Understanding intervals

There are two ways to understand intervals: the right way, and the way I learned them. Before we get into that, let me point you to some good resources for learning the right way. I like the online tutorials by Robert Hutchinson, Chelsey Hamm and Bryn Hughes, musictheory.net and musicca.com. I really love Nate May’s visual approach. And if you like learning from videos (which I don’t), this one by Saher Galt is good. I find it most helpful to visualize the intervallic structure of the diatonic scale on a circle, and if you like to think that way too, try the aQWERTYon.

I am an aural learner who needs concrete examples, so I made this track for my students to sing along with in class, and for practice outside of it. It combines a soothing tanpura drone with the Apache break.

Here’s an interactive Noteflight score.

I said at the top of the post that I learned intervals the wrong way. Like many self-taught guitarists, I approached intervals by counting semitones (guitar frets.) A minor second is one semitone, a major second is two semitones, a minor third is three semitones, and so on. What, you may ask, is wrong with that? Aren’t intervals comprised of semitones? Sure they are, but counting semitones is no help if you want to name your intervals correctly. If you go up a tritone (six semitones) from C, do you land on F-sharp or G-flat? In other words, is the tritone an augmented fourth or a diminished fifth? In blues and related music, it doesn’t matter, a tritone is a tritone. But in Western tonal music, it does matter, especially if you want to produce readable notation.

The right way to think about intervals is to start by counting staff positions. If you see C to F-sharp, that is a fourth: C, D, E, F. Adding the sharp makes it an augmented fourth. If you see C to G-flat, that is a fifth: C, D, E, F, G. Adding the flat makes it a diminished fifth. If you are coming into music theory class with weak reading skills (like me), the main challenge in thinking this way is to internalize where on the staff the semitones are. In treble clef, the bottom line to the bottom space is E to F. The top space to the top line is also E to F. The middle line to the next space up is B to C. I don’t have any good mnemonics for higher and lower octaves or other clefs; you have to just practice.

So here is how I recommend that you approach intervals. Start with generic intervals: unisons, seconds, thirds, fourths, fifths, sixths, sevenths and octaves. Unisons could more logically be called “firsts” and octave could more logically be called “eighths”. Unless it says otherwise, you can assume that these intervals are based on the ascending major scale. In C:

  • Unison is the interval between C and the same C (imagine two people singing the same note). They’re on the same staff line or space.
  • A second is the interval between C and D. Move up one slot on the staff. (Yes, a second is one slot, not two. The counting is ordinal, not cardinal.)
  • A third is the interval between C and E. Move up two slots on the staff.
  • A fourth is the interval between C and F. Move up three slots on the staff.
  • A fifth is the interval between C and G. Move up four slots on the staff.
  • A sixth is the interval between C and A. Move up five slots on the staff.
  • A seventh is the interval between C and B. Move up six slots on the staff.
  • An octave is the interval between C and the next C. Move up seven slots on the staff. 

You make all the other intervals by modifying the generic ones. Seconds, thirds, sixths and sevenths are major by default, meaning that moving up from some root note by these intervals gives you notes from the major scale. If you narrow major intervals by a semitone (flatten them), you get minor intervals.

  • A minor second is the interval between C and D-flat.
  • A minor third is the interval between C and E-flat.
  • A minor sixth is the interval between C and A-flat.
  • A minor seventh is the interval between C and B-flat.

Going up from some root by the minor intervals gives you notes from Phrygian mode. Interesting.

Things get to be tricky when you want to think about descending intervals rather than ascending ones. Descending “flips the polarity” of everything.

  • Seconds become sevenths, and vice versa.
  • Thirds become sixths, and vice versa.
  • Fourths become fifths, and vice versa.
  • Major intervals become minor, and vice versa. (Perfect intervals stay perfect, how nice.)

So if you descend by a major second, that’s the same as ascending by a minor seventh (and then dropping an octave). If you descend by a minor second, that’s the same as ascending by a major seventh (and then dropping an octave). The weird thing is that if you descend from C by any major interval, you land on a degree of C Phrygian mode. If you descend from C by any minor interval, you land on a degree of C major. I do not have any useful mnemonics for any of this. I suggest just working it through on an instrument, preferably a keyboard. Learn my interval song, then play it from roots other than C. Do as many as you can stand.

Now that you know about major, minor and perfect intervals, the color scheme of the aQWERTYon might make more sense. Major intervals above the root are green, minor intervals are blue, and perfect intervals are purple. If you look at the scale menu, the “darker” scales have more minor intervals in them, while the “brighter” scales have more major intervals.

It gets worse! Fourths and fifths don’t actually have to be perfect. You can make them wider and narrower too. Expanding a fourth or fifth by a semitone gives you an augmented fourth or fifth. Narrowing a fourth or fifth by a semitone gives you a diminished fourth or fifth. You will be unsurprised to learn that diminished chords contain diminished fifths, and that augmented chords contain augmented fifths. Let’s see what notes we get by augmenting and diminishing our perfect fourths and fifths. 

  • An augmented fourth is the interval between C and F-sharp.
  • A diminished fourth is the interval between C and F-flat. Yes, F-flat is a real note; it’s the third of a Dbm chord.

But isn’t a diminished fourth the same thing as a major third? Not on the page. Noah Kahrs gave me a lovely example of a diminished fourth from the very beginning of Bach’s Fugue in C-sharp Minor from the Well-Tempered Clavier Book 1, BWV 849. 

The first three notes in the fugue are C-sharp, B-sharp, and E. C-sharp is the tonic, B-sharp is the leading tone, and E is the minor third. There are four semitones from B-sharp to E, and counting naively, you might think it would be a major third. But no! From any kind of B to any kind of E is always a fourth. Raising the B makes it narrower by a semitone, which gives you a diminished fourth.

Anyway:

  • An augmented fifth is the interval between C and G-sharp. Isn’t that just a minor sixth, though? It’s the same piano key as A-flat, but C to G is always a fifth. The sharp makes it augmented.
  • A diminished fifth is the interval between C and G-flat. Isn’t that the same as an augmented fourth? Again, it’s the same piano key, but C to G has to be a fifth, and the flat makes it diminished.

It gets even worse from here. In special circumstances, the major intervals can sometimes be augmented, and the minor intervals can sometimes be diminished.

  • Augmenting a major interval means making it a semitone wider. If a major sixth above C is A, then an augmented sixth above C is A-sharp. Isn’t that just B-flat, though? As far as the piano is concerned, yes, but notation distinguishes them.
  • Diminishing a minor interval means making it a semitone narrower. If a minor seventh above C is B-flat, then a diminished seventh above C is B-flat-flat. Yes, it is a real note. And yes, it’s the same piano key as A. But isn’t it nutty to call a note B-flat-flat when you use the A key to play it? Why even maintain such an arcane naming system? We will get to that.

There are a lot of rules about correct spelling of enharmonic equivalents. To begin with, major and minor scales should only use each letter name once.

  • The D major scale uses F-sharp rather than G-flat, because otherwise the scale would have both G-flat and G.
  • The D-flat major scale uses G-flat rather than F-sharp, because otherwise the scale would have both F and F-sharp.
  • The E minor scale uses F-sharp rather than G-flat, because otherwise the scale would have both G-flat and G.
  • The E-flat minor scale uses G-flat rather than F-sharp, because otherwise the scale would have both F and F-sharp.

What if you are dealing with a non-diatonic note? That is, what if the note is outside the current major or minor key? Now you have to look at the voice leading context. If the note is moving upward, it should be sharped. If the note is moving downward, it should be flatted.

  • If you are approaching G chromatically from below, the note should be F-sharp.
  • If you are approaching F chromatically from above, the note should be G-flat.

You could also think of this in terms of ease of reading. It’s harder to read G-flat to G than it is to read F-sharp to G. It’s harder to read F-sharp to F than it is to read G-flat to F.

At this point, your eyes may have long ago glazed over, and rightly so. If F-sharp and G-flat are the same note, who even cares what you call it? Who cares if you use a letter more than once in a scale? Yeah, notation, but what if you aren’t using notation? This is why guitarists who play by ear tend to be casual about note naming. Is the second fret on the E string F-sharp or G-flat? It’s the second fret on the E string. Jazz musicians often name notes incorrectly for the sake of simplicity. John Coltrane’s handwritten chart of “Naima” uses A where it would be more accurate to say B-flat-flat. If Coltrane didn’t care, why should anyone? In the blues, nothing is within the diatonic system at all, so there is no ” correct” way to name notes to begin with. 

If you are a DAW producer, you couldn’t spell your notes correctly in the piano roll even if you wanted to. In fairness to the developers, note naming is not at all easy to represent in software. When we were designing the aQWERTYon, it took a lot of hours to iron out all the edge cases. Many DAWs “solve” this problem by decreeing that every black key note is a sharp. Ableton did that for its first ten versions of Live, until finally they heard the cries of anguish from music theory teachers and gave you the option to toggle sharps and flats. That’s a big improvement, but even Ableton won’t let you call a note B-sharp or C-flat, much less B-flat-flat.

The thing that finally made me care about note spelling wasn’t notation, it was learning about the history of tuning. The Western standard tuning system is a relatively recent development. In older systems, D-sharp and E-flat were not just different spellings for the same sound or piano key or guitar fret; they were different pitches, requiring different piano keys or guitar frets. They were very close together, but not close enough to be interchangeable. Listen for yourself!

And so how do you learn and practice all this? There are all kinds of ear training apps and flash cards and exercises. I did not use any of those things. I learned tunes, many many tunes. I learned them in multiple keys on multiple instruments. I improvised in all the keys. I wrote and arranged and produced. After a couple of decades, I just kind of picked all of it up. It is not a good strategy for a one semester class, but it works.