Superparticular number
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Superparticular numbers, also called epimoric ratios, are improper vulgar fractions of the form
- <math> {n + 1 \over n} = 1 + {1 \over n}. <math>
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.
Examples:
| Ratio | Name | Related musical interval |
|---|---|---|
| 2:1 | duplex | diapason (P8) |
| 3:2 | sesquialterum | diapente (P5) |
| 4:3 | sesquitertium | diatessaron (P4) |
| 5:4 | sesquiquartum | ditonus (M3) |
| 6:5 | sesquiquintum | semiditonus (m3) |
| 9:8 | sesquioctavum | tonus (M2) |
| 18:17 | (super)sesquiseptimus decimus | semitonium (m2) |
See also
External links
- The new Arithmonic Mean — Preliminaries (http://mipagina.cantv.net/arithmetic/roots.htm#I2) by D. Gómez.
- An Arithmetical Rubric (http://perso.club-internet.fr/daschour/micromegas/terpstra.html) by Siemen Terpstra, about the application of superparticular numbers to harmony.
- Superparticular numbers (http://www.redshift.com/~dcanright/super/) applied to construct pentatonic scales by David Canright.