Music Performance
Calculate cents and interval names between two frequencies.
What this calculator does
Musical intervals are measured in cents, a logarithmic unit that divides one semitone (the smallest interval in Western music) into 100 equal parts. Cents provide precise measurements for tuning, microtonal composition, and comparative acoustics. Understanding cents is essential for musicians working with alternative tuning systems, electronic instruments, synthesizers, and sound design. Two frequencies that differ by even 1 cent are imperceptibly close to the human ear, yet microtonal music and historical temperaments may differ by 5-50 cents from equal temperament, creating distinctly different harmonic characters. This calculator converts frequency ratios into cents, revealing the exact intervallic distance between any two pitches.
How it works
The calculator converts frequency ratios into cents using logarithmic mathematics. The formula is: cents = 1200 × log₂(frequency₂/frequency₁), where 1200 is the number of cents in an octave. The logarithmic base-2 reflects octave relationships; doubling frequency equals one octave (1200 cents). The calculator also derives the semitone count and identifies the nearest equal-tempered interval name. This reveals both the precise deviation and the closest conventional interval, helping musicians understand if a frequency pair represents a standard interval or a non-standard microtonal relationship.
Formula
cents = 1200 × log₂(f₂/f₁); semitones = cents/100. The 1200 constant reflects one octave's cent value; log₂ ensures octave doubling equals 1200. This logarithmic conversion makes linear measurements possible for exponential frequency relationships.
Tips for using this calculator
- Equal temperament uses 100-cent semitones; pure intervals based on simple ratios (3:2 fifth, 5:4 major third) often deviate 1-14 cents, creating noticeable tonal differences in harmonic music
- Synthesizer detuning typically ranges ±5-50 cents; small detunes (5-10 cents) create subtle width, while larger amounts produce noticeable pitch variations
- Microtonal composition systems use cent divisions: Arabic music (~50 cents quarter-tones), Indian classical (~20-40 cents), and 31-tone equal temperament (≈39 cents per step)
- When tuning by ear, most musicians can perceive 5-10 cent differences; deviations under 5 cents usually go unnoticed
- Use cent measurements to analyze historical temperaments or microtonal scales, comparing their harmonic content to equal temperament
Frequently asked questions
Why measure intervals in cents instead of just using frequency ratios?
Cents provide a linear measurement system for logarithmic pitch relationships. Frequency ratios are non-linear and difficult to compare—a 3:2 ratio and 5:3 ratio don't directly show their relative sizes. Cents linearize this; 700 cents (perfect fifth) and 400 cents (major third) clearly show their relationship. Additionally, the equal-tempered semitone of 100 cents provides a familiar reference point for musicians accustomed to standard 12-tone intervals.
What's the difference between a 700-cent interval in different temperaments?
In equal temperament, 700 cents is a perfect fifth. However, the mathematically pure fifth (just intonation, 3:2 ratio) is approximately 701.96 cents—about 2 cents higher. Pythagorean tuning's fifth is closer to just intonation. These differences are subtle individually, but across multiple intervals in a chord, they accumulate, creating different harmonic colors. Just intonation sounds 'warmer' and more resonant; equal temperament sounds more uniform and modern.
How accurate do frequency measurements need to be for useful cent calculations?
Frequency accuracy directly affects cent precision. For perceivable cent differences (5+ cents), you need frequency accuracy within about 0.5%. A 1-cent difference requires roughly 0.042% frequency accuracy. For synthesis or tuning, this is easily achievable with digital tools, but acoustic instruments are more challenging. Measurement to the nearest Hz is usually sufficient for practical applications.
Can cents help me compose in alternative tuning systems?
Absolutely. By calculating cent values for desired frequency ratios or microtonal divisions, you create a tuning map. For example, quarter-tone music uses 50-cent intervals; 31-tone equal temperament uses ~38.7-cent steps. Many DAWs and synthesizers allow pitch offsets in cents, letting you directly implement these systems. This calculator helps you determine exact offset values for your compositional goals.
