Superpartient ratio

In mathematics, a superpartient ratio or superpartient number is a rational number that is greater than one and is not superparticular. The term has fallen out of use in modern pure mathematics, but continues to be used in music theory and in the historical study of mathematics.

Mathematically, a superpartient number is a ratio of the form

$\frac{n + a}{n}\, ,$

where a is greater than 1 (a > 1) and is also coprime (reduced to its lowest terms) to n. Ratios of the form $\tfrac{n+1}{n}$ are also greater than one, but are called superparticular ratios and are not superpartient.

Examples

Ratio	n+a/n	Related musical interval	Audio
5:3	3+2/3	Major sixth	<phonos file="Just major sixth on C.mid">Play</phonos>
7:4	4+3/4	Harmonic seventh	<phonos file="Harmonic seventh on C.mid">Play</phonos>
8:5	5+3/5	Minor sixth	<phonos file="Just minor sixth on C.mid">Play</phonos>

See also

• Mathematics of musical scales
• Superparticular number

Further reading

• Partch, Harry (1979). Genesis of a Music, p.68. ISBN 978-0-306-80106-8.