5/4

In 5-limit just intonation, 5/4 is the frequency ratio between the 5th and 4th harmonics. It has been called the just major third, classic(al) major third, or ptolemaic major third to distinguish it from other intervals in that neighborhood. Measuring about 386.3 ¢, it is about 13.7模板:C away from 12edo's major third of 400模板:C. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for 5-limit harmony. It is distinguished from the Pythagorean major third of 81/64 by the syntonic comma of 81/80, which measures about 21.5模板:C, and from the Pythagorean diminished fourth of 8192/6561 by the schisma, which measures about 1.95模板:C. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".

In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in File: 5-4.mp3 melodically in singing into a resonant udderbot (from the fundamental up to 5 and then noodling between 5 and 4).

Approximations by edos

平均律	5/4的步数	绝对误差(ie)	相对误差
3	1	+126.4941	+3.42%

See also

• 8/5 – its octave complement
• 6/5 – its fifth complement
• 16/15 – its fourth complement
• 5/2 – the interval up one octave which sounds even more consonant
• Ed5/4
• Gallery of just intervals
• List of superparticular intervals