The intervals of Pythagorean tuning are just intervals involving only powers of two and three. The cent is a logarithmic unit of measure used for musical intervals. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... In music, Just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ...

The fundamental intervals are the superparticular ratios 2/1, 3/2, and 4/3. 2/1 is the octave or diapason (Greek for "across all"). 3/2 is the perfect fifth, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth, diatessaron ("across four"), or sesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances of the Pythagorean system. All other intervals have varying degrees of dissonance, ranging from smooth to rough. Superparticular numbers, also called epimoric ratios, are improper rational fractions of the form Superparticular numbers were written about by Nicomachus in his treatise Introduction to Arithmetic. They are useful in the study of harmony: many musical intervals can be expressed as a superparticular ratio. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... The perfect fourth or diatessaron, abbreviated P4, is one of two musical intervals that span four diatonic scale degrees; the other being the augmented fourth, which is one semitone larger. ... In music, a consonance (Latin consonare, sounding together) is a harmony, chord, or interval considered stable, as opposed to a dissonance, which is considered unstable. ...

The difference between the perfect fourth and the perfect fifth is the tone or major second. This has the ratio 9/8, and it is the only other superparticular ratio of Pythagorean tuning, as shown by Størmer's theorem. A major second is one of three commonly occuring musical intervals that span two diatonic scale degrees; the others being the minor second, which is one semitone smaller, and the augmented second, which is one semitone larger. ...

Two tones make a ditone, a dissonantly wide major third, ratio 81/64. The ditone differs from the just major third (5/4) by the syntonic comma (81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises in meantone temperament. A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ... The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... A minor third is the smaller of two commonly occurring musical intervals that span three diatonic scale degrees. ... Meantone temperament is a system of musical tuning. ...

The difference between the minor third and the tone is the minor semitone or limma of 256/243. The difference between the tone and the limma is the major semitone or apotome ("part cut off") of 2187/2048. Although the limma and the apotome are both represented by one step of 12-pitch equal temperament, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma. Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... When one ascends by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, one eventually reaches a note around seven octaves above the note one started on, which, when lowered to the same octave as the starting point, is 23. ...

External links

• Tonalsoft encyclopedia entry
• Neo-Gothic usage by Margo Schulter