# Superparticular number

**Superparticular numbers**, also called **epimoric ratios**, are ratios of the form

Thus:

A superparticular number is when a great number contains a lesser number, to which it is compared, and at the same time one part of it. For example, when 3 and 2 are compared, they contain 2, plus the 3 has another 1, which is half of two. When 3 and 4 are compared, they each contain a 3, and the 4 has another 1, which is a third apart of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc.

—Throop (2006),^{[1]}

Superparticular numbers were written about by Nicomachus in his treatise "Introduction to Arithmetic". They are useful in the study of harmony: many musical intervals can be expressed as a superparticular ratio. In this application, Størmer's theorem can be used to list all possible superparticular numbers for a given limit; that is, all ratios of this type in which both the numerator and denominator are smooth numbers.

In graph theory, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise as the possible values of the upper density of an infinite graph.

These ratios are also important in visual harmony. Most flags of the world's countries have a ratio of 3:2 between their length and height. Aspect ratios of 4:3 and 3:2 are common in digital photography. Aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.

Ratio | Name | Related musical interval | Audio |
---|---|---|---|

2:1 | duplex | octave | Play (help·info) |

3:2 | sesquialterum | perfect fifth | Play (help·info) |

4:3 | sesquitertium | perfect fourth | Play (help·info) |

5:4 | sesquiquartum | major third | Play (help·info) |

6:5 | sesquiquintum | minor third | Play (help·info) |

9:8 | sesquioctavum | major second | Play (help·info) |

10:9 | sesquinona | minor tone | Play (help·info) |

16:15 | just diatonic semitone | Play (help·info) | |

25:24 | just chromatic semitone | Play (help·info) | |

81:80 | syntonic comma | Play (help·info) | |

4375:4374 | ragisma | Play (help·info) |

The root of some of these terms comes from Latin *sesqui-* "one and a half" (from *semis* "a half" + *-que* "and") describing the ratio 3:2.

## See also

## Sources

- ↑ Throop, Priscilla (2006).
*Isidore of Seville's Etymologies: Complete English Translation, Volume 1*, p.III.6.12,n.7. ISBN 978-1-4116-6523-1.

## Further reading

- {{#invoke:Citation/CS1|citation

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## External links

- An Arithmetical Rubric by Siemen Terpstra, about the application of superparticular numbers to harmony.Template:Dead link
- Superparticular numbers applied to construct pentatonic scales by David Canright.
*De Institutione Arithmetica, liber II*by Anicius Manlius Severinus Boethius