Introduction to EDO tunings
By Marc SabatThe chromatic scale found on most keyboard instruments is made from 12 semitonesIn 12-tone equal temperament, a semitone is the interval between any two adjacent pitches.Visit the link to learn more, which are repeated in every octavePitches an octave apart have a 1:2 (“one-to-two”) frequency ratio. If one pitch is at 200 Hz, a pitch an octave higher is at 400 Hz. A pitch an octave lower is at 100 Hz.Visit the link to learn more. These semitones sound equal in size because for each rising step frequencyFrequency is the rate of repetition of a waveform, measured in cycles per second or Hertz (Hz). As frequency increases or decreases, we perceive the pitch rising or falling.Visit the link to learn more increases by the same proportion. After 12 notes, the frequency is doubled, making an octave. This tuning system is commonly called 12edo or 12-TET. "EDO" stands for "equal divisions of an octave". and "-TET" stands for "-tone equal temperament". In general, EDOs are tuned by dividing the octave into different numbers of equal-sounding steps.
Some EDOs, like 12, 19, 31, 41, 43, 53, 55, 72, closely approximate certain consonant intervalsAn interval is the distance between two pitches. It can be measured as a ratio between their frequencies or in cents.Visit the link to learn more (small-number ratios). These divisions are often used as temperaments approximating JI (rational or just intonation). Other EDOs, like 5, 7, 11, 13, 35, do not include a consonant perfect fifthPitches a fifth apart have a 2:3 frequency ratio (in just intonation) or a difference of 700 cents (in 12 tone equal temperament).Visit the link to learn more, and are therefore part of a family of less familiar or "xenharmonic" tunings. Harmony in EDOs can be treated both "tonally," by evoking the harmonic series, and "atonally," by evoking patterns of combinations and symmetries that can be transposed to any scale degree.
EDOs belong to a larger family of tunings that take a single interval and stack it repeatedly across the range of frequencies used in music. This is done by choosing a reference frequency (for example A4 = 440 Hz) and repeatedly multiplying or dividing by the same factor. In EDOs this factor is chosen so that a certain number of intervals stack to make an octave. But in fact, any interval, consonant or dissonant, might be divided in a similar way. The Wendy Carlos tunings alpha, beta, and gamma, for example, are based on dividing a perfect fifth into various parts, and the Bohlen-Pierce tuning divides a perfect twelfth (a perfect fifth plus an octave).