5/1

5/1, the 5th harmonic, pentave or quintuple, is the harmonic past 4/1 and before 6/1. It is two octaves above 5/4, and is the basis of 5-limit harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.

Interval information

Ratio	5/1
Factorization	5
Monzo	[0 0 1⟩
Size in cents	2786.314¢
Names	5th harmonic, pentave, quintuple
Color name	ccy3, cocoyo 3rd
FJS name	[math]\displaystyle{ \text{M17}^{5} }[/math]
Special properties	harmonic, prime harmonic
Tenney norm (log2 nd)	2.32193
Weil norm (log2 max(n, d))	4.64386
Wilson norm (sopfr(nd))	5
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5/1 is the third prime harmonic, after 3/1 and before 7/1.

5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the syntonic comma.

Intervals of 5

5/1 is the first prime harmonic greater than 3, and is thus the first that adds interval qualities other than those found in Pythagorean tuning. Simple ratios of 5 tend to differ from Pythagorean intervals by the syntonic comma 81/80 (21.5 ¢), which contains a single factor of 5 in its denominator. In diatonic interval classification, simple ratios with a factor of 5 in the numerator, such as 5/4, 10/9, and 15/8, are major intervals, while simple ratios with a factor of 5 in the denominator, such as 6/5, 16/15, and 9/5, are minor intervals. Major intervals are flat of their corresponding Pythagorean interval by 81/80; for example, 5/4 is flat of 81/64, the Pythagorean major third, by 81/80. Minor intervals, on the other hand are sharp of their corresponding Pythagorean interval by 81/80; for example, 6/5 is 81/80 sharp of 32/27, the Pythagorean minor third. A simple ratio of 5 iscan generally more consonant than the Pythagorean interval which was modified by 81/80 to reach it; for example, 5/4 is more consonant than 81/64, and 6/5 is more consonant than 32/27. Meantone tempers out 81/80 so that the Pythagorean and 5-limit thirds are equated.

The octave-reduced 5th harmonic, the classical major third 5/4, combines nicely with the octave-reduced 1st and 3rd harmonics, being the unison and perfect fifth with ratios 1/1 and 3/2 respectively, to form the major triad 1–5/4–3/2 (4:5:6), which consists of the 5/4 major third and the 6/5 minor third stacked on top of each other, and is otonal. Its inverse is the minor triad 1–6/5–3/2 (1/(6:5:4) = 10:12:15), which consists of 6/5 and 5/4 stacked on top of each other, and is utonal. More extended chords can be built with these intervals; for example the dominant seventh chord with ratios 1–5/4–3/2–9/5 (20:25:30:36), which is widely used in classical and contemporary music.

See also

• 5/4 – its octave reduced form
• Ed5 – equal divisions of the 5th harmonic