3/9 3. Intervals
A step-by-step guide to naming, comparing, extending, and inverting musical intervals using letter names, semitones, spelling, and register.
The two parts of an
An is the distance between two pitches. It may be heard melodically when the pitches sound one after another or harmonically when they sound at the same time.
Every has two parts:
Its , which describes the letter-name distance.
Its , which describes the precise semitone size.
For example, a major third contains the number third and the quality major. A complete name therefore combines both pieces of information.
Finding the
Start by counting letter names inclusively from the lower note to the upper note. Include both endpoints, and ignore accidentals while finding the number.
Examples:
C–C is a unison, or 1st.
C–D is a 2nd.
C–E is a 3rd.
C–F is a 4th.
C–G is a 5th.
C–A is a 6th.
C–B is a 7th.
C–C, with the upper C in the next register, is an octave, or 8th.
Thus, B–F♯ is a 5th because B–C–D–E–F contains five letter names. B♭–F♯ is also a 5th: the accidentals alter the quality, not the number.
Takeaway: Count letter names first. Do not count only spaces, keyboard keys, or semitones when determining the number.
Determining
Once the number is known, count the semitones and compare the result with the standard for that number. The standard reference sizes are:
Perfect unison: semitones
Major 2nd: semitones
Major 3rd: semitones
Perfect 4th: semitones
Perfect 5th: semitones
Major 6th: semitones
Major 7th: semitones
Perfect octave: semitones
The quality families are:
Perfect family: unisons, 4ths, 5ths, and octaves.
Major/minor family: 2nds, 3rds, 6ths, and 7ths.
Augmented and diminished: possible for either family.
Quality changes follow these relationships:
A perfect or major enlarged by one semitone becomes augmented.
A major reduced by one semitone becomes minor.
A perfect or minor reduced by one semitone becomes diminished.
Examples:
D–F♯ is a 3rd with semitones: major 3rd, or M3.
D–F is a 3rd with semitones: minor 3rd, or m3.
F–B is a 4th with semitones: augmented 4th, or A4.
F♯–B♭ is a 4th with semitones: diminished 4th, or d4.
Takeaway: The number comes from letter names; the quality comes from comparing semitones with the standard size.
Spelling, diatonic distance, and chromatic distance
A and a emphasize different aspects of pitch distance.
A is identified through its letter-name spelling and its written number and quality. A is measured by counting semitones. The two descriptions can agree about the sound while differing about the notation.
For example:
C–D is a diatonic whole step and a major 2nd.
E–F is a diatonic half step and a minor 2nd.
C–C♯ is a chromatic half step, written as an augmented unison, A1.
C–D♭ is a diatonic half step, written as a minor 2nd, m2.
C–F♯ and C–G♭ both contain semitones, but their spellings produce different names:
C–F♯ has four letter names, C–D–E–F, so it is an augmented 4th, A4.
C–G♭ has five letter names, C–D–E–F–G, so it is a diminished 5th, d5.
These may sound the same on an equal-tempered keyboard, but the notated functions are different. This distinction matters for scales, voice leading, and harmonic analysis.
Takeaway: Semitones describe chromatic size, while letter-name spelling determines the written structure.
Simple and compound intervals
A is smaller than or equal to an octave. A is larger than an octave and usually preserves the quality of its simple equivalent.
To find the compound number, add to the corresponding simple number:
2nd becomes 9th.
3rd becomes 10th.
4th becomes 11th.
5th becomes 12th.
6th becomes 13th.
7th becomes 14th.
Octave becomes 15th, or double octave.
Examples:
C4–E4 is a major 3rd; C4–E5 is a major 10th.
D4–A4 is a perfect 5th; D4–A5 is a perfect 12th.
E4–F5 is a minor 9th because E–F is a minor 2nd plus an octave.
For analysis, reduce a by one or more octaves when useful. A major 10th reduces to a major 3rd, and a perfect 12th reduces to a perfect 5th.
Applying
moves one pitch by an octave so that the original lower pitch becomes the upper pitch and the original upper pitch becomes the lower pitch. The pitch classes remain the same, but their registral positions are exchanged.
Use two rules:
The original and inverted numbers add to .
The quality changes according to its family.
Number pairs:
Unison and octave:
2nd and 7th
3rd and 6th
4th and 5th
Quality pairs:
Perfect becomes perfect.
Major becomes minor.
Minor becomes major.
Augmented becomes diminished.
Diminished becomes augmented.
Examples:
P5 inverts to P4.
M3 inverts to m6.
m3 inverts to M6.
M7 inverts to m2.
A4 inverts to d5.
d7 inverts to A2.
C–E is a major 3rd. When inverted, it becomes E–C, a minor 6th: the number changes from 3 to 6, and major changes to minor.
For compound intervals, first reduce to the simple equivalent. A major 10th reduces to M3, which inverts to m6; restored to compound form, the corresponding inversion is m13.
Takeaway: Inversion changes both the number and the quality; changing only one of them is incomplete.
A complete identification strategy
Use this reliable workflow for any fully spelled :
Identify the lower and upper written pitches.
Count the letter names inclusively to determine the number.
Count semitones to determine the quality.
Check whether the is simple or compound.
If needed, reduce a by an octave before analyzing its behavior.
For inversion, make the numbers add to and apply the quality-pair rule.
Common mistakes to avoid:
Counting only semitones and ignoring letter names.
Forgetting that accidentals change quality but not number.
Calling C–C♯ a 2nd; repeated letter names make it an augmented unison.
Treating A4 and d5 as interchangeable written intervals merely because they can sound alike.
Forgetting inclusive counting, as in C–E being a 3rd rather than a 2nd.
Treating every above an octave as unrelated to a .
Inverting only the quality without changing the number.
A compact model is: letters determine the label's number, semitones determine its quality, register determines whether it is simple or compound, and inversion pairs the number and quality by rule.