Superparticular ratio
Superparticular numbers are ratios of the form [math]\displaystyle{ \frac{n+1}{n} }[/math], or [math]\displaystyle{ 1+\frac{1}{n} }[/math], where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."
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 difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.
- The 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.
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).