're called "perfect." He believed the celestial planets actually "sang" in perfect intervals as they orbited the Earth.
• The perfect fourth shows up in a lot of Christmas songs and religious hymns. Coincidence?
• Intervals greater than an octave have names, too. Can you guess what a major ninth would be?

Level: Beginner

This article presents an introduction to the way musical notes are named and referred to. If you already know even a little music theory, this may seem pretty basic for you, so you may want to skip ahead or come back for more advanced lessons. If you're a "play by ear" musician, you may not see the need to learn any theory at all. In many ways, music theory is giving names and complicated explanations to things we already know instinctively. Still, the next time you're jamming and someone asks you to play an A sharp, it would be good to understand what that means. In addition, this lesson provides a base for later, more interesting, lessons.

Even those who have never studied music or an instrument can appreciate that music is made up of different notes, different pitches. The musical "alphabet" has twelve notes. All of the music that we care about can be spelled using these twelve notes. There are twelve notes on a guitar fretboard (and a piano keyboard, and a flute). Twelve unique notes, that is, that repeat themselves at higher and lower pitches. This distance of twelve notes is called an octave. I'll discuss in a future lesson why it's twelve and not some other number, but for now, just remember that there are twelve notes in the musical alphabet.

Now here's where things get kooky. Because of history, we're stuck with a pretty illogical method of naming these twelve notes. Probably one of the reasons so many people get bored with music theory is because of the silly naming conventions that we have to deal with. Trust me, though, that you get used to it.

If logic prevailed, the notes be named 1 through 12, or A through L, or even I through XII, but they're not; they're named A through G. That presents a slight problem, since A through G only gives seven letters. Some of the notes need to be named using two letters. Well, actually a letter plus another symbol: either a flat symbol, which kinda-sorta looks like a lower-case letter b, or a sharp symbol, which looks pretty much like a pound sign, #. So, for example, A# is pronounced "A sharp," and Bb is pronounced "B flat."

Here's how the notes are named, from low to high starting from A, and using sharp symbols:

Then it starts over at A an octave higher. Here are the exact same notes, named using flat symbols:

There are a couple things worth noticing there. The first thing is that five of the notes are named twice. For example, A# is the same note as Bb. That's because a sharp symbol just means to go up one, and a flat symbol means to go down one, so they end up in the same place. Every sharped note can also be named as a flatted note.

The next thing to notice is that not every pair of letters has a sharp/flat note in between. There is nothing between B and C, and there is nothing between E and F. It's a silly way to arrange things, but when we study scales it will make (slightly) more sense. There is nothing magical about the choice of the letters A through G and the sharp and flat symbols to represent these twelve notes, or about which notes get decorated with a b or a #, and in fact at different times in history and in different cultures, different systems have applied. It's good to learn this system, though, since it's the one you're most likely to encounter.

So what do you suppose the distance between two notes is called? Once again, logic seems to be lacking. In a logical system, maybe it would be called a step. Nope, it's a "half-step." (Also sometimes "semitone.") Yeah, there is a reason for this silliness, but it's a historical thing and really not very interesting at this point. Just accept it: two adjacent notes are separated by half-step. Two half-steps make a whole-step, which actually makes sense, surprisingly. So there is a half-step between A and A#, and there is a whole-step between A and B. And, as discussed above, a half-step between B and C and between E and F.

So that's the musical alphabet. Yes, this has been a pretty basic lesson, but stick around, as this is just a foundation on which we'll build more of what you need to know. Next up: intervals and chords!