ordinal numbers - Is there a straightforward word for "The thing in between first and second"?

Sometimes when writing I find myself looking for a word to describe something in the “³⁄₂^th” place — exactly between first and second. I would like to ask, does there exist an easy expression for this? If so, does it generalise to other “in-between” places as well?

(I mostly encounter this problem when writing mathematics, where I wish to write the “³⁄₂” order derivative, and normally have to endure clumsy or imprecise expressions (“Taking the fractional derivative. . . .”).)

Answer

The word you are looking for is the sesquialter, sesquialteral, sesquialterate, sesquialteral, or sesquialterous element.

You can also use sesquialter as an ordinal noun, for the midpoint between first and second:

I’ve skipped the zeroth, but then jumped straight from the first to second, so I’m am still missing the sesquialter I expected midway between those two.

That’s because sesquialter means “one-and-a-halfth”,¹ but is substantially easier to say.² It is one of those sesquipedalian terms surpassing both in erudition and utility alike. Per the OED, it means:

1. Of a proportion: That is as 1½ is to 1. Of an object: Proportionate to another object as 1½ is to 1; that is such a multiple of.

Here are a few of its more recent citations:

• 1698 Phil. Trans. XX. 81 ― We assign to a Fifth··the Sesquialter Proportion (or that of 3 to 2).


• 1711 H. Needler in J. Duncombe Lett. (1773) I. 90, ― 6 is only sesquialter of 4.


• 1715 Cheyne Philos. Princ. i. 222 ― In all the Revolutions of the Planets about the Sun,··the periodical Times is [sic] in a Sesquialter Proportion to the middle Distances.


• 1784 J. Keeble Harmonics 29 ― The sesquialter chromatic.


• 1846 Penny Cycl. Suppl. II. 369/2 ― The following ratios are super‐particular: 15 to 10, which is sesquialter.

And its etymology is:

Etymology: L., f. sesqui- (see prec.) + alter second. For the formation cf. ONor. hálfr annarr, OE. óþer healf, G. anderthalb.

The sequi- prefix is today most familiar in terms like sesquicentennial, for the 150-year anniversary of some event. It gives rise to a delightful multitude of derived terms.

1. a. With designations of measure or amount, denoting one-and-a-half times the unit; as sesquihōra an hour and a half; sesquipēs a foot and a half (see sesquipedalian); so † sesquiˈhoral a., lasting an hour and a half; ˌsesquioˈcellus Ent. (see sesquialterous); † ˈsesquitone Mus., an interval consisting of a tone and a semitone, a minor third; also used loosely in † sesquiˈdecuman a., consisting of fifteen; † sesquiˈdecury, a set of fifteen.

The OED provides not only senses 1b through 1d, with which we need not here concern ourselves, but also the operative sense 2a:

2. a. With an ordinal numeral adjective, denoting the proportion 1 + ¹⁄_n:1, i.e. n + 1:n, where n is the corresponding cardinal number, as sesquioctāvus, bearing the ratio 1⅛:1, i.e. 9:8; so sesquialter, -altera, etc., sesquitertia, etc.; † ˌsesquibiˈtertial, involving a proportion of 5:3; † ˌsesquiˈdecimal, of 11:10; sesquiˈnonal, of 10:9; ˌsesquiocˈtaval, -ˈoctave, of 9:8; ˌsesquiˈquartal, -quartan, of 5:4; ˌsesquiˈseptimal, of 8:7.

Sense 2b also gives several nice words for harmonic situations:

b. in Music, after sesquialtera and sesquitertia; sesquiquarta, -quinta, -sexta, -octava (-octave), -nona, applied

• (i) to harmonic intervals producible by sounding four-fifths, five-sixths, etc. of a given string;


• (ii) rhythmic combinations of four notes against five, five against six, etc.

As you correctly perceive, having a word that means ³⁄₂^th is exceedingly convenient at times, which no doubt why are ancestors invented sesquialter way back in the 16^th century. This is the first citation given for the term in the OED:

• 1570 Dee Math. Pref. c j b, ― A Cylinder, whose heith, and Diameter of his base, is æquall to the Diameter of the Sphære, is Sesquialter to the same Sphære.

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Footnotes:

1. 
   

   Regarding the productivity of the -th suffix for creating ordinals from cardinals in English, see this question.

   
   


2. 
   

   Insofar as words that have end in -fth or -xth are often considered difficult to pronounce by non-native speakers, like fifth, sixth, twelfth, and if they find twelfth tough, they seem likely to find halfth tough as well.