Music Performance
Convert cent offsets to frequency ratios and target frequencies.
Ideal for tuning systems and microtonal work.
A microtonal interval temperament converter is an advanced tool that translates pitch information between different tuning systems and temperaments. Temperaments are mathematical schemes for dividing the octave into usable musical intervals; common examples include equal temperament (12 equal divisions), just intonation (based on whole-number frequency ratios), meantone temperament, and various historical or experimental systems. This calculator converts between systems using standard units like cents (1/100th of a semitone), frequency ratios, and Hz values. Musicians, composers, and researchers use this tool when working across tuning traditions or adapting music between temperaments. Understanding temperament conversion is crucial for historically informed performance and cross-cultural musical exploration.
The converter accepts input in multiple formats: cents offset from a reference pitch, frequency ratio (e.g., 5/4), note names in specific tuning systems, or absolute frequencies in Hz. It compares against reference temperaments (12-TET, just intonation, Pythagorean, meantone, etc.) and calculates the equivalent interval across different systems. The tool displays conversions in cents, ratios, frequency differences, and visual fretboard positions. Some versions include built-in tables of historic temperaments for reference.
Convert frequency ratio to cents: Cents = 1200 × log₂(frequency ratio). Convert cents to frequency ratio: Frequency Ratio = 2^(cents/1200). Equal temperament semitone = 100 cents. Just intonation fifth = 701.96 cents (vs 700 in ET). Pythagorean comma = 23.46 cents (frequency ratio 531441/524288). Temperament conversion calculates offset from equal temperament reference.
Tuning refers to the specific pitches assigned to individual notes, while temperament is the mathematical system for dividing the octave into intervals. For example, equal temperament is a specific temperament where the octave divides into 12 equal semitones. Equal temperament is our standard tuning system, but it's historically recent - earlier temperaments like meantone or Pythagorean created different sonorities.
Just intonation is acoustically pure - intervals are based on simple frequency ratios (3/2 for a fifth, 5/4 for a major third). However, you cannot stack just intonation intervals and close a cycle (e.g., 12 just fifths don't equal 7 octaves). Equal temperament sacrifices purity in each interval slightly to allow all keys to work equally well. The compromises are nearly imperceptible to most listeners but enable the chromatic versatility we use today.
A cent is 1/100th of an equal temperament semitone, totaling 1200 cents per octave. Cents quantify pitch differences in units that match human hearing perception (logarithmic). A 5-cent difference is barely perceptible; 10 cents is obvious; 50+ cents is clearly out of tune. Using cents allows precise comparison across different temperaments and tuning systems.
Theoretically yes, but practically it's very difficult. Just intonation works perfectly for single melodies or isolated chords, but when musicians must play together across different chords, the tuning system breaks down mathematically. String players and singers can approximate just intonation through careful listening, but keyboards and fretted instruments are locked into equal temperament (or custom microtonal frets). Some orchestras experiment with just intonation for specific repertoire.
