Equal Temperament Calculator

About the Equal Temperament Calculator

The Equal Temperament Calculator generates frequency tables for any equal division of the octave (EDO). Standard Western music uses 12-TET, but microtonal musicians explore systems like 19-TET, 24-TET, 31-TET, and beyond to access intervals not available in the standard chromatic scale. It is handy when you need a note table for tuning, composition, or instrument setup instead of calculating each step by hand.

In equal temperament, each step has the same frequency ratio, calculated as the Nth root of 2 where N is the number of divisions per octave. This calculator lets you set the number of divisions, choose a reference frequency, and see all note frequencies across multiple octaves.

The tool also compares how well different EDO systems approximate pure intervals like perfect fifths, major thirds, and minor thirds. This helps composers and theorists choose the right tuning system for their musical goals — whether maximizing consonance, exploring exotic scales, or matching historical tuning practices.

When This Page Helps

Use this calculator when you need frequency tables for a tuning system other than 12-TET, or when you want to compare equal temperaments against just intervals. It is useful for microtonal composition, tuning research, and instrument setup when you need to see the whole octave layout at once. That is especially helpful when you want to compare multiple EDO systems side by side before retuning an instrument or synth.

How to Use the Inputs

1. Enter the number of equal divisions per octave (12 for standard, or any custom number).
2. Set the reference frequency (A4 = 440 Hz is standard).
3. Choose the octave range to display.
4. View the complete frequency table with note numbers and cent values.
5. Compare interval approximations against just intonation ratios.
6. Use preset buttons for common tuning systems.

Example Calculation

Tips & Best Practices

• 12-TET is a compromise: its fifths are 2 cents flat and thirds are 14 cents sharp compared to just intonation.
• 19-TET has a step size of ~63.2 cents — better major thirds than 12-TET.
• 31-TET provides excellent approximations of most just intervals.
• 53-TET is often cited as the best EDO for approximating just intonation ratios.
• Use Scala (.scl) files to load custom tunings into most software synths.
• Quarter-tone (24-TET) notation is standardized with half-sharp and half-flat accidentals.

History of Equal Temperament

Equal temperament was first described mathematically by Zhu Zaiyu in China (1584) and Simon Stevin in Europe (1585). Before equal temperament, various unequal temperaments like meantone and well-temperament were used, each favoring certain keys at the expense of others.

J.S. Bach's "Well-Tempered Clavier" is often mistakenly cited as promoting equal temperament, but it actually demonstrated well-temperament — a system where all keys are usable but have distinct characters. True equal temperament became standard only in the 19th century with the rise of chromatic harmony and modulation to distant keys.

Microtonal Music

Microtonal music uses intervals smaller than the semitone. Many world music traditions naturally use microtonal intervals — Arabic maqam, Turkish makam, Indian raga, Indonesian gamelan, and Thai classical music all feature scales that don't align with 12-TET.

In Western music, composers like Charles Ives, Alois Haba, and more recently Sevish and Brendan Byrnes have explored microtonal tunings. The rise of electronic instruments has made microtonal music more accessible than ever.

Choosing an EDO System

The ideal EDO depends on your priorities. If you want better thirds, try 19 or 31. If you want quarter tones, use 24. If you want maximum consonance, 53-EDO approximates most just intervals within 1-2 cents. For experimental sound, prime-numbered EDOs like 13, 17, or 23 create unfamiliar interval palettes.

Frequently Asked Questions

• What does TET or EDO mean?

  TET stands for Equal Temperament, and EDO stands for Equal Division of the Octave. 12-TET and 12-EDO are identical — 12 equal steps per octave.

• Why use anything other than 12-TET?

  12-TET compromises interval purity for versatility. 31-TET, for example, has nearly pure major thirds (386 cents vs 387 in just intonation). Different EDOs suit different musical aesthetics.

• What is 24-TET used for?

  24-TET (quarter-tone system) is used in Arabic maqam music and some 20th-century Western art music. It adds pitches exactly between each standard semitone.

• How do I play microtonal music?

  Software synths, refretted guitars, specially tuned keyboards, and fretless instruments all support microtonality. Many DAWs support custom tuning via Scala files.

• Which EDO best approximates just intonation?

  53-EDO is famous for closely matching many just intervals. 31-EDO excels at thirds. 19-EDO balances thirds and fifths well. The "best" depends on which intervals matter most to you.

• Is A440 the only standard?

  A440 was standardized in 1955 but is not universal. Baroque ensembles often use A415. Some modern orchestras tune to A442-443. Alternative tuning advocates prefer A432.