Music Interval Calculator

Pitch & frequency calculator • 2026 audio tools

Music Interval Formula:

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\( f_2 = f_1 \times 2^{\frac{n}{12}} \)

Where:

  • \( f_1 \) = Starting frequency
  • \( f_2 \) = Resulting frequency
  • \( n \) = Number of semitones between notes

This formula calculates frequency ratios in equal temperament tuning system.

Common Intervals:

  • Unison: 0 semitones (1:1 ratio)
  • Minor 2nd: 1 semitone (16:15)
  • Major 2nd: 2 semitones (9:8)
  • Perfect 4th: 5 semitones (4:3)
  • Perfect 5th: 7 semitones (3:2)
  • Octave: 12 semitones (2:1)

Example: From A4 (440 Hz) to E5 (perfect 5th):

\( f_2 = 440 \times 2^{\frac{7}{12}} = 440 \times 1.4983 = 659.26 \text{ Hz} \)

Thus, E5 is approximately 659.26 Hz.

Interval Input

Tip: A4 = 440 Hz is standard concert pitch.

Advanced Options

Results

E5
Target Note
659.26
Target Frequency (Hz)
1.4983
Frequency Ratio
700
Interval Size (cents)

Comprehensive Music Theory Guide

What is a Music Interval?

A music interval is the distance in pitch between two notes. It can be described in terms of semitones (half steps) or whole tones (whole steps). Intervals are fundamental to music theory, forming the basis for chords, scales, and harmonic relationships. They define the harmonic content and melodic movement in music.

Interval Calculation Formula

The frequency relationship in equal temperament is calculated using:

\(f_2 = f_1 \times 2^{\frac{n}{12}}\)

Where:

  • \(f_1\) = Starting frequency
  • \(f_2\) = Target frequency
  • \(n\) = Number of semitones between notes

Common Intervals
1
Unison: 0 semitones (same note), frequency ratio 1:1.
2
Minor 2nd: 1 semitone, frequency ratio 16:15.
3
Major 2nd: 2 semitones, frequency ratio 9:8.
4
Perfect 4th: 5 semitones, frequency ratio 4:3.
5
Perfect 5th: 7 semitones, frequency ratio 3:2.
Tuning Systems
  • Equal Temperament: 12 equally spaced semitones per octave (modern standard)
  • Just Intonation: Pure frequency ratios (more consonant, limited key changes)
  • Pythagorean: Based on perfect 5ths (used historically)
  • Meantone: Compromise between pure intervals and key availability
Production Applications
  • Harmony: Building chords using interval relationships
  • Counterpoint: Creating independent melodic lines
  • Modulation: Changing keys using pivot intervals
  • Microtonal: Using intervals smaller than semitones
  • Frequency Analysis: Identifying harmonic content

Interval Fundamentals

What is an Interval?

Distance in pitch between two musical notes.

Formula

\(f_2 = f_1 \times 2^{\frac{n}{12}}\)

Where f₂=target frequency, f₁=start freq, n=semitones.

Key Rules:
  • 1 octave = 12 semitones
  • Perfect intervals are most consonant
  • Intervals form the basis of chords

Production Tips

Equal Temperament

Modern tuning system with equal semitone spacing.

Interval Construction
  1. Select root note and frequency
  2. Determine interval size in semitones
  3. Apply frequency formula
  4. Verify harmonic relationship
Considerations:
  • Perfect 5th: 3:2 frequency ratio
  • Octave: 2:1 frequency ratio
  • Consonance vs dissonance varies by interval

Music Theory Learning Quiz

Question 1: Multiple Choice - Interval Definition

What is a musical interval?

Solution:

The answer is B) The distance in pitch between two notes. A musical interval measures the difference in pitch between two notes. It can be described in terms of semitones (half steps) or whole tones (whole steps). For example, the interval between C and G is a perfect fifth, which spans 7 semitones.

Pedagogical Explanation:

Understanding intervals is fundamental to music theory. They form the building blocks of scales, chords, and melodies. The size of an interval is determined by the number of semitones between the two notes. This concept is essential for harmony, counterpoint, and composition.

Key Definitions:

Interval: Distance in pitch between two notes

Semitone: Smallest interval in Western music (half step)

Whole Tone: Two semitones (whole step)

Important Rules:

• Intervals measure pitch distance, not time

• Size measured in semitones

• Form basis for chords and scales

Tips & Tricks:

• Count half steps between notes to identify intervals

• Memorize common intervals (4th, 5th, octave)

Common Mistakes:

• Confusing intervals with rhythm or duration

• Mixing up interval size with volume

Question 2: Interval Formula Application

Calculate the frequency of the note that is a perfect fifth above A4 (440 Hz). Use the formula f₂ = f₁ × 2^(n/12) where n = 7 semitones for a perfect fifth. Show your work.

Solution:

Using the interval formula: \(f_2 = f_1 \times 2^{\frac{n}{12}}\)

Given:

  • f₁ = 440 Hz (A4)
  • n = 7 semitones (perfect fifth)

Step 1: Calculate the exponent

\(\frac{n}{12} = \frac{7}{12} = 0.5833\)

Step 2: Calculate the multiplier

\(2^{0.5833} = 1.4983\)

Step 3: Calculate the resulting frequency

f₂ = 440 × 1.4983 = 659.25 Hz

Therefore, E5 (perfect fifth above A4) has a frequency of approximately 659.25 Hz.

Pedagogical Explanation:

This calculation demonstrates the mathematical relationship between musical intervals and frequencies. The equal temperament system divides the octave into 12 equal parts on a logarithmic scale, ensuring that the same interval produces the same frequency ratio regardless of the starting note.

Key Definitions:

Equal Temperament: Modern tuning system with equal semitone spacing

Logarithmic Scale: Scale where each step is a multiple of the previous

Perfect Fifth: Consonant interval spanning 7 semitones

Important Rules:

• Use the formula f₂ = f₁ × 2^(n/12)

• Perfect 5th = 7 semitones

• Verify calculations with standard ratios

Tips & Tricks:

• Remember: Perfect 5th ≈ 1.5× frequency

• Perfect 4th = 5 semitones

• Octave = 12 semitones

Common Mistakes:

• Forgetting to divide by 12 in the exponent

• Using linear instead of exponential calculation

• Confusing the number of semitones for different intervals

Question 3: Word Problem - Chord Construction

A major triad consists of a root note, a major third (4 semitones above root), and a perfect fifth (7 semitones above root). If the root is C4 (261.63 Hz), calculate the frequencies of all three notes in the chord. Show your work.

Solution:

Given: Root note C4 = 261.63 Hz

Step 1: Calculate major third (E4, 4 semitones above root)

f₂ = 261.63 × 2^(4/12) = 261.63 × 2^(0.3333) = 261.63 × 1.2599 = 329.63 Hz

Step 2: Calculate perfect fifth (G4, 7 semitones above root)

f₂ = 261.63 × 2^(7/12) = 261.63 × 2^(0.5833) = 261.63 × 1.4983 = 392.00 Hz

Step 3: List the chord notes

C Major Triad: C4 (261.63 Hz), E4 (329.63 Hz), G4 (392.00 Hz)

Therefore, the C major chord consists of frequencies 261.63 Hz, 329.63 Hz, and 392.00 Hz.

Pedagogical Explanation:

This problem demonstrates how intervals combine to form chords. A major triad is built using specific intervals: root, major third, and perfect fifth. These intervals create the characteristic consonant sound of major chords. Understanding these relationships is essential for composition, harmony, and sound synthesis.

Key Definitions:

Major Triad: Chord with root, major 3rd, perfect 5th

Major Third: Interval spanning 4 semitones

Perfect Fifth: Interval spanning 7 semitones

Important Rules:

• Major triad = root + major 3rd + perfect 5th

• Major 3rd = 4 semitones

• Perfect 5th = 7 semitones

Tips & Tricks:

• Major 3rd ≈ 1.26× frequency

• Perfect 5th ≈ 1.5× frequency

• Use intervals to build any chord type

Common Mistakes:

• Confusing major 3rd with perfect 5th

• Forgetting the semitone counts for intervals

• Arithmetic errors with exponential calculations

Question 4: Application-Based Problem - Just vs Equal Temperament

In just intonation, a perfect fifth has a frequency ratio of exactly 3:2 (1.5). In equal temperament, it's 2^(7/12) ≈ 1.4983. Calculate the difference in cents between these two tunings for a perfect fifth. (Hint: cents = 1200 × log₂(ratio), where log₂(x) = ln(x)/ln(2))

Solution:

Step 1: Calculate cents for just intonation (3:2 ratio)

Cents = 1200 × log₂(3/2) = 1200 × ln(1.5)/ln(2) = 1200 × 0.4055/0.6931 = 1200 × 0.5850 = 702 cents

Step 2: Calculate cents for equal temperament (2^(7/12) ratio)

Cents = 1200 × log₂(2^(7/12)) = 1200 × (7/12) = 700 cents

Step 3: Calculate the difference

Difference = 702 - 700 = 2 cents

Therefore, the equal temperament perfect fifth is 2 cents flatter than the just intonation perfect fifth.

Pedagogical Explanation:

This demonstrates the compromise inherent in equal temperament tuning. While it allows for modulation to any key, individual intervals are slightly out of tune compared to their just intonation counterparts. The difference of 2 cents for a perfect fifth is barely perceptible to most listeners, but adds up across multiple intervals in complex harmonies.

Key Definitions:

Just Intonation: Tuning system with pure frequency ratios

Equal Temperament: System with equal semitone spacing

Cent: Unit of musical interval (1/100 of semitone)

Important Rules:

• Cents = 1200 × log₂(ratio)

• Just perfect 5th = 702 cents

• Equal tempered 5th = 700 cents

Tips & Tricks:

• Equal temperament = compromise system

• Just intonation = pure but key-limited

• 1 cent ≈ smallest perceivable difference

Common Mistakes:

• Forgetting the 1200 multiplier in cent calculation

• Confusing log₂ with log₁₀

• Misapplying the logarithm formula

Question 5: Multiple Choice - Interval Quality

Which of the following correctly orders these intervals from smallest to largest in terms of semitones?

Solution:

The answer is B) Major 2nd, Minor 3rd, Perfect 4th. Here are the semitone counts:

  • Major 2nd = 2 semitones
  • Minor 3rd = 3 semitones
  • Perfect 4th = 5 semitones

So the correct order from smallest to largest is: Major 2nd (2) < Minor 3rd (3) < Perfect 4th (5)

Pedagogical Explanation:

Understanding the size of intervals in semitones is crucial for music theory. The pattern of semitones determines the quality of the interval. A major 2nd is smaller than a minor 3rd, which is smaller than a perfect 4th. This knowledge is essential for scale construction, chord formation, and harmonic analysis.

Key Definitions:

Major 2nd: 2 semitones (whole step)

Minor 3rd: 3 semitones

Perfect 4th: 5 semitones

Important Rules:

• Major 2nd = 2 semitones

• Minor 3rd = 3 semitones

• Perfect 4th = 5 semitones

Tips & Tricks:

• Memorize semitone counts for common intervals

• Major intervals are larger than minor

• Perfect intervals are pure and consonant

Common Mistakes:

• Confusing the size of major/minor intervals

• Forgetting that major intervals are larger

• Mixing up semitone counts for different intervals

Music Interval Calculator

FAQ

Q: What's the difference between just intonation and equal temperament?

A: Just intonation uses pure frequency ratios (like 3:2 for perfect 5th), creating perfectly consonant intervals in specific keys. Equal temperament divides the octave into 12 equal semitones, allowing modulation to any key but making intervals slightly impure.

Mathematically, in just intonation:

Perfect 5th = 3:2 = 1.5 ratio

In equal temperament:

Perfect 5th = 2^(7/12) ≈ 1.4983 ratio

The difference is only 2 cents, but just intonation sounds more consonant in its intended key while equal temperament allows for key changes.

Q: How do I calculate the frequency of a note that's several octaves away?

A: Each octave doubles the frequency. If A4 is 440 Hz:

  • A5 (one octave higher) = 440 × 2 = 880 Hz
  • A3 (one octave lower) = 440 ÷ 2 = 220 Hz
  • A6 (two octaves higher) = 440 × 2² = 1760 Hz

For any note, use the formula: \( f_n = f_0 \times 2^{\frac{n}{12}} \) where n is the number of semitones from the reference note.

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This calculator was created by our Music & Audio Team , may make errors. Consider checking important information. Updated: April 2026.