## Tuning as an approximation

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:

Notes with frequencies multiplied by various numbers

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.

A clock with the circle of fifths

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.

## Overtones and harmonics

We have discussed the circle of fifths as a way in which the various pitches, major keys, and minor keys can all be arranged in a circle. But certainly there must be something else special about the interval of a perfect fifth to make it a worthy basis for all of this discussion. Sure, the perfect fifth (or its inverted mirror image, the perfect fourth) is the only interval other than the boring half step that you can make a circle out of. But why? What would cause someone to decide to do this? Was the circle of fifths invented by a mathematician-turned-musician-turned-clockmaker whose business was slow and needed some way to pass the time?

Well, there is a sense in which keys (and chords, as it happens) that are near each other in the circle of fifths are closely related. But why? We can begin to guess why the human ear hears such things (or at least ears that are accustomed to Western music) by examining the way that things vibrate.

## The vibration of things

If you had a thick rope, or perhaps a long old-fashioned telephone handset cord (its little helical circular squiggles make it sort of rope-like) you could hold one end still (this was easy in the old days when the telephone was generally attached to the wall) and wiggle the other end. If you wiggle it at a certain speed, the whole rope or cord will vibrate up and down together. It’s as if the rope/cord has a natural “resonant” speed that it wants to vibrate at.

Helical telephone handset cord

It turns out that the cord wants to vibrate at other speeds as well, in different ways. If you wiggle the end twice as fast as the original resonant speed, it will vibrate in halves. The middle of the cord will stay still, and the two halves will vibrate in opposite directions. You can wiggle the end three times as fast as earlier and get the cord to vibrate in thirds as well. (This was a convenient way to pass the time when stuck in a monotonous telephone conversation.)

In the picture below, imagine the string vibrating between the red and blue positions in the three styles of vibration shown: all at once, in halves, and in thirds.

A string vibrating all at once, in halves, and in thirds

Now, it happens that if a shorter string is vibrating (or a smaller thing of any sort), all other things being equal, it will tend to vibrate faster and produce a higher note. (This is the reason for the shape of grand pianos and harps; the strings for the higher notes get gradually shorter.) So when the end of the cord is wiggled twice as fast and the string vibrates in halves, each piece of the string that vibrates is half as long, so this follows the above rule. In fact, things generally “like” to vibrate at some basic speed, or frequency (corresponding to the whole-string vibration) and also at even multiples of that frequency (corresponding to the faster vibration in halves, thirds, quarters, etc.).

## Ratios and hearing intervals

It so happens that we hear a certain ratio of frequencies as a certain interval. What does that mean? Well, for example, if you hear something vibrating at a certain speed, and then something else vibrating at twice that speed, the second thing will sound one octave higher than the first. This means that if you sat at the piano and hit all the Cs, starting with C1, and then C2, and then C3, and so on up to the highest note on the piano, C8, each note you play has its strings vibrating twice as fast as the strings did for the previous note.

You may wonder what sorts of intervals you may hear with the other ratios of frequencies. Starting with a general disclaimer that in the usual Western style of tuning used nowadays, the pitches may not be precisely in tune with those written here, the picture below shows you the answer (up to multiplying the frequency by 8):

Notes with frequencies multiplied by various numbers

As expected, doubling the frequency gives an interval of an octave. But when you triple the frequency, the interval from the original bottom note is an octave plus our old friend, the perfect fifth! Let’s keep going: quadrupling gives you a note two octaves higher (it would have to, since quadrupling is equivalent to doubling something twice). But quintupling gives us a note two octaves higher plus a major third. Multiplying by six goes up two octaves plus a perfect fifth (same as doubling and then tripling). Multiplying by seven goes up two octaves plus a minor seventh (it happens, though, that it’s a rather out-of-tune minor seventh).

## Points of interest

First, it happens that the notes C-E-G are the notes in a C-major chord, and those are the first notes that you see in the picture (although they appear in the order C-G-E, or, if you count duplicates, C-C-G-C-E).

Second, there is something very basic about the fifth for it to appear so early in this series of notes. If you were to look at pieces in the key of C, by far the most common final notes in the bass part are G-C. The last bass note is almost always C, and the chord is almost always C-major, because then you’re “home” in your comfortable resting place for the key of C. The penultimate bass note is usually G because that note suggests being “almost home”; that G wants to go down to a C, and it’s satisfying to the ear when it does so.

Third, it’s common to space the notes in a chord very much like in the series of notes, with bigger intervals near the bottom and close-together notes at the top. If you asked some piano player out of the blue to sit at the piano and play a C-major chord, it’s probable that he’ll just naturally play an octave C in the left hand, and play the other notes in the right hand, as shown in “Example 1” below. Depending on his mood, he might sometimes decide to put a fifth on the bottom as in “Example 2” below. But for him to put a bunch of close-together notes at the bottom as in “Example 3” would be quite unnatural. If this happens, then the jig is up; he’s just messing with you because he knows you’re looking to see how he’s going to play that chord.

Ways to play a C-major chord

## Vocabulary

Several words are used to describe this particular series of pitches. The bottom note (in this case, the low C) is called the fundamental. The other notes are called overtones and the whole bunch of them together are called harmonics or partials.

## Understanding key signatures

Once you have the major keys arranged in the circle of fifths, and you know why they only go around the circle seven steps either way, so they have up to seven flats or seven sharps, it’s only a small leap to understanding key signatures.

The basic concept is that if your music is in a major key, most of the notes are going to be the notes in the major scale. Suppose, for example, you have a piece in the key of D-major. Now, the D-major scale has two sharped notes, F-sharp and C-sharp. So your piece is going to have a lot of F-sharps and C-sharps and not so many F-naturals and C-naturals. So wouldn’t it be great to remove some excess clutter from your music? Why not just remind the musician at the beginning that all Fs and Cs are going to be sharped unless otherwise specified? Then you don’t have to sprinkle all those extra sharp-symbols all over your music. You’d save ink, help the environment, and contribute to saving our planet by decreasing your carbon footprint.

If you’re not a musician, this probably sounds like a huge nuisance. It would be hard enough to read those notes off the page, and now you have to keep remembering extra sharps (or flats) that aren’t written! Nevertheless, this is what musicians do. At the left of every staff there’s a key signature which reminds you of these “default” accidentals that are to be assumed unless otherwise specified.

So without further ado, here’s a circle of fifths with the key signatures included. The names of the major keys are given, and also the names of the minor keys (which we will discuss momentarily):

The circle of fifths, with major and minor keys indicated

## Some keys are more equal than others

You may wonder: what should you do in the cases where you have two possible key signatures? For example, if the key of B-major and the key of C-flat-major are composed of the same pitches written differently, if you’re writing music in this key, which key signature should be used? In theory, you could use either, but in practice, musicians often choose the one with less accidentals for convenience. So it’s common to see music notated in B-major but it’s very rare to see music in C-flat-major, and similarly it’s more common to see music written out in D-flat-major than C-sharp-major. As far as F-sharp-major and G-flat-major with six accidentals each, both are used. I think there is a bias towards F-sharp-major, however. F-sharp feels more “ordinary” because F-sharp is the very first sharp in many key signatures, while G-flat is a more “obscure” note.

## Minor keys

Every major key has a relative minor key that starts on the sixth note of the major scale. For example, the notes in the C-major scale are C-D-E-F-G-A-B-C. And the sixth note in this scale is A, so the relative minor key of C-major is a-minor. (Often you’ll see minor keys written in lowercase letters, and major keys in capitals.) The “regular” or “natural” a-minor scale has these notes: A-B-C-D-E-F-G-A. (To keep things simple we won’t discuss the other “less natural” minor scales just yet.) Having the same notes, the minor key has the same key signature as its relative major key.

So from the key signature alone, you can’t tell whether a piece of music is in a major key or a minor key. If there are no accidentals in the key signature, it could be in C-major or in a-minor. (A musician would be able to look at the notes and chords used to determine which key the piece is in, but to keep things simple we won’t discuss the details of that just yet.)

Previously we listed the notes of all the major scales, and now that you know how to figure out the relative minor key of any major key (just take the sixth note of the major scale, or, if you’re feeling lazy, start at the top and go two notes down) and how to construct the natural minor scale (it has the same notes as its relative major, but starts on the major scale’s sixth note), you could construct all of the natural minor scales yourself if you so desired.

You could even go in reverse; if you start with the minor scale, its relative major will be its third note. For example, the third note of the a-minor scale is C, and a-minor’s relative major key is C-major.

## Some minor keys are also more equal than others

As you’d expect, you rarely see music written in a-sharp-minor or in a-flat-minor; musicians would rather use five accidentals in the key signature instead of seven, and so they’d generally substitute b-flat-minor and g-sharp-minor respectively. You’d imagine that d-sharp-minor and e-flat minor would be equally common, both having six accidentals in their key signatures, but here the “feel” of the notes comes into play even more than with the relative major scales. D-sharp feels like an “obscure” note to base a scale on, while E-flat feels more ordinary. After all, E-flat is the second flat in many key signatures, and there is even a key of E-flat-major (but no key of D-sharp-major).

## Touchy-feely stuff

Now all this touchy-feely stuff about the six sharps or flats may seem a bit abstract, but yet I can cite a musical example that illustrates exactly this point. It did strike me as odd when I encountered it.

There is a particular piano piece that I have been playing which starts in a minor key and then modulates (changes keys) to the relative major key. This type of key change is not uncommon, and since you know that each major key and its relative minor key have the same key signature, you’d imagine that the key signature would stay the same. Normally you’d be right.

But here’s what happened: the piece started out in e-flat-minor, and then modulated to F-sharp-major! The composer (or whoever notated this music) may have used this logic of the “obscurity” of the notes to make such a choice. It must have seemed better to use e-flat-minor than d-sharp-minor, and similarly better to use F-sharp-major than G-flat-major. But by doing this, the key signature had to change from six flats to six sharps. Had the composer (or copyist) instead used e-flat-minor and G-flat-major, or d-sharp-minor and F-sharp-major, the key signature would not have had to change. (If you must know, the piece is Louis Moreau Gottschalk’s Souvenir de Porto Rico, Opus 31.)

Yet another example: I have the Edition Peters of J. S. Bach’s Well-Tempered Clavier, Book I, and Fugue VIII is notated in d-sharp-minor. But as an appendix, this edition includes another copy of the exact same piece notated in e-flat-minor. I suppose they thought that some people would be uncomfortable having to read music written in d-sharp-minor.

## Notes on the staff

The time has finally come to show some notes on a musical staff.

The scheme is rather straightforward. The notes on the staff follow the white keys on the piano keyboard. Each successive white key goes up half of a staff line. A note can be on a staff line or in the space between staff lines. The range of notes is not limited by the height of the staff; notes can go right off the top or bottom of the staff. In these cases, you draw in little pieces of the staff lines (ledger lines) that would have been there if only the staff were taller. A picture should make this perfectly clear:

The notes on the staff

But how do you know what notes they actually are? You could guess that higher notes on the staff represent higher pitches, but unless I told you a letter name of one of the notes, you’d still be lost. Once you knew one note’s name, however, you could figure out the rest easily enough by just going through the letters from A to G until you reached your note. For example, if I told you that the note straddling the bottom line of the staff was E, and asked you what note straddles the top line of the staff, you could count up the lines and spaces: E, F (space between the bottom two lines), G (second line from the bottom), A, B, C, D, E, F. So the note straddling the top line of the staff would be F.

By the way, generally the lines are counted from bottom to top, so if someone says “there was a note on the first line of the staff” they mean that the note was on the bottom line. I can imagine jazz musicians using this terminology all the time: “So, dig this, man, I went to this jam session, and they gave me the chart, and there was, like, this note on the first line of the staff.”

So is the bottom line’s note really E? It depends. This is where clefs come into play. Essentially, a clef is a goofy-looking symbol at the left of the staff, and some key feature of the clef’s symbol tells you what one of the notes is. Here is a picture of some clefs:

Musical clefs and their names

On the left we have the treble clef, also referred to as a G-clef. (There are other G-clefs, but the treble clef is by far the most commonly used nowadays.) The spiral part of the symbol appears to wrap around the second line from the bottom, and, as you are about to guess, that note is G. Usually this is the G above middle C, or G4. So if you saw a note straddling the bottom line of the staff and to its left you saw a treble clef, you could interpret that note as E4.

Next let’s skip over to the far right of the picture above. This clef is a bass clef, also known as an F-clef. (There are also other F-clefs but only the bass clef is commonly used.) The two dots in the symbol surround the third line from the bottom, and the note on that third line is F. This is usually the F below middle C, or F3.

The other clefs shown in the middle are C-clefs (the alto and tenor clefs shown are the most commonly used of the C-clefs), and the central notch of the symbol indicates the line for the note C (generally middle C, or C4).

Now you know enough to verify that the note names listed in the diagram below are correct. For each set of notes, the nearest clef to its left is used:

Some notes and their names

## More about the Circle of Fifths

Now we can explain why the circle of fifths looks the way it does. Recall that we had an ambiguous area from 5 o’clock to 7 o’clock:

A clock with the circle of fifths

We’re going to examine the circle of fifths in terms of major scales. To start out, you need to know that the scale you get by going up the white keys of the piano starting from C is a major scale. Thinking about half steps and whole steps, it’s a mixture of both. You can see from this half-step proportional wacky stretched keyboard that sometimes the white keys have black keys between them that get skipped over in our C-major scale, and sometimes they don’t:

Wacky stretched piano keyboard

What if we wanted to construct a G-major scale? Way back when we talked about transposing, we first introduced this stretched piano keyboard, and we said that transposing is like taking the pattern of notes and sliding it up or down the keyboard. So let’s take two of these keyboards and slide one over so that C on one lines up with G on the other:

Transposing between C-major and G-major

Now we can look at the white keys on the bottom keyboard (the ones in the C-major scale) and just read off the notes in the G-major scale from the top keyboard. Look at the diagram, start at C on the bottom keyboard and follow along on the top, reading off each note at the top above a white key on the bottom: G, A, B, C, D, E, F#, G. Note that it’s proper musical etiquette to keep the letter names in a scale in sequence, which is why we use the name F# rather than Gb here. It would be a serious faux pas to use Gb here.

This rule of etiquette actually has a good reason: when transposing, one strives to preserve all the intervals. In the C-major scale, the C up to the B is a major seventh, and in the G-major scale, the G up to F# is also a major seventh. G up to Gb would be a diminished octave. (Intervals were discussed earlier.)

Notice that the seventh note of the G-major scale is sharped (namely, F#) while the other notes are the same old natural white keys that appeared in the C-major scale. If it seems cumbersome to count up to the seventh note, you can just go down one note from G to get F, and that’s the one that gets sharped. The F was raised by a half step to get F#, and we cleverly verbed the noun “sharp” to mean “raise by a half step.”

Certainly transposing major scales up by a perfect fifth ought to do the same thing every time, namely sharp the seventh note and keep the other notes the same. And it does just that. Now we can go ahead all the way from 12 o’clock to 7 o’clock in the circle of fifths and instantly figure out which notes are sharped in all those major scales:

 Major scale Notes C major C, D, E, F, G, A, B, C G major G, A, B, C, D, E, F#, G D major D, E, F#, G, A, B, C#, D A major A, B, C#, D, E, F#, G#, A E major E, F#, G#, A, B, C#, D#, E B major B, C#, D#, E, F#, G#, A#, B F# major F#, G#, A#, B, C#, D#, E#, F# C# major C#, D#, E#, F#, G#, A#, B#, C#

The note in red is the note that’s different from the scale above it.

The same example worked backwards can help us understand how going counterclockwise works. To go from G-major to C-major, one note got un-sharped, namely F. F was the seventh note of the G-major scale, but it’s the fourth note of the C-major scale. Let’s verb the word “flat” to mean “lower by a half step.” It would seem that to go counterclockwise in the circle of fifths, each successive major scale flats its fourth note:

 Major scale Notes C major C, D, E, F, G, A, B, C F major F, G, A, Bb, C, D, E, F Bb major Bb, C, D, Eb, F, G, A, Bb Eb major Eb, F, G, Ab, Bb, C, D, Eb Ab major Ab, Bb, C, Db, Eb, F, G, Ab Db major Db, Eb, F, Gb, Ab, Bb, C, Db Gb major Gb, Ab, Bb, Cb, Db, Eb, F, Gb Cb major Cb, Db, Eb, Fb, Gb, Ab, Bb, Cb

Again, the note in red is the note that’s different from the scale above it. (If you are ambitious, you can go back and look at the diagram above with the two keyboards, and look at the notes on the bottom keyboard that line up with the C-major scale on the top keyboard; you should see the F-major scale on the bottom, with Bb as its fourth note.)

In each of the tables we can see that the last scale in the table (C# and Cb) has all its notes sharped or flatted. In order to sharp or flat any notes further, we would have to start using double-sharps or double-flats. In theory, you could indeed construct a scale of Fb major containing a Bbb, but in practice, musicians tend not to do that.

And that is why the circle of fifths has those two “tracks” that go seven steps in either direction. It relates to sharps and flats in the major scale. When we later talk about key signatures, that will help to tie this all together even more.

## A review

Previously we discussed a clock in which the numbers going clockwise from the 12 o’clock position went up by sevens. Of course, we want to keep the numbers between 1 and 12, so when the number gets larger than 12, we subtract the 12 out. We start with the number 7 at the 1 o’clock position. When we add 7 to that, we get 14, but that’s bigger than 12, so we subtract 12 from 14 and the next number becomes 2. Notice that adding 7 and subtracting 12 is the same thing as subtracting 5, so we could also have explained what we’re doing by saying that for each step around the clock face, we either add 7 or subtract 5, whichever keeps us in the 1-to-12 range. When you do this, you get all the way around the clock, using each number exactly once. It sounds more complicated than it is; look at this picture:

Wacky clock face by 7s

We then realized we could do the same thing with musical notes, as there are 12 hours on the clock and also 12 different notes in an octave, and we ended up with this:

Clock face with notes in seven-half-step jumps

We mentioned that this was the equivalent of something called “The Circle of Fifths”. When we discussed intervals and inversions and all that, we determined that a perfect fifth is seven half steps in size, so you’d think we could keep making the notes go up by perfect fifths (or, to stay within the same octave, go down by the inversion of that interval, the five-half-step perfect fourth) and get all the way around our clock face.

## Making a true circle of fifths

But there’s a catch! Recall that the designation “fifth” comes strictly from the note letter names. So C4 to G4 is a perfect fifth, but C4 to Abb4 is a diminished sixth, even though the pitches are the same. With this in mind, something odd happens if we try to go up by twelve perfect fifths.

Let’s try going clockwise. We start with C at the 12 o’clock position. Then, going up by perfect fifths (or, equivalently, down by perfect fourths) we have G, D, A, E, B, F#, C#, G#, D#, A#, E#, and finally B#. We did get to the same pitch, but not to the same note name.

Let’s try going counter-clockwise now, going up by perfect fourths (or, equivalently, down by perfect fifths). Here we go: C, F, Bb, Eb, Ab, Db, Gb, Cb, Fb, Bbb, Ebb, Abb, Dbb. Once again: same pitch, different note name.

So it would seem that we can’t literally create a circle of perfect fifths. Is music theory toying with us? How do musicians resolve this and represent a circle of fifths? Basically, they start from C and start going in both directions, making two tracks around the circle. If you want to travel all the way around the circle back to C, somewhere near the 6 o’clock mark you have to hop onto the other track. You can imagine in our original diagram that the clockwise direction continues on the outside track where the counter-clockwise direction moves on the inside. But in the approved musical version, the track ends after seven steps in either direction from C, so somewhere from 5 to 7 o’clock, you had better jump tracks. A picture should make this clear:

A clock with the circle of fifths

The tracks of the same color move by perfect fifths/fourths. Going clockwise from C, you’d follow the blue track, but to go all the way around, you must eventually jump to the red track near the bottom. Similarly, going all the way counter-clocwise, you’d have to jump from the red track to the blue track to get back around to C.

Why does the crossover occupy three positions? Why does each direction go seven steps? Why didn’t they just keep it simple and make you jump tracks at the 6 o’clock position? There are actually good reasons why the tracks go as far as they do and no farther, which we’ll explore next time.

## Up to six hours at a time

Suppose I have a clock and it reads 12 o’clock. If I set the clock forward one hour, then it will say 1 o’clock. If I do it again, the clock will say 2 o’clock, and if I keep setting it forward by one hour, after twelve times it will go back around to 12 o’clock. In the process of doing that, I have set the time to each hour exactly once.

Now suppose that I do this again, but I set the clock two hours ahead each time. Starting at 12 o’clock, I’ll then set the time to 2 o’clock, then to 4 o’clock, then to 6, 8, 10, and finally back to 12 o’clock again after six settings. In the process, I didn’t set the time to every possible hour. No matter how many times I set the clock forward two hours at a time, I’ll never hit the odd hours: 1, 3, 5, 7, 9, and 11 o’clock.

Similarly I could set the clock ahead three hours at a time. Now the time goes from 12 o’clock to 3, 6, 9, and back to 12 again. It took only four settings to get back to 12 o’clock, and I skipped a whole bunch of numbers.

Similarly, if I take four-hour steps from 12 o’clock I go to 4, 8, and then back to 12, taking only three settings and again skipping lots of numbers.

Let’s forget about five-hour steps for now.

If I use six-hour steps, I just go from 12 o’clock to 6 and back to 12 again, so it takes only two settings to get back to 12.

## Beyond six hours

Let’s also forget about seven-hour steps for now.

If I take eight-hour steps starting at 12 o’clock, I first set the clock forward to 8 o’clock. In the next eight-hour step I move right past 12 and end up at 4 o’clock. When I set the clock forward another eight hours from 4 o’clock, I’m back at 12 o’clock again. So using eight-hour steps from 12, the sequence was: 8, 4, and 12.

Let’s choose to ignore AM and PM, and whether or not I pass over 12 o’clock with one of the settings. By doing this, going forward by eight hours becomes exactly the same as going backward by four hours. If I turned the clock face away from you while I was setting it and just showed you the clock face as it appeared after each setting, you’d have no way of knowing whether I went forward eight hours or backward four hours; the result is exactly the same. (Assume you’re not watching my fingers while I turn the little knob on the back of the clock.)

If I use nine-hour steps, I go from 12 o’clock to 9 o’clock, 6, 3, and finally back to 12. Here we find that going forward nine hours is the same as going backward three hours.

Ten-hour forward steps are similarly like going backward in two-hour steps, and eleven-hour steps are like going backward in one-hour steps, and there you have it.

Note that each X-hour step larger than 6 hours is equivalent to going backward by (12 – X) hours. So: forward 8 = backward 4, forward 9 = backward 3, forward 10 = backward 2, and forward 11 = backward 1.

But what do we have? And what about the five-hour and seven-hour steps that we omitted? What was the point of all this?

## Summary

Let’s agree that we can ignore steps greater than six hours, because moving forward X hours is the same as moving backward (12 – X) hours. Here’s what we have:

• One-hour steps: gets back to 12 after twelve steps (hits every hour once)
• Two-hour steps: gets back to 12 after six steps
• Three-hour steps: gets back to 12 after four steps
• Four-hour steps: gets back to 12 after three steps
• Five-hour steps: mysteriously omitted
• Six-hour steps: gets back to 12 after two steps

## The mysterious five-hour and seven-hour steps

If you imagine going forward by five hours at a time, starting from 12 o’clock, here are the hours that you’d hit: 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12. Notice that it took twelve steps to get back to 12 o’clock, and we hit each hour exactly once!

Of course, a seven-hour step would have done the same thing in reverse from 12 o’clock: 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 12. This is because going seven hours forward is equivalent to going five hours backward, so the sequence is simply reversed.

Except for the five-hour steps, none of the other step sizes up to six hours could possibly hit all the hours in this way (except for the obvious case of one-hour steps) because it happens that their sizes all divide evenly into twelve. For example, six is half of twelve, so after two six-hour steps you’re already back at 12 o’clock. Similarly four is one third of twelve, three is one fourth, and two is one sixth. But five does not go evenly into twelve. Similarly, except for seven-hour steps, the step sizes greater than six hours (except for the obvious case of eleven-hour steps) when considered as their equivalent backward steps also divide evenly into twelve hours. Eight hours forward = four hours backward, and four is one third of twelve, and so on.

There’s one more thing to observe here. In addition to seven hours forward being equivalent to five hours backward, five hours forward is equivalent to seven hours backward. So suppose that for some reason we didn’t want to have any of the settings make the hour go past 12 o’clock. Look again at our five-hour stepping sequence: 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12. The five-hour forward jumps that pass through 12, such as 10 to 3, could equally well be replaced by a seven-hour backward jump. So to get from one number to the next, we add 5, or, if the result would be greater than 12, we can instead subtract 7.

To conform with musical orthodoxy (you’ll see later what this all has to do with music) we’ll consider the seven-hour sequence going from 12: 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 12. This can be considered as seven-hour forward jumps combined with some five-hour backward jumps if you don’t care to pass through 12 o’clock.

## The wacky clock

Now suppose that you had a twisted mind and wanted to make a messed-up clock face. With the regular clock face, you put 12 at the top and use one-hour steps to go to each successive number until after twelve steps you get back to 12 at the top. But someone with a twisted mind could use the seven-hour sequence instead. The numbers wouldn’t be in order, but since every number from 1 to 12 is used exactly once, it would still “work” in that you could make such a clock face and each number would have its own place. It would look like this:

Wacky clock face by 7s

You could actually construct such a clock that tells time if you made your hour hand move faster than normal. It just happens that 1 is the seventh number on this wacky clock, so the hour hand would move seven times as fast as normal, and it would be confusing to read most of the time. For example, the hour hand would pass over several other numbers to get from 12 to 1, so recognizing a time such as 12:15 would take some getting used to. But every hour the clock would be easy to read: you could just look at what number the hour hand points to and that would be the time. In the picture it’s 9 o’clock. Truly this is a wacky clock.

## The musical connection

Notice that the use of the word “wacky” has cleverly foreshadowed how this ties in with music. We had previously considered a wacky stretched keyboard where each half step was equal in physical size. There are twelve hours on the clock, but there are also twelve different pitches within an octave. Let’s assign the hours from 1 to 12 to the various notes. Since we have been generally using C as our starting point, we’ll let C be 12 o’clock. Here’s what we get (for simplicity, we’ll omit the double-sharp and double-flat note names):

• 12 o’clock = C
• 1 o’clock = C#/Db
• 2 o’clock = D
• 3 o’clock = D#/Eb
• 4 o’clock = E
• 5 o’clock = F
• 6 o’clock = F#/Gb
• 7 o’clock = G
• 8 o’clock = G#/Ab
• 9 o’clock = A
• 10 o’clock = A#/Bb
• 11 o’clock = B

Now, we can make a regular clock face, but instead of the numbers, we can put note names:

Clock face with musical notes

Each note within the octave clearly appears once on this clock face.

But we know that if we used the seven-hour step sequence of hours to make a clock face, it would also use the numbers exactly once. We can take that seven-hour-step clock face and substitute the corresponding musical notes, and we get this:

Clock face with notes in seven-half-step jumps

If you have some background in music theory, you already recognize what we have ended up with: the circle of fifths! A perfect fifth is seven half steps. Going clockwise we move up seven half steps each time (or, equivalently, down five half steps). Going counter-clockwise we move down seven half steps each time (or, equivalently, up five half steps). A perfect fourth has five half steps, and it’s the inversion of a perfect fifth which has seven half steps. And at this point it should seem practically self-evident that you can’t make such a circle with any other musical intervals and hit each note exactly once (except for the trivial case of going one half step each time, as in the “regular” clock face). Technically there are four such intervals that would make a circle, but two are inversions of the other two, so their circles are simply mirror-images of the others:

1. One half step up (“regular” clock face)
2. Five half steps up (perfect fourth), mirror-image of #3 below
3. Seven half steps up (perfect fifth, making the wacky clock face or the standard circle of fifths)
4. Eleven half steps up (or one half step down), mirror-image of #1 above

In the future we’ll discuss the circle of fifths in more detail. It’s a key element of music theory.

## Intervals fully named

Previously we named intervals using the numeric portion of their names, and we introduced scientific pitch notation. Now we’re going to finish the job by using the full designation of each interval. To make sense of what follows, you’ll need to know a bit about sharps and flats as well.

## “Near-major” and “near-minor”

We considered intervals that start on C4 (middle C) and end on some white key. It so happens that if you play notes going up the white keys starting with C4 and ending up at C5, you are playing a major scale.

You can imagine someone trying to name the intervals saying, “These are the intervals in a major scale, so I’ll call them ‘major’ intervals.” Actually, they’re not, so let’s for now call them “near-major” intervals.

When you invert the near-major intervals, most of them end up with different sizes than their non-inverted number-mates as measured in half steps. (The table below clarifies this.) If they miss, they always end up on the low side. In the previous article we showed a diagram with two stretched keyboards back-to-back (which appears later in this article too) and explained why this happens.

So our interval namer now says, “These inversions of [near-]major intervals are mostly smaller in half-step size than the [near-]major intervals, so I’ll call them ‘minor’ intervals.” As before, they’re not really, so we’ll call them “near-minor” intervals. Look at this table to see what we’re talking about:

 “Near-major” interval “Near-minor” inversion C4-to-C4, unison (0 half steps) C4-to-C5, octave (12 half steps) C4-to-D4, second (2 half steps) D4-to-C5, seventh (10 half steps) C4-to-E4, third (4 half steps) E4-to-C5, sixth (8 half steps) C4-to-F4, fourth (5 half steps) F4-to-C5, fifth (7 half steps) C4-to-G4, fifth (7 half steps) G4-to-C5, fourth (5 half steps) C4-to-A4, sixth (9 half steps) A4-to-C5, third (3 half steps) C4-to-B4, seventh (11 half steps) B4-to-C5, second (1 half step) C4-to-C5, octave (12 half steps) C5-to-C5, unison (0 half steps)

For an example, look at the near-major third C4-to-E4 in the left-hand column; its size is 4 half steps. The corresponding near-minor third in the right-hand column, A4-to-C5, has only 3 half steps. So the near-minor third is smaller than the near-major third. You’ll find that all of the near-minor intervals are either the same or one half step smaller than the near-major intervals with the same numerical designation.

## A diversion about the minor scale

It would have been really cool if the near-minor intervals were the ones you found within a minor scale, but that’s not quite so. If you took the near-minor inverted intervals and transposed them so they all go up from C4, you’d get this sequence of notes topping off the intervals: C4, Db4, Eb4, F4, G4, Ab4, Bb4, C5. The note D-flat-4 is the one not in the minor scale (the minor scale has D4 instead; the scale with Db4 is a Phrygian scale). This is not obvious to see, so you can look at the picture below (the same one that appeared in the previous article); the keys on the top that line up with the white keys on the bottom are the ones that form the inverted “near-minor” scale:

Stretched piano keyboard, with another one below it upside-down

Since most of the inverted intervals are in the minor scale and they’re smaller than the major intervals, the name “minor” seems appropriate, more or less.

## Perfect intervals

Now our interval namer hits a glitch. There are four near-major intervals that match other near-major intervals in size when you invert them. They’re the unisons, fourths, fifths, and octaves (you can see this in the table above, or see the previous article if you want to know why). It would have been a convenient rule to just say the major scale has all major intervals and the inverted near-minor scale has all minor intervals, but four of the intervals are in both scales. It wouldn’t make any sense, for example, to have a major fourth and a minor fourth actually be the same size. If you saw the interval outside of a scale, how would you know what to call it? Actually, based on the musical context, someone probably could have invented a method of determining the appropriate name for these intervals, but fortunately another solution was chosen.

Intervals of a unison, fourth, fifth, or octave that occur in the major scale are called “perfect” because, well, they’re just that good. And when you invert a perfect interval, the result is still a perfect interval.

So let’s amend our table with the full interval names:

 Major-scale interval Inversion C4-to-C4, perfect unison (0 half steps) C4-to-C5, perfect octave (12 half steps) C4-to-D4, major second (2 half steps) D4-to-C5, minor seventh (10 half steps) C4-to-E4, major third (4 half steps) E4-to-C5, minor sixth (8 half steps) C4-to-F4, perfect fourth (5 half steps) F4-to-C5, perfect fifth (7 half steps) C4-to-G4, perfect fifth (7 half steps) G4-to-C5, perfect fourth (5 half steps) C4-to-A4, major sixth (9 half steps) A4-to-C5, minor third (3 half steps) C4-to-B4, major seventh (11 half steps) B4-to-C5, minor second (1 half step) C4-to-C5, perfect octave (12 half steps) C5-to-C5, perfect unison (0 half steps)

For this next bit we need to introduce some sharps and flats. Let’s start with a major third, C4 to E4. The table tells us that this is four half steps in size. Now let’s flat the E, so the interval becomes C4 to Eb4. Since flatting lowers the E by a half step, this new interval must be three half steps in size. But notice that in the table, we already have a third that is three half steps in size, namely A4 to C5, and it’s a minor third. So C4 to Eb4 is a minor third as well. We could do the same thing with the other major intervals, and we’d find that C4 to Db4, C4 to Ab4, and C4 to Bb4 are all minor intervals.

None of the above applies to the perfect intervals. If you flatted the top note of a perfect interval, the result doesn’t match any interval in the table. For example, C4 to Gb4 is a fifth with six half steps, and all of the fifths in the table are perfect fifths with seven half steps, so the above technique can’t help us. In the next section we’ll define what this interval is called.

## Augmented and diminished intervals

What if you go one step further? We flatted the top note of a major third and ended up with a minor third. What if you flat the top note yet again? In the example we used above, C4 to E4 was a major third, C4 to Eb4 was a minor third, but what is C4 to E-double-flat-4?

It turns out that the interval C4 to Ebb4 is called a diminished third. This makes sense because “diminished” sounds like it would refer to an interval that was made smaller. Similarly the other minor intervals can all be reduced by a half step to give diminished intervals.

There is an important point to be careful of here. The numeric designation only refers to letter names, not considering sharps and flats. One might be tempted to say that since Ebb is the same note as D, that the interval C4 to Ebb4 ought to be the same interval as C4 to D4. But this would be wrong. It is the same size in half steps, but it is not the same interval.

So for the technically nitpicky types, if someone plays C4 and D4 on the piano and asks you what interval they’re playing, the strictly correct answer would be, “I don’t know.” For all you know they could be “actually” playing Ebb4. Actually, though, you’d come across as quite the dork if you said “I don’t know” and tried to use this justification. You could instead choose the erudite path and say, “it’s a major second or a diminished third.” Now the questioner will be impressed by your sophisticated musical knowledge.

But you can go the other way also. What if you raise the top note of a major interval? What kind of a third is C4 to E-sharp-4? It happens that it’s called an “augmented” third, which makes sense because “augmented” sounds like an interval that was made larger.

Perfect intervals don’t have major or minor variations, so when you increase a perfect interval you go straight to “augmented” and when you decrease a perfect interval you go straight to “diminished”. So our previous example of C4 to Gb4 is a diminished fifth. It has the same size as an augmented fourth, C4 to F#4, but they’re considered different intervals. If someone played these two notes on a piano and asked you what interval it was, you could say, “a diminished fifth or an augmented fourth.”

## Going even further

By the time all the work of naming major, minor, perfect, diminished, and augmented intervals was done, our interval naming person just didn’t care anymore, being quite worn out with all the work of coming up with these names. If you raise the top note of an augmented interval, it’s “doubly augmented.” And if you lower the top note of a diminished interval, it’s “doubly diminished.” At this point the interval namer just said, “Yeah, whatever.”

## Naming any interval

Technically, with the techniques above and the technique of transposing that was previously discussed, you now know enough to properly name any interval. But in a future article we’ll take a closer look at this technique. We can use some foreshadowing here to summarize some rules, which should look reasonable at this point:

• Intervals from the first note of the major scale up to one of the other notes in the scale are perfect for fourths, fifths, unisons, or octaves; otherwise they’re major
• If you decrease a major interval by a half step by flatting or sharping a note (you could flat the top note or sharp the bottom note), you get a minor interval
• When you invert a perfect interval, you get another perfect interval
• When you invert a major interval, you get a minor interval, and vice versa
• When you decrease a minor or a perfect interval by a half step by flatting or sharping a note, you get a diminished interval
• When you increase a major or a perfect interval by a half step by sharping or flatting a note, you get an augmented interval
• When you invert a diminished interval, you get an augmented interval, and vice versa (we didn’t discuss this, but it should seem reasonable)
• You can sharp and flat things more to get doubly diminshed and doubly augmented intervals, yada yada yada…

## Octave numbering

The scientific pitch notation system assigns a number to each octave. The numbers go up as the octaves go higher. For example, the note C4 is an octave higher than C3, which in turn is an octave higher than C2. For fun, you can remember that C4 is middle C on the piano, in addition to being a well-known explosive material.

There’s a slight twist to scientific pitch notation: the octaves start with C! You’d think they’d start with A, because that’s the first letter, but you’d be mistaken. So the white-key notes in ascending pitch sequence from A0 (the lowest key on the piano) are as follows: A0, B0, C1, D1, E1, F1, G1, A1, B1, C2, D2, etc.

## Intervals and inversions

Previously we discussed the numeric names of intervals. The interval of C4 up to E4 is a third, and you can count out C-D-E as 1-2-3 to figure this out. The interval of C4 up to B4 is a seventh: C-D-E-F-G-A-B is 1-2-3-4-5-6-7. C4 to C4 itself is not called a “first” but rather a “unison” and C4 to C5 is not an “eighth” but rather an “octave.”

Suppose that note X is the note corresponding to any white key on the keyboard from C4 up to C5. We’ll consider intervals from C4 up to X and their inversions which go from X up to C5. For example, X could be E4: the inversion of C4-to-E4 would be E4-to-C5. There’s nothing special about using C4 and C5; it works just the same for any octave Cs, but C4 is convenient what with it being in the middle of the piano and all.

If the inversion of C4-to-X is X-to-C5, the inversion of a unison is an octave, and the inversion of an octave is a unison. How? Suppose X is C4: the inversion of C4-to-C4 (unison) is C4-to-C5 (octave). Now suppose X is C5: the inversion of C4-to-C5 (octave) is C5-to-C5 (unison). These intervals obey the same rule of nine that the others do; the numbers of the interval and its inversion add up to 9 (1 + 8 = 9).

So here is a list of all of the intervals we are considering, with the “regular” intervals C4-to-X and their inversions X-to-C5:

 “Regular” Interval Inversion C4-to-C4 (unison) C4-to-C5 (octave) C4-to-D4 (second) D4-to-C5 (seventh) C4-to-E4 (third) E4-to-C5 (sixth) C4-to-F4 (fourth) F4-to-C5 (fifth) C4-to-G4 (fifth) G4-to-C5 (fourth) C4-to-A4 (sixth) A4-to-C5 (third) C4-to-B4 (seventh) B4-to-C5 (second) C4-to-C5 (octave) C5-to-C5 (unison)

## The obvious question

Recall that in a certain previous article we mentioned that intervals can have the same numeric indication but a different size in half steps. So, for example, all thirds are not equal in their half-step size.

This begs the obvious question: Which intervals of the same number in the two columns of the table above contain the same amount of half steps? For example, does the third in the left-hand column (C4-to-E4) have the same amount of half steps as the third in the right-hand column (A4-to-C5)?

To understand this, we’ll use a picture with a wacky stretched keyboard at the top, just the way you’d normally look at it, with the notes getting higher from left to right. Another wacky stretched keyboard will be placed below it upside-down, having been flipped around by 180 degrees. Notes on the bottom keyboard go the other way: they get higher from right to left. Arrows show the direction in which the notes get higher. The two keyboards are placed so that C4 on one keyboard lines up with C5 on the other. Recall that wacky stretched keyboards allot the same physical distance to each half step, so the picture will tell us how the half steps of intervals and inversions match up.

Look at the picture below and consider the interval from C4-to-E4 on the top keyboard. Notice that E4 on the top keyboard lines up with the black key between G4 and A4 on the bottom keyboard (which is called A-flat-4). Because the bottom keyboard is backwards, this is showing us that the C4-to-E4 interval on the top lines up with A-flat-4-to-C5 on the bottom. Thus the size in half steps of C4-to-E4 equals that of A-flat-4-to-C5. Since A-flat-4 is not a white key, that means: C4-to-E4 in the left-hand column of the table does not have the same number of half steps as any of the intervals in the right-hand column!

To make this even clearer, let’s consider at an interval that does line up with a white key. The F4 on the top keyboard lines up with G4 on the bottom. Therefore the distance C4-to-F4 equals the distance G4-to-C5, and so the number of half steps of an interval in the left-hand column (namely C4-to-F4) does equal that of some interval in the right-hand column (namely G4-to-C5).

Stretched piano keyboard, with another one below it upside-down

## Answer to the obvious question

I put it to you that exactly four intervals in the left-hand column of the table are equal to intervals in the right-hand column. Why? Because in the picture, between C4 and C5, exactly four of the white keys on the top (including the two Cs) line up with white keys on the bottom.

You can see how the white keys get misaligned. The pairs of white keys without black keys in-between occur in different places on the top and bottom keyboards. Moving from left to right, the bottom keyboard “skips” a black key first and the white keys on the top and bottom are “out of sync” until the top keyboard later “skips” a black key as well.

## Inversions and half steps, proving the answer

An octave contains twelve half steps (this seems reasonable since each octave goes up by seven white keys and passes through five black keys on the way, and seven plus five is twelve). Since each interval goes partway up the octave, and its inversion goes from there the rest of the way up the octave, the size of the interval in half steps plus the size of its inversion in half steps must always equal twelve. Look at this:

 “Regular” Interval Inversion C4-to-C4 (0 half steps) C4-to-C5 (12 half steps) C4-to-D4 (2 half steps) D4-to-C5 (10 half steps) C4-to-E4 (4 half steps) E4-to-C5 (8 half steps) C4-to-F4 (5 half steps) F4-to-C5 (7 half steps) C4-to-G4 (7 half steps) G4-to-C5 (5 half steps) C4-to-A4 (9 half steps) A4-to-C5 (3 half steps) C4-to-B4 (11 half steps) B4-to-C5 (1 half step) C4-to-C5 (12 half steps) C5-to-C5 (0 half steps)

Which numbers in the right-hand column match any of the numbers in the left-hand column? Only four: 0, 5, 7, and 12, corresponding to the C4, F4, G4, and C5 intervals and their inversions. I rest my case.

## The complete list of match-ups

So here’s our complete list of all the C4-to-X intervals that match, in half step size, the inversion of one of those intervals:

• The inversion of the unison of C4-to-C4 matches the octave of C4-to-C5
• The inversion of the fourth of C4-to-F4 (namely the fifth of F4-to-C5) matches the fifth of C4-to-G4
• The inversion of the fifth of C4-to-G4 (namely the fourth of G4-to-C5) matches the fourth of C4-to-F4
• The inversion of the octave of C4-to-C5 (namely the unison of C5 with itself) matches the unison of C4 with itself

So something is different about unisons, fourths, fifths, and octaves! There’s this set of intervals from C up to each of the white keys, and when you invert only these four intervals, your result is equal in half-step size to another interval in the set. All the others are unequal in half-step size: C4-to-D4 doesn’t equal B4-to-C5, C4-to-E4 doesn’t equal A4-to-C5, C4-to-A4 doesn’t equal E4-to-C5, and C4-to-B4 doesn’t equal D4-to-C5.

## The final question

Now that you know why unisons, fourths, fifths, and octaves in our white-key intervals that start or end with C are unique, you probably have this question in mind: What’s the significance of this? That will be a topic for another article.

## Intervals and the rule of nine

Recall that the letter names of the white keys on a keyboard cycle through the letters A, B, C, D, E, F, G, and then back to A, like this:

The letter sequence of the note names

So what are we talking about when we’re speaking of intervals? We are talking about two notes, and trying to name the distance between them.

Also, I’ll tell you ahead of time that the name of each interval has a numeric portion, and some word that identifies it more specifically. For now, we’ll just consider the numeric part. To do this, you can ignore any sharps or flats, and just consider the letter names. So to keep things simple, we’ll just consider the notes named with plain letters, without any sharps or flats. This is peculiar, in a way that will be explained later.

### Numeric designations

Let’s start with the easiest case; the “empty” interval from a note to itself. Suppose the first note is C, and the second note is that identical C. Obviously there ain’t no distance between the two notes at all, since they’re both the same note, and this interval gets assigned the number one. In music-speak, this is called a “unison” which is simply a more erudite way of saying “first.”

Now let’s take a C and the D just above that. The interval between the C and D is called a “second.” Similarly, the interval of a C going up to the nearest E is a “third”, and so on. We can count up the letters to get there: C-D-E is like 1-2-3, so C to E is a third. When we get up to eight, the interval is not called an “eighth” (which is probably a good thing because there’s an eighth note in music already, having nothing to do with intervals). So upon reaching the next C, the interval name changes to help add erudition to our terms, as you can see in the list below (erudite terms are highlighted):

• C to same C: unison
• C to closest D above: second
• C to closest E above: third
• C to closest F above: fourth
• C to closest G above: fifth
• C to closest A above: sixth
• C to closest B above: seventh
• C to next C above: octave
• C to second-closest D above: ninth
• C to second-closest E above: tenth
• C to second-closest F above: eleventh
• C to second-closest G above: twelfth
• C to second-closest A above: thirteenth
• C to second-closest B above: fourteenth

What happens when your interval leaps up by two Cs? It could be a fifteenth, but doesn’t it feel like some fancy erudite term is called for? After all, the interval starts on a C and ends up back on a C, similar to the erudite unison and octave intervals. It is actually called a fifteenth, but you can also use the term “double octave” if it makes you feel more scholarly. Here’s a picture to help further cement the concept of how the numbers work.

The intervals, by their numbers

If you are any sort of computer nerd, you are well aware that had these interval names been invented in the past 30 years or so, they almost certainly would have started their numbers at zero for the two identical notes. It is a bit mathematically odd that the interval containing zero distance has the number 1.

Jumping up two Cs would seem to be twice as far as jumping up one C. But jumping up one C gives us an octave, and jumping up two Cs gives us a fifteenth. Clearly 15 is not 8 doubled. How can this be? The answer is that to move up an octave, we advanced up from C by seven notes; we got one extra count for free because we already had a count of one when we were still on the same note. So although an octave is supposedly an interval of eight, it moves up from the bottom by seven notes. So two octaves are a fifteenth (1-for-nothing + 7 + 7). Three octaves would then be an interval of a twenty-second (1 + 7 + 7 + 7), but let’s not even think about that for now.

### Inversion and the rule of nine

What does it mean to invert an interval? For intervals one octave or less, we’ll define inversion as what happens when you move the lower note up by an octave.

For example, suppose you have the interval from C to E, which as we saw earlier is a third. To invert it, we move the C up by an octave, so that now the C is above the E.

So what do we end up with? If you count up E-F-G-A-B-C you get 1-2-3-4-5-6, so E to C is a sixth. The C-to-E interval is a third, and its inversion E-to-C is a sixth. The numbers for the interval and its inversion will always add up to nine. Look at the following:

Interval inversion: a third and a sixth

The first interval was C to E, as shown on the left above. To invert it, we changed from the C on the left to the other C on the right. The original interval from the C to the E goes part of the way up the octave, and the inverted interval goes from the E up to C, the rest of the way up the octave.

So if an octave is supposedly eight notes, why don’t the two intervals add up to eight? It’s our mathematical oddity again; counting one for a unison puts us off by one from the get-go. So the number for the sum of two intervals is one more than it seems like it should be, because each extra interval in the sum gives you an extra “one” for free. (Actually, the octave goes up by seven notes from its starting point, so its count of eight already includes one for free. Having two intervals in the octave is adding a second extra one for free, bring the total to nine.)

To illustrate this point further, imagine the sum of three intervals:

Interval sum: a third, a third, and a fourth

Consider the intervals indicated in the picture above: C to E, E to G, G to C. Counting them out, we get C-D-E (a third), E-F-G (another third), and G-A-B-C (a fourth). So they’re a third, a third and a fourth respectively. But what do they add up to? 3+3+4=10! So by including yet another interval in the sum, yet another one for nothing gets added.

Hence when inversion splits up one octave into two intervals, the second interval adds one for nothing and the sum of the intervals is nine.

### Why this is peculiar

Previously we talked about half steps and whole steps. But at the time we also mentioned that all of the white keys on the keyboard (the ones with just plain letter names that we’ve been talking about) are not the same distance apart. Some adjacent white keys are a whole step apart, and some differ by only a half step. So though we have given the intervals labels according to the letter distance, these labels do not tell us how many half steps or whole steps comprise the interval. Whether the letter distances are whole steps or half steps doesn’t affect the way we count the interval numbers.

So if someone asked you how many half steps are in a third, the correct answer would be, “I don’t know.” With the knowledge in the earlier article you can easily figure out that, for example, the number of half steps in the interval of a third from C up to E is not the same as the number of half steps in the interval of a third from D up to F.

So the numeric naming of intervals is totally independent from how many half steps the interval contains, and that is rather peculiar. In fact, it is possible to have two intervals with different number designations that have the same number of half steps! But that is a topic for another article.