5/1: Difference between revisions
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{{Infobox Interval | |||
| Name = 5th harmonic, pentave, quintuple | |||
| Color name = ccy3, cocoyo 3rd | |||
| Sound = jid_5_1_pluck_adu_dr110.mp3 | |||
}} | |||
'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s 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. | |||
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5/1 is the third [[prime harmonic]], after [[3/1]] and before [[7/1]]. | |||
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< | 5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80|syntonic comma]].<ref>{{OEIS|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</sup> decreases monotonically to zero</ref><ref>{{OEIS|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''</ref> | ||
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== Intervals of 5 == | |||
5/1 is the first prime harmonic greater than [[3/1|3]], and is thus the first that adds [[interval quality|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{{C}}), 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 [[4:5:6|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 [[10:12:15|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 [[20:25:30:36|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 | |||
== References == | |||
<references/> | |||
Latest revision as of 15:38, 20 March 2026
| Interval information |
pentave,
quintuple
prime harmonic
[sound info]
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.
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.[1][2]
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