It's just my guess, but I think compatibility has to do with it. The alternatives to decimal that aren't dozenal tend to be multiples of the dozen, which says a lot about the matter. Obviously dozenalists think an alternative to decimal needs to have all or most of the advantages of the dozen. Where the other bases differ from dozenal, it's in the degrees of compatibility they offer with the decimal world out there. As in the computing world, when a new standard is proposed, there are revolutionaries who want a clean break with all past legacies, there are stick-in-the-muds who think the new standard will never catch on as is but must be set as part of the old-standard world, and there's every shade of grey in between. Let us explore those ranges, if only briefly.

**Revolutionaries**

The purest in dozenalism is to look upon decimal in contempt as a technological dinosaur awaiting extinction. With the decimal

*ancien regime*there should be no connection maintained, no bridges allowed to stand. Of necessity for dozenal purism is an anti-quinary stance: the prime 5 is believed to be objectively as useless as 7, inflated in its importance only due to decimalism. Therefore, any accommodation for 5 in dozenal divisibility testing and representation of fractions is a waste of time at best.

Of the dozenal-multiple bases that burn the bridges with decimal totally, those that fulfil 5nÂ±2 are to be chosen: 48 (quadranunqual), 72 (hexanunqual), 108 (ennunqual) or 132 (levanunqual). The fundozmentalist believer, the member of the Ultra-Orthodoz community waiting for 10th Imam (that's dozenth, not tenth, ye of little faith!) to shine the light on the benighted decimal world, would likely champion base four-dozen as a pure manifestation of dozenal fervour, the kind that leaves no bridge to decimal and its dastardly divisor 5. Of course, dozenal itself might be suitable, but the SPD test for 5 devised by one of the compromisers would have to be avoided; in base *40 such a test is out of reach, so it would be the ideal choice for the one who wishes to avoid temptation.

Satire aside, a dozenal purist would not actually reject something like the ability to test for divisibility by 5 in dozenal, yet he would consider it an extra, something nice to have but not that important. He would not stress himself over the fact that fifths will never be clean in dozenal. Compatibility with decimal is, as far as he's concerned, nothing to be striven for. If achieved, well and good, but dozenal is so much superior to decimal that the one who truly appreciates it no longer cares about keeping in touch with the decimal world. Once he's settled on names and symbols and a system of metrology, the pure dozenalist never looks back.

**Minimal Compatibilists**

Less stringent than the revolutionaries are those who think there needs to be some handy provisions for decimal compatibility. They hold that the factor 5, though not so important that it needs to be a divisor of the base, should not be treated in such hostility as 7 is in dozenal or decimal. It should be a neighbour-totative of the base, in order for it to have an easy divisibility test and to avoid maximal recurrence for its fractions.

The choices here depend on which neighbour relationship one would prefer and how many powers of 5 one is to be able to test for. The Î±-totative relationship gives dozenal multiples like 24, 84 and 624, and the Ï‰-totative relationship, 36, 96, 276 and 876. Since the base chosen as one's main base tends not to be too large, the choice will be between 24 (double dozen) and 36 (triple dozen), the former winning because of its smaller size, the latter because of the benefits of the omega relationship (the digit-sum test and fractions with only one recurring digit). Both require a lot of thinking about symbols and names for them, which is why dozenal purism might be the easier route to take after all; but they offer much more in the way of compatibility with decimal than dozenal does, especially in the case of fractions.

**Decimal Auxiliarists**

The next level of compatibility is bases that give decimal (fivefoldness) and dozenal (threefoldness) equal stature. With those multiples of the dozen that also have the prime factor 5 in them, we get to have that gorgeous trailing zero in both representations: *50 = 60, X0 = 120. We have no need to compromise on either side, because prime factors 3 and 5 are both first-class. Another advantage is that we can maintain compatibility with decimal names and symbols up to a much further point than with other dozenal-multiple bases. For example, in the usual representation of base 120 (dozenal-on-decimal encoding), all numbers up to 99 inclusive have the same names and meanings as in decimal, and only after that do we have to break out of the mainstream with 'teenty', 'elfty' and so on.

These bases are large. The smallest of them, sexagesimal, already needs to be encoded as two sub-bases for each digit. The alternating-radix encoding means we have a different carry rule for each slot, so that for example in the Long Hundred there is a carry after 9 in one slot but after E (eleven) in the other. That is the price to be paid for the good measure of decimal compatibility gained.

**Timid Accommodationists**

And then are those who think decimal is here to stay, differing from those who write out dozenalism completely only by their agreement that the advantages of dozenalism are real. They do not think any change of number base in the main, meaning any change of names and symbols, will ever catch on; what they propose in order to gain the benefits of dozenal in an incurably decimal world is to integrate twelve-valued scales into niches.

They point out that we already have thirds represented as 4 inches of a foot ruler or 8 hours of a 24-hour day, yet we do not change our decimal numerals and names one tiny bit for those gains; this, they say, is the way forward for dozenalism. Theirs is a proposal to use base-12 modular arithmetic in a decimal world: count up to 11, and write it thus (not something like E as the other dozenalists do), and just carry to a new unit above when adding to it. So 11 plus 4 inches equal 1 foot 3 inches, and that's the spirit. In numeration, their scheme would have the dozenal 6E4 (decimal 1000) represented as 6,11,4. It is the type of dozenalism that demands the least change of its adherents, at the price of the lack of the benefits of base-n calculation. Also, in a decimal world that is itself not much predisposed towards compromise (look at SI and its roughshod attitude to anything outside the confines of straight decimal multiples), attempts to integrate dozenal scales do not produce a more palatable result than real dozenalism, except for established practices like the foot/inch ratio.

**Summary**

The outline above shows that the gradation of decimal compatibility advocated by various dozenalists is not primarily a technological, arithmetic issue; the biggest factor here is one's attitude, the measure of one's optimism about the ability of dozenal to become mainstream. The first two groups are of those who believe that the vision of

*Douze notre dix futur*('Twelve, our future ten', the title of Jean Essig's 1955 book) can be achieved, and decimal, if accommodated at all, needs just a little help from the base that is going to replace it. Members of the other two groups think decimal must stay one's main base as it will always be so in the world, and the benefits of dozenal are to be gained through optional-use auxiliary superbases or special-purpose dozenal scales. Yours truly has had times of belonging to all four groups, and as of this writing is vacillating between the first and the third, occasionally flirting with the second group by pondering the triple dozen as a base. Hopefully harmless...