Superparticular ratio

In mathematics, a superparticular ratio, also called an epimoric ratio or delta-1 ratio, is the ratio of two consecutive integer numbers.

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Superparticular ratio

More particularly, the ratio takes the form:

[math]\displaystyle{ \frac{n + 1}{n} = 1 + \frac{1}{n} }[/math] where [math]\displaystyle{ n }[/math] is a positive integer.

In music, superparticular ratios describe intervals between consecutive harmonics in the harmonic series.

A ratio greater than 1 which is not superparticular is a superpartient ratio.

Etymology

The word superparticular has Latin etymology and means "above by one part". The equivalent word of Greek origin is epimoric (from επιμοριος, epimórios).

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.

Definitions

In Thomas Taylor's Theoretic Arithmetic, in Three Books (1816), superparticular ratios are defined as those for which the denominator divides into the numerator once, leaving a remainder of one.

In almost every case, this checks out with the popular definition of superparticular, i.e. ratios of the form [math]\displaystyle{ \frac{n + 1}{n} }[/math]. In only one case does it deviate: that of 2/1. According to Taylor, 2/1 is not superparticular, because 1 divides into 2 twice, leaving a remainder of 0.

Properties

Superparticular 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 superparticular ratios is always a superparticular ratio.
• The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio.
• Every superparticular ratio can be split into the product of two superparticular ratios.
  • One way is via the identity: [math]\displaystyle{ 1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1}) }[/math]; e.g. [math]\displaystyle{ \frac{9}{8} \times \frac{10}{9} = \frac{10}{8} = \frac{5 \times 2}{4 \times 2} = \frac{5}{4} }[/math].
  • Other splitting methods exist; e.g. [math]\displaystyle{ \frac{12}{11} \times \frac{33}{32} = \frac{396}{352} = \frac{9 \times 44}{8 \times 44} = \frac{9}{8} }[/math].
• 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 superparticular.
• The ratio between two successive members of any given Farey sequence is 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.

Generalizations

Taylor's book further describes generalizations of the superparticulars:

• superbiparticulars are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3)
• double superparticulars are those where the denominator divides into the numerator twice, leaving a remainder of one (such as 5/2)
• one can go on and on, with e.g. triple supertriparticulars, where both the divisions and the remainder are 3 (such as 15/4).

See also

• List of superparticular intervals

External links

• Generalisation of the terms "epimoric" and "superparticular" as applied to ratios on the Sagittal forum