Intervals

Two notes at once, or one after the other, make a distance, and naming that distance is the single most useful skill in reading a score. This lesson builds the naming system, which needs two numbers rather than one, and shows where the names come from.

Why one number is not enough

Play C4 then E4, then play C4 then E flat. Both pairs cover three letter names, C to E, and both are called some kind of third. They do not sound the same: the first is four semitones wide and bright, the second is three semitones wide and dark, and the whole character of a piece can turn on which one is used.

So an interval name has two parts. The number counts letter names, ignoring sharps and flats entirely. The quality counts semitones, and separates the intervals that share a number. C to E is a major third; C to E flat is a minor third. Neither part alone identifies the interval, and neither is redundant: the number tells you what the notes look like on the staff, and the quality tells you what they sound like.

Counting the number

The count is inclusive, meaning both end notes are counted. C up to E covers C, D, E, so it is a third. C up to G covers C, D, E, F, G, so a fifth. C up to the C above covers eight letters and is an eighth, which is the octave, and octave is simply the Latin for eighth. A note with itself is a first, called a unison.

Inclusive counting has one consequence that surprises everyone once and then never again: interval numbers do not add. Stack a third on a third and you do not get a sixth. C up to E is a third, E up to G is a third, and C up to G is a fifth, because the note E was counted twice, once as the top of the lower interval and once as the bottom of the upper. The rule is $a+b-1$: two thirds make $3+3-1=5$, a third plus a fourth makes a sixth, and two fifths make a ninth.

Accidentals never change the number. F to A is a third and so are F sharp to A, F to A flat, and F sharp to A sharp, because all four span three letters. Only the quality moves.

Quality, and where the two families come from

Rather than memorising a table, generate it. Take the lower note of the interval, build the major scale on it (the next lesson derives that scale properly; for now, C major is the white keys), and look for the upper note. If the upper note belongs to that scale, the interval is perfect when the number is 1, 4, 5 or 8, and major for 2, 3, 6 and 7.

Applying that to C, the intervals up to each white key are: unison 0 semitones, major second 2, major third 4, perfect fourth 5, perfect fifth 7, major sixth 9, major seventh 11, octave 12. That is the full reference table, and it is worth knowing cold.

The split into two families is not arbitrary bookkeeping. The perfect intervals are the ones that appear earliest in the harmonic series and sit at the simplest ratios: 1:1, 4:3, 3:2 and 2:1. They also have the property that inverting one gives another perfect interval, which the major and minor pairs do not. Medieval theory treated exactly these as stable, which is why they inherited the word perfect, and their names have outlived the theory.

From the two families, four more qualities are reachable by moving one semitone at a time. Lower a major interval by a semitone and it becomes minor. Lower a minor interval further, or a perfect interval, by a semitone and it becomes diminished. Raise a major or a perfect interval by a semitone and it becomes augmented. The chains run

for seconds, thirds, sixths and sevenths, and

for unisons, fourths, fifths and octaves. There is no such thing as a minor fifth or a major fourth, and using those names is the fastest way to tell a reader you have not done this lesson.

Example. Name the interval from E flat 4 up to C5.

Count letters first: E, F, G, A, B, C is six, so it is a sixth. Now count semitones: E flat is 3 semitones above C and the C above is 12, so the distance is $12-3=9$ semitones. A major sixth is 9 semitones. So it is a major sixth.

Now you. Name the interval from A3 up to F4.

Building an interval upward

The reverse operation, given a note and an interval name, is the one that appears in every harmony exercise, and doing it in the right order prevents almost all mistakes. Get the letter first, from the number, and then adjust with accidentals until the semitone count is right. Never pick the pitch first and then look for a spelling, or you will write a diminished fourth where an augmented third was wanted.

Example. Write the note a minor sixth above F sharp 4.

The number is 6, so counting six letters from F gives F, G, A, B, C, D: the letter is D, and no other letter is permitted. A minor sixth is 8 semitones. F sharp 4 is 6 semitones above C4, so the target is 14 above C4, which is 2 above C5, which is D5 natural. The answer is D5, and it checks: F sharp to D is a sixth of 8 semitones.

Now you. Write the note an augmented fourth above B flat 3.

The tritone

One interval deserves its own paragraph. Six semitones is exactly half of twelve, and it is the only interval that inverts into itself. It arises as an augmented fourth, from F up to B, and as a diminished fifth, from B up to F, which are the same six semitones spelled two ways, and the neutral name for both is the tritone, so called because it spans three whole tones.

In equal temperament its ratio is $2^{6/12}=\sqrt{2}=1.4142$, which is irrational, so it is as far from a simple whole-number ratio as an interval can get. The nearest small ratios are 45:32 at 590.2 cents and 7:5 at 582.5 cents, neither of them close and neither of them simple. Counterpoint treatises restricted it from the Middle Ages onward, and it later picked up the nickname diabolus in musica, the devil in music; nineteenth-century composers then used it for exactly the reason it had been restricted. When progressions arrive it turns out to be the engine inside the dominant seventh chord, which is where most of the harmonic motion in tonal music comes from.

Inversion halves the work

Move the lower note of an interval up an octave, or the upper note down one, and you have its inversion. C up to E, a major third, becomes E up to C, a sixth. Two rules cover every case.

The numbers add to 9. This falls straight out of inclusive counting: the two intervals together span an octave, whose eight letters are counted twice at the shared note, so $8+1=9$. A third inverts to a sixth, a second to a seventh, a fourth to a fifth, a unison to an octave.

The quality flips: major becomes minor, minor becomes major, augmented becomes diminished, diminished becomes augmented, and perfect stays perfect. Check it in semitones, which must add to 12: a major third is 4, a minor sixth is 8, and $4+8=12$. A perfect fifth is 7 and a perfect fourth is 5, so the perfect family maps onto itself, which is the property the word perfect was pointing at.

Inversion is worth using because it halves what has to be recognised at speed. Sevenths and sixths are hard to judge by eye on the staff; thirds and seconds are easy. Invert mentally, name the easy one, convert.

Example. What is the inversion of a diminished fifth, and does the semitone arithmetic agree?

$9-5=4$, so it is some kind of fourth, and diminished becomes augmented: an augmented fourth. A diminished fifth is 6 semitones and an augmented fourth is 6, and $6+6=12$. The tritone inverting into itself, as promised.

Now you. What is the inversion of a minor seventh, and what are the two semitone counts?

Bigger than an octave

Intervals wider than an octave are compound. Their names continue the count, so an octave plus a second is a ninth, an octave plus a third is a tenth, an octave plus a fifth is a twelfth. To get the simple interval inside, subtract 7 rather than 8, because of inclusive counting again: a ninth reduces to a second, an eleventh to a fourth, a thirteenth to a sixth.

Quality carries over unchanged, so a major tenth is a major third plus an octave. In practice only ninths, elevenths and thirteenths are named as compounds, because they appear as chord extensions; anything else is usually described as "a tenth" or just as its simple form with the octave taken for granted.

Consonance, and what the ear is actually doing

The first lesson showed that simple frequency ratios share upper partials. Now the intervals have names, the list can be written down properly. The perfect consonances are the octave at 2:1, the fifth at 3:2 and the fourth at 4:3. The imperfect consonances are the major third at 5:4, the minor third at 6:5, the major sixth at 5:3 and the minor sixth at 8:5. Everything else, the seconds, the sevenths and the tritone, is treated as dissonant, and the ratios show why: a major second is 9:8 and a major seventh 15:8, which share almost nothing in the audible part of the spectrum.

Two honest qualifications. First, there is a physical mechanism underneath, and it is not the arithmetic itself: partials that fall close together in frequency, within roughly a critical band of the ear, beat against each other and produce the sensation called roughness. Simple ratios avoid this because partials either coincide exactly or separate cleanly, and that is why the perceptual ranking tracks the numerical one.

Second, the categories are historical as much as acoustic, and the fourth is the proof. It has the third-simplest ratio of all, yet counterpoint from the fifteenth century onward treats a fourth above the bass as a dissonance requiring resolution, while a fourth between upper voices is fine. No acoustic account will produce that rule, because it is not an acoustic fact; it is a stylistic one, in a style that thought in terms of the bass. Expect the ratios to explain the broad ranking and expect the fine print to be cultural.

Intervals name distances between any two notes at all. Music does not use any two notes: it draws from a fixed selection, seven of the twelve, chosen so that the intervals inside it come out a particular way. That selection is the scale, and it is next.