Superparticular ratio

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Superparticular ratio

In mathematics, a superparticular ratio, also called an epimoric ratio or delta-1 ratio, is the ratio of two consecutive integer numbers (1:2, 2:3, 3:4...).

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.

Superparticular ratios appear frequently in just intonation and harmonic series music. Consecutive harmonics are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio 21/20. As the harmonics get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common commas are superparticular ratios.

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

Kite Giedraitis has proposed a delta-N terminology (where delta means difference, here the difference between the numerator and the denominator). Thus delta-1 is an alternative term for superparticular, delta-2 is for ratios of the form [math]\displaystyle{ \frac{n+2}{n} }[/math], likewise delta-3, delta-4, etc.

Kyle Gann's 1992 composition Superparticular Woman bears the namesake of this term, and indeed originated from a melody which uses superparticular ratios 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, and 5/4, "seven pitches which lie within only 221 cents (a slightly large whole-step)".

Etymology

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

Definitions

In ancient Greece and until around the 19th century, superparticular ratios are defined as follows: "When one number contains the whole of another in itself, and some part of it besides, it is called superparticular." In other words, a ratio is superparticular if, when expressed as an irreducible fraction, the denominator divides into the numerator once and leaves a remainder of 1.

In almost every case, this matches the modern definition of superparticular, i.e. ratios of the form [math]\displaystyle{ \frac{n + 1}{n} }[/math], where [math]\displaystyle{ n }[/math] is a positive integer. In only one case does it deviate: that of 2/1. According to traditional Greek arithmetic, 2/1 is not a superparticular ratio, but rather a multiple: 1 divides into 2 twice, leaving a remainder of 0. Multiples and superparticulars are considered as distinct categories of numbers from that perspective. In musical terms, this would imply considering that 2/1 is not superparticular because it describes a multiple of the fundamental, which other superparticular ratios do not.

Properties

Superparticular ratios have some peculiar properties:

• The difference tone of the interval 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.
• Størmer's theorem states that, in each limit, there are only a finite number of superparticular ratios.

Generalizations

Taylor describes generalizations of the superparticulars:

• superbiparticulars (or odd-particulars) are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3)
• supertriparticulars (or throdd-particulars) are those where the denominator divides into the numerator once, but leaves a remainder of three (such as 25/22)
• 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).

Generalisation in the "meta" direction gives rise to square superparticulars and then ultraparticulars, under the idea that if a superparticular is the difference between two adjacent harmonics then a square superparticular is the difference between two adjacent superparticulars and an ultraparticular is the difference between two adjacent square superparticulars. This gives rise to descriptions of infinite comma families of which many known commas are examples. A notable property is that just as "all superpartient ratios can be constructed as products of [consecutive] superparticular numbers", all ratios between two superparticular intervals (e.g. (8/7)/(11/10) = 80/77) can be constructed as a product of consecutive square superparticular numbers (e.g. (64/63)(81/80)(100/99) = S8 × S9 × S10), for the same algebraic reason as in the corresponding case of superpartient ratios. (There is a corresponding analogy with ultraparticulars too, for the same reason.)

"Subparticular" is a natural generalization of the idea to ratios which are n/(n + 1).

See also

• List of superparticular intervals
• Square superparticular

External links

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