Measuring Distance

Introduction: The Syllables of Music

If individual notes are the letters of the musical alphabet, and chords are the words, then intervals are the syllables. An interval is the exact distance between two pitches.

Every time you write a melody for a soprano or plot a bass line for a cello, you are stringing intervals together. Measuring them quickly and accurately is the most important mechanical skill in this course. Every chord, scale, and voice-leading rule you meet from here on is built out of intervals.

Throughout this chapter, click any example to hear it: first one note after the other, then both together.

Melodic and Harmonic Intervals

When the two notes sound one after the other, the interval is melodic. When they sound at the same time, it is harmonic. A melodic interval is part of a tune; a harmonic interval is part of a chord. We name both the same way.

Step 1: Size (the Number)

Every interval has a two-part name: a size (a number) and a quality (major, minor, perfect, augmented, or diminished). Find the size first.

To find the size, count the letter names from the bottom note to the top note, counting both ends. C up to E is C–D–E: three letters, so a 3rd. C up to C is an octave (an 8th), and two identical pitches form a unison (a 1st).

Two shortcuts make this nearly instant on the staff:

• Odd sizes (3rd, 5th, 7th) go line to line, or space to space.
• Even sizes (2nd, 4th, 6th, octave) go line to space, or space to line.

Accidentals never change the size. C to E, C to E♭, and C♯ to E♭ are all 3rds, because they all span the letters C, D, E. Accidentals change only the quality, which is Step 2.

Step 2: Quality (the Exact Distance)

The size alone isn't precise enough: C up to E and C up to E♭ are both 3rds, but they sound different. The quality tells us exactly how many half steps the interval contains.

Sizes fall into two families, and each family uses different qualities:

• The perfect family (unison, 4th, 5th, octave) can be perfect, augmented, or diminished. These intervals are never major or minor.
• The major/minor family (2nd, 3rd, 6th, 7th) can be major, minor, augmented, or diminished. These intervals are never perfect.

Moving one step along a ladder adds or removes one half step without changing the size:

Here is every common interval with its half-step count. You don't need to memorize the whole table, but the half-step counts are worth knowing cold.

Two Ways to Find the Quality

Method 1: The major-scale method. Imagine the bottom note is the tonic of a major scale. If the top note belongs to that major scale, the interval is major (2nds, 3rds, 6ths, 7ths) or perfect (unisons, 4ths, 5ths, octaves). If the top note doesn't fit the scale, count how far it is from the note that does, and move along the ladder.

This method works beautifully as long as the bottom note is a comfortable key (C, G, F, D, B♭…). It breaks down when the bottom note is something like D♯ or F♭, because nobody wants to think in the key of D♯ major. For those, use Method 2.

Method 2: The white-key method. First, learn the intervals between white keys (no accidentals). There are only a few exceptions to memorize:

Then, to name any interval:

1. Cover up the accidentals and name the white-key interval.
2. Put the accidentals back one at a time and adjust:
   • Raising the top note, or lowering the bottom note, makes the interval larger.
   • Lowering the top note, or raising the bottom note, makes the interval smaller.
   • If both notes get the same accidental (both sharp, or both flat), the quality doesn't change.

Enharmonic Intervals

Two intervals that sound identical but are spelled differently are enharmonic. F up to B and F up to C♭ both span six half steps and sound exactly the same on a piano. But F–B is a 4th (augmented), while F–C♭ is a 5th (diminished), because they span a different number of letter names.

Spelling matters because it tells a performer, and you, where the music is going. You'll meet the most important example in harmonic minor (Chapter 3): the step from the 6th to the raised 7th scale degree is an augmented 2nd, not a minor 3rd, even though the two sound the same.

Compound Intervals

An interval larger than an octave is a compound interval. A 9th is a 2nd plus an octave, a 10th is a 3rd plus an octave, and so on. To reduce a compound interval to its simple form, subtract 7 from its number. The quality stays the same: a major 10th is a major 3rd plus an octave.

In four-part writing, the distance between the bass and the tenor is often compound. Most of the time we name it by its simple equivalent: a bass on C3 and a soprano on E5 form "a 3rd" for the purposes of harmony.

Consonance vs. Dissonance

Intervals have distinct physical characters, rooted in the overtone series. When two notes sound together, we sort the result into three families:

• Perfect consonances (P1, P5, P8): hollow, pure, and very stable.
• Imperfect consonances (M3, m3, M6, m6): sweet and full, stable but more colorful. These are the bread and butter of four-part choral writing.
• Dissonances (M2, m2, M7, m7, and every augmented or diminished interval, including the tritone): they clash and sound unstable, demanding motion.

The special case of the perfect 4th. Acoustically, the P4 is a consonance. In common-practice harmony, though, a P4 that sits above the bass is treated as a dissonance and must resolve. Between two upper voices it is perfectly fine. This is why second-inversion chords, which have a 4th above the bass, need special handling later in the course.

Dissonance isn't bad. Music without it is boring. Dissonance creates the tension that pushes the music forward into a consonant resolution, and almost every voice-leading rule in this course is about controlling how dissonances are introduced and resolved.

Inverting Intervals

When you move the bottom note of an interval up an octave, so it becomes the top note, you have inverted the interval.

The sizes follow the Rule of 9: an interval and its inversion always add up to 9. A 3rd inverts into a 6th; a 2nd inverts into a 7th; a 4th inverts into a 5th; a unison inverts into an octave.

The qualities flip on a predictable axis:

• Major ↔ minor
• Augmented ↔ diminished
• Perfect stays perfect

Inversion is also everyday practical knowledge for pianists and guitarists: you can rearrange the spacing of a chord, flipping its intervals, while keeping the same harmony.

Try It: The Interval Explorer

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