Superparticular ratio

In

More particularly, the ratio takes the form:

${\displaystyle {\frac {n+1}{n}}=1+{\frac {1}{n}}}$ where n is a

positive integer

.

Thus:

A superparticular number is when a great number contains a lesser number, to which it is compared, and at the same time one part of it. For example, when 3 and 2 are compared, they contain 2, plus the 3 has another 1, which is half of two. When 3 and 4 are compared, they each contain a 3, and the 4 has another 1, which is a third part of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc.

— Throop (2006), ^[1]

Superparticular ratios were written about by

The Wallis product

${\displaystyle \prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdots =2\cdot {\frac {8}{9}}\cdot {\frac {24}{25}}\cdot {\frac {48}{49}}\cdots ={\frac {\pi }{2}}}$

represents the irrational number π in several ways as a product of superparticular ratios and their inverses. It is also possible to convert the Leibniz formula for π into an Euler product of superparticular ratios in which each term has a prime number as its numerator and the nearest multiple of four as its denominator:^[5]

${\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{16}}\cdots }$

In graph theory, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise via the Erdős–Stone theorem as the possible values of the upper density of an infinite graph.^[6]

Other applications

In the study of

Every pair of adjacent positive integers represent a superparticular ratio, and similarly every pair of adjacent harmonics in the harmonic series (music) represent a superparticular ratio. Many individual superparticular ratios have their own names, either in historical mathematics or in music theory. These include the following:

Examples

Ratio	Cents	Name/musical interval	Ben Johnston notation above C	Audio
2:1	1200	duplex:[a] octave	C'
3:2	701.96	sesquialterum:[a] perfect fifth	G
4:3	498.04	sesquitertium:[a] perfect fourth	F
5:4	386.31	sesquiquartum:[a] major third	E
6:5	315.64	sesquiquintum:[a] minor third	E♭
7:6	266.87	septimal minor third	E♭
8:7	231.17	septimal major second	D-
9:8	203.91	sesquioctavum:[a] major second	D
10:9	182.40	sesquinona:[a] minor tone	D-
11:10	165.00	greater undecimal neutral second	D↑♭-
12:11	150.64	lesser undecimal neutral second	D↓
15:14	119.44	septimal diatonic semitone	C♯
16:15	111.73	just diatonic semitone	D♭-
17:16	104.96	minor diatonic semitone	C♯
21:20	84.47	septimal chromatic semitone	D♭
25:24	70.67	just chromatic semitone	C♯
28:27	62.96	septimal third-tone	D♭-
32:31	54.96	31st subharmonic , inferior quarter tone	D♭-
49:48	35.70	septimal diesis	D♭
50:49	34.98	septimal sixth-tone	B♯-
64:63	27.26	septimal comma, 63rd subharmonic	C-
81:80	21.51	syntonic comma	C+
126:125	13.79	septimal semicomma	D
128:127	13.58	127th subharmonic
225:224	7.71	septimal kleisma	B♯
256:255	6.78	255th subharmonic	D-
4375:4374	0.40	ragisma	C♯-

The root of some of these terms comes from Latin sesqui- "one and a half" (from semis "a half" and -que "and") describing the ratio 3:2.

Notes

Citations