thumb|right|Just perfect fifth on D[[File:Just perfect fifth on D.mid The perfect fifth above D (A+) is a syntonic comma higher than the (A) that is a just major sixth above C, assuming C and D are apart.

The syntonic tempering dates to Didymus the Musician, whose tuning of the diatonic genus of the tetrachord replaced one 9:8 interval with a 10:9 interval (lesser tone), obtaining a just major third (5:4) and semitone (16:15). This was later revised by Ptolemy (swapping the two tones) in his "syntonic diatonic" scale (, , from + ). The term was based on Aristoxenus, and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to (, from ), translated as "relaxed" (conventional "soft"), referring to looser strings (hence flatter or "softer").

This was rediscovered in the late Middle Ages, where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds (as opposed to the harder-to-reach diminished fourths and augmented seconds) could be made consonant. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C–E (a major third), and E-G (a minor third) become just. Namely, C–E is narrowed to a justly intonated ratio of

:<math> {81\over64} \cdot {80\over81} = = {5\over4}</math>

and at the same time E–G is widened to the just ratio of

:<math> {32\over27} \cdot {81\over80} = = {6\over5}</math>

The drawback is that the fifths A–E and E–B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C–G stays consonant, since only E has been flattened (C–E&nbsp;×&nbsp;E–G =&nbsp;5/4&nbsp;×&nbsp;6/5 =&nbsp;3/2), and can be used together with C–E to produce a C-major triad (C–E–G). These experiments eventually brought to the creation of a new tuning system, known as quarter-comma meantone, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by narrowing each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complex texture, such as polyphonic music, or melody with instrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to the syntonic temperament continuum, including meantone temperaments.

Comma pump

[[File:Comma pump Benedetti.png|300px|thumb|Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.

</references> <!-- end "refs=" -->

  • Indiana University School of Music: Piano Repair Shop: Harpsichord Tuning, Repair, and Temperaments: "What is the Syntonic Comma?"
  • Tonalsoft: "Syntonic-comma"
  • Explanation of comma drift