Syntonic comma

In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes.

This is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments[clarification needed].

Mathematically, by Størmer's theorem, 81:80 is the closest superparticular ratio possible with regular numbers as numerator and denominator.

A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose prime factors are limited to 2, 3, and 5.

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 consonant.

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.

Therefore, the above-mentioned sequence is equivalent to: or, by grouping together similar intervals, This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four P5 plus one m6 minus three P8).

In Helmholtz-Ellis notation, a syntonic comma is indicated with up and down arrows added to the traditional accidentals.