Superparticular ratio
Superparticular numbers are ratios of the form [math]\displaystyle{ \frac{n+1}{n} = 1+\frac{1}{n} }[/math], where n is a whole number greater than 0.
The word "superparticular" has Latin etymology and means "above by one part". The equivalent word of Greek origin is "epimoric" (from επιμοριος, epimorios).
These ratios have some peculiar properties:
- The difference tone of the dyad is also the virtual fundamental.
- The first 6 such ratios (3/2, 4/3, 5/4, 6/5, 7/6, 8/7) are notable harmonic entropy minima.
- The logarithmic difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.
- The logarithmic sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.
- Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: [math]\displaystyle{ 1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1}) }[/math], but more than one such splitting method may exist.
- If a/b and c/d are Farey neighbors, that is if a/b < c/d and bc - ad = 1, then (c/d)/(a/b) = bc/ad is epimoric.
- The ratios between successive members of any given Farey sequence will be superparticular.
Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a multiple of the fundamental (the same rule applies to all natural harmonics in the Greek system). Another explanation for the exclusion of 2/1 can be found on the Generalized superparticulars page.
Kite Giedraitis has proposed the term delta-1 (where delta means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form [math]\displaystyle{ \frac{n+2}{n} }[/math], likewise delta-3, delta-4, etc.