Most base enthusiasts should know jan Misali and their base naming system. I personally think it's a wonderful idea, but the execution could have been better. And less of a shitpost.

Along this page, I will describe my modified version of the Misalian base naming system, while going over the flaws of Misali's. If you don't know or don't remember well the original, don't worry, this is enough to understand everything.

The basic concept behind the Misalian system is the factorization of a number. Not only does it never change, no matter which base you use, it is also a base's most important factor in determining whether it's good or not (except for possibly size).

A simple example is 22, expressed as 2×11 (biseptimal). A more complex one is 1②2, expressed as 11×2×11 (heptabiseptimal).

The numbers one through ten already have base neutral enough names:

Number

Name

1

Unary

2

Binary

3

Ternary

1②

Quaternary

1①

Quinary

10

Senary

11

Septimal

12

Octal

13

Nonary

2②

Decimal

It's ternary, not trinary, Misali. Ternary has enough unironical use to keep its name, no matter how much you think it doesn't look like a number.

Speaking of keeping names, the following three bases keep their names from usage:

20

Dozenal

3②

Hex

1③2

Vigesimal

No niftimal (literally a name only misali ever uses) nor centesimal (decimal centric).

Misali then adds three more names for three prime bases: 2① (elevenary), 21 (baker's dozenal) and 3① (suboptimal). Why would anyone think that is a good idea?! Not only are these the stupidest possible names (just English word-nary, a reference to baking knowledge and an uncalled for offense) they're also barely necessary anyways! Misali's prefixes shorten names, but only starting at 1③21 (2①^{2}), since their base forms are actually longer than mine! Plus, do you really see a serious mathematical paper calling anything "baker's dozenal"?

To refer to bases without basic names, we combine latinogreekish prefixes to multiply the basic names:

Number

Prefix

×2

bi-

×3

tri-

×1②

tetra-

×1①

penta-

×10

hexa-

×11

hepta-

×12

octo-

×13

ennea-

×2②

deca-

×20

doza-

×3②

tesser-

×1③2

icosi-

(Yes, Misali, it should be ennea!)

All these are greek, except bi- (latin), doza- (probably from dozen, Misali's idea) and tesser- (typo of tessara-, corruption of a greek four associated with tesseract just enough to be its own thing.)

Some examples of combining prefixes: 22 is 2×11, so biseptimal. 23 is 3×1①, so triquinary. 30 is 3×10, so trisenary. 2②0 is 10×2②, so hexadeci-- hexagesimal. Decimal turns to -gesimal after the multiplicative prefixes.

For bases that have more complex breakdowns, break them up into two chunks of as close sizes as possible, leaving the larger one to the right. For example, base 10①0 breaks up into 22 and 23, which break up into 2×11 and 3×1①, making the end name biheptatriquinary. That's not the only way to factorize it (you could use bitripentaseptimal or decaheptaternary, for example) but it's the only one I've managed to get the translator to reliably follow.

Since there's no root for 2① (or any primes above it), and we can't break it any further into factors, we need to introduce an extra prefix.

Un- is a prefix meaning "plus one". This means that base 2① is undecimal (uncoincidentally matching the usual name), base 21 is undozenal and base 3① is unhex. This can still be used in larger bases, such as untrisenary (1③1, 1 + 3×10), and unbundecimal (1②①, 1 + 2×(1+2②)).

Misali has rules for what to do with vowel encounters to make names more pronounceable. I made my own rules too. They're super simple: If two vowels (from different prefixes, otherwise ennea- wouldn't exist) meet, remove the first one. Examples:

bi-un- = bun-

tetra-octo = tetrocto-

ennea-un- = enneun-

sna-un- = snun-

radi-ennea- = radennea-

I believe that the first letter is usually more recgonizeable, therefore more important. Un- and octo- aren't really the same as -n- and -cto-.

The un- prefix holds up until you get to 1③21 (2①²), where we need a multiplicative for undecimal. Un-'s multiplicative form is the circumfix hen--sna, creating hendecasna-, hendozesna- and hentessersna- for 2①, 21 and 3①. (Yes -sna is ex nihilo but I don't want to make a change just for the sake of latinogreek purity.)

The -sna is there because that kind of prefix needs a close bracket. Otherwise, cases like hentrihexa- (1③1) would be technically ambiguous, and differentiating cases like hendecasnaundozenal ((1+2②)×(1+20))and undecaundozenal (1+ 2②×(1+20))would be harder.

With that, we are able to write any positive integer base! However, we are not done yet.

Most people, Misali included, use nega- for negative bases (i.e. negabinary = base ②), which I keep.

Misali uses -vöt- to represent reciprocal bases, but who on Arceus's green earth has heard of VötGil outside of Misali watchers? If you want to use a language that writes numbers technically backwards why not Arabic? I decided to replace it with the infix -fra-, from fraction. Frabinary is base 1/2, trifrabinary is base 3/2 (the multiplicative prefixes before the -far- multiply normally.)

A thing Misali neglected is root bases. They're not that useful, but if you need them you can use -radi-. It is a unique infix, in the fact that it uses the number to its right to define what kind of root it is: Radibinary is √2, triradibinary is ³√2.

To make roots more useful, you can use the -kai- infix to add numbers! Which makes base phi potentially unfrabikairadipentafrabinary!

Finally, for the imaginary bases, -ima- behaves similarly to -nega-. Imabinary is base 2i, negimaternary is base -3i.

-nary and -imal are generic suffixes, which can be used on any word to create a base, such as pinary (-nary is used with numbers up to 10) or taunimal (-imal is used with numbers 10 and above).

Nullary is reserved for base zero, which is useless.

My main gripe with Misali's system is how its shortenings are opaque. It's impossible to get a shortening from a name, or a name from a shortening. What's this, English spelling? They are somewhat recognizeable and short, I'll give them that, but I think it would be easier if shortenings were actually human computable.

The Cal Modified System fixes that assigning each value a letter, which stands in for both its prefix and base (for example, Q is both tetra- and quaternary.) The table of letters, which also doubles as a helpful recap, is as follows:

#

Letter

Prefix

Base

2

B

bi-

binary

3

T

tri-

ternary

1②

Q

tetra-

quaternary

1①

P

penta-

quinary

10

H

hexa-

senary

11

S

hepta-

septimal

12

O

octo-

octal

13

E

ennea-

nonary

2②

D

deca-

decimal/-gesimal

20

Z

doza-

dozenal

3②

X

tesser-

hex

1③2

V

icosi-

vigesimal

+1

U

un-

unary

(U

hen-

Ø

)

)

-sna

Ø

×①

N

nega-

Ø

^①

F

-fra-

Ø

√

R

-radi-

Ø

i

I

-ima-

Ø

+

K

-koi-

Ø

The letters for one through nine were picked based on IUPAC's convention for writing unnamed element symbols. Dozenal is used with a subscript Z by the DSA, hex numbers come with a 0x prefix in C, hen and sna show their true forms as basically a bracket for an un- and the other ones just use their initial.

Examples:

Binary = B

Undecimal = UD

Trisenary = TH

Untrisenary = UTH

Unbundecimal = UBUD

Hendecasnundecimal = (UD)UD

Hentrihexasnuntrisenary = (UTH)UTH

Negadecimal = ND

Trifrabinary = TFB

Radibinary = RB

Bimabinary = BIB

Trikoiradibinary = TKRB

If you're confused by the U without brackets, imagine they have an open bracket immediately to its right and the close bracket at the end of the name.

Like usual, I made a converter toy below. It can turn numbers into bases and shortenings (so far I only figured out decimal input tho) and shortenings into numbers and bases. However, only the integer bases are implemented so far.