Editing Syntonic comma (section)

==Syntonic comma in the history of music==
{{multiple image
| align     = right
| direction = vertical
| width     = 300
| image1    = Syntonic comma minor third Cuisenaire rods just.png
| caption1  = Syntonic comma (the mismatch at the top)
| image2    = Syntonic comma major third Cuisenaire rods ET.png
| caption2  = is tempered out in 12TET (bottom)
}}
{{multiple image
| align     = right
| direction = vertical
| width     = 300
| image1    = Syntonic comma major and minor tone Cuisenaire rods just.png
| image2    = Septimal and syntonic comma whole tones Cuisenaire rods ET.png
| footer    = Syntonic comma, such as between the 9/8 (203.91 approximate cents) and 10/9 (182.40 approximate cents) major and minor tones (top), is tempered out in 12TET, leaving one 200 cent tone (bottom).
}}
The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the [[perfect fifth]] and its inversion, the [[perfect fourth]]. The Pythagorean [[major third]] (81:64) and [[minor third]] (32:27) were [[Consonance and dissonance|dissonant]], and this prevented musicians from using [[Triad (music)|triad]]s and [[Chord (music)|chord]]s, forcing them for centuries to write music with relatively simple [[Texture (music)|texture]].

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 (συντονόν διατονικός, ''syntonón diatonikós'', from συντονός + διάτονος). The term ''syntonón'' was based on [[Aristoxenus]], and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to μαλακόν (''malakón'', from μαλακός), 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 could be made [[Consonance and dissonance|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 [[just intonation|justly intonated]] ratio of

:<math> {81\over64} \cdot {80\over81} = {{1\cdot5}\over{4\cdot1}} = {5\over4}</math>

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

:<math> {32\over27} \cdot {81\over80} = {{2\cdot3}\over{1\cdot5}} = {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 interval|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 chord|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 (music)|texture]], such as [[Polyphony|polyphonic music]], or melody with [[Homophony|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 temperament]]s.