superparticular

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Superparticular ratio

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From Late Latin superparticulāris.

superparticular (not comparable)

1. (mathematics, obsolete) Of a natural number, being larger than another natural number by a unit fraction of that smaller number.
   • 1624, Democritus Junior , The Anatomy of Melancholy: , 2nd edition, Oxford, Oxfordshire: John Lichfield and James Short, for Henry Cripps, →OCLC:
     , I.iii.1.4:

     'Tis superparticular, sesquialtera, sesquitertia all those geometric proportions are too little to express it.

   • 1816, Thomas Taylor, Theoretic Arithmetic, pages 37–38:

     When a number contains the whole of another in itself, and some part of it besides, it is called superparticular. And if the part of the less it contains is the half, it is called sesquialter; if the third part, sesquitertian; if the fourth, sesquiquartan; if the fifth, sesquiquintan. And the like names being employed to infinity, the form of superparticular numbers will also proceed infinitely.

2. (mathematics, music) Pertaining to a ratio of any whole number to the next below it (e.g. ${\displaystyle {\tfrac {3}{2}}}$, ${\displaystyle {\tfrac {8}{7}}}$, etc.).

It seems that the meaning of the term superparticular has narrowed over time. According to the definition of Thomas Taylor, e.g. 144 is superparticular (sesquitertian, to be specific) in relation to 108, because it contains exactly the number 108 and a third part (36) of it. Similarly, 144 is also superparticular in relation to 96, 120, 126, 128, 132, 135, 136, 138, 140, 141, 142 and 143. Current sources limit the meaning to whole numbers that are larger than another number by 1. The definitions are not contradictory, since n + 1 always consists of n plus one nth part of n. If defined in the latter way, 144 is superparticular only in relation to 143.

• epimoric

• multiple superparticular
• superparticular number