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数学における隣接整数比 (英: superparticular ratio, epimoric ratio) は、連続する2つの整数による比である。

次のような形をとる:

[math]\displaystyle{ \frac{n + 1}{n} = 1 + \frac{1}{n} }[/math]

ここで [math]\displaystyle{ n }[/math] は正整数。ただし 2/1 (n=1) を含まないという流儀もあり、必要なら都度定義するのがよい。

（英語と違って"隣接整数比"では1より大きい数ということが明示されていないが、訳の工夫で対応する。reciprocal of a superparticular ratioは直訳して隣接整数比の逆数としてもよいが、1未満の隣接整数比とか、下隣接整数比といった用語を用意するのがよさそう）

隣接整数比は純正律に頻出する。倍音列の連続する2音は隣接整数比音程となる。例えば第20倍音と第21倍音は 21/20 だけ隔たっている。上のほうに行くほど倍音の間隔（周波数の差ではなく比として）は狭まっていくので、隣接整数比も小さくなっていく。このため、隣接整数比を調べることは整数比調律システムの中の簡単かつ小さい音程について調べることを意味する。実に、全てではないが多くのコンマが隣接整数比となっている。

既約で分母と分子の差が2以上の分数をsuperpartient ratioという。

分母と分子の差を一般化する用語が提案されている。デルタ-N 比 (en) は分子が分母より N だけ大きい比である。なのでデルタ1比は隣接整数比を意味する。

語源

superparticularという単語はラテン語から来ていて、"1パーツ分だけ超過している"という意味になる。相当するギリシャ語由来の単語はepimoric (希: επιμοριος, epimórios) である。

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.)

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