12 Quintal (center C)

For a general introduction to Just (Rational) Intonation and these presets, please visit Introduction to Just or Rational Intonation (JI).

About this tuning

Commonly known as Ptolemaic tuning, after Claudius Ptolemy, who identified the sequence of proportions 8:9, 9:10, 15:16. The two tones of slightly different size combine to produce a major third in the proportion 4:5, which combines with the semitone to produce a perfect fourthPitches a fourth apart have a 3:4 frequency ratio (in just intonation) or a difference of 500 cents (in 12 tone equal temperament).Visit the link to learn more 3:4. The diatonic major scale with central pitch C combines two conjunct "tetrachords," one rising from G to C, and another rising from C to F, followed by a 8:9 wholetone between F and G to reach an octave 1:2. This tuning produces concordant major triads in the proportion 4:5:6 on C, F, and G. Similarly, the melodic minor scale (Aeolian mode) centered on C is produced by tuning three concordant minor triads in the proportion 10:12:15 on C, F, and G, deriving the notes E♭, A♭, B♭. The two remaining notes of the 12-tone chromatic scale are derived as the Neapolitan D♭ (diatonic minor 2nd, a perfect fifth below A♭) and the secondary Dominant F♯ (augmented fourth, a major third above D). Note that the interval from D to A is NOT a 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. To obtain a concordant major triad on D, A must be raised by a Syntonic commaA comma is a small difference in pitch between the same interval in different tuning systems.Visit the link to learn more in the proportion 80:81 (approximately 21.5 cents).

Learning more

Legend of ASCII Notations

The following ASCII are used to write note names in rational intonation:

For EDO systems, the standard notation uses a degree and division, i.e. 1\31 for the first step of 31edo. If another interval is being divided, it may be placed in brackets or angle brackets afterwards, i.e. 1\11 (3/2) for Wendy Carlos beta.