5/1: Difference between revisions
ratios, not steps |
→See also: link 5/4 |
||
| (One intermediate revision by the same user not shown) | |||
| Line 5: | Line 5: | ||
}} | }} | ||
'''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. | '''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. | ||
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 [[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> | 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> | ||
| Line 14: | Line 16: | ||
== See also == | == See also == | ||
* [[5/4]] – its [[octave reduced]] form | |||
* [[Ed5]] – equal divisions of the 5th harmonic | * [[Ed5]] – equal divisions of the 5th harmonic | ||
== References == | == References == | ||
<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