An Introduction to TGM

TGM is a coherent dozenal metric system - a complete metrology devised for the dozenal system, better designed and integrated than is the decimal metric system. It was devised and developed by Tom Pendlebury, a member of the Dozenal Society of Great Britain (DSGB).

In its original form it was a trial booklet issued with Dozenal Review *26, and no more than twenty pages long, but later reissued in an enlarged form with tables and additional units, running to over six dozen pages. Our new Dozenal Forum on the Web has aroused more interest in the topic, so we have created a pdf version of the TGM booklet (slightly revised) and this can be found at TGM pdf version and a supplementary booklet of TGM units at TGM units

Our audience being so much greater on the Web, we hope there will be more comments and feedback than were possible in the past.

Reviews of TGM and comments (for and against) also appear in this section.

This Introduction tells you how the system is constructed.
The main details of the system will be presented as PDF documents (most of which have yet to be rewritten before they can be linked to these pages) which you can download and read at your leisure. The reason for using PDF documents is quite simply that the symbols for ten and eleven, and the font in which they are used, can be embedded in the documents.
There is no reason why the ones we use in our work should be considered sacrosanct; we need someone who can design us two brand new symbols that will fit in with the present set and be acceptable to all those who use dozenals. Not an easy task - but there are many people out there on the Web!

Note: we need to distinguish dozenal numbers from decimal; it has long been our custom to use the asterisk (*) to mark a dozenal number, and so, for convenience, we will mark the dozenal numbers where necessary and leave the decimal unmarked.

TGM - the basic system.

Many who suggest new systems of metrology start by defining the unit of length. Tom Pendlebury considered this a mistake. He began with the unit of time.


The regular recurrence of night and day is mankind's chief notion of the passage of time. The day is already divided into two dozen hours, written dozenally as *20, and for reasons which will become apparent later, the hour is the major time unit in TGM. Traditional subdivisions of the hour proceed in sixties, with minutes and seconds; in a dozenal system they proceed in twelfths - one twelfth of the hour corresponding to five minutes. Traditional clock and watch faces are marked off in twelfths of the hour.
Fractions, which are so easy to visualise using a clock face, have a familiar appearance when written in dozenal instead of decimal - "quarter past" the hour has the minute hand at the 3 on the clock face, and one quarter in dozenals is *0·3. Similarly for half past the minute hand is at the 6, and a half in dozenals is *0·6.
In order to develop a dozenal metrology and derive a practical system of weights and measures from the unit of time, that time unit has to be chosen carefully. The most suitable - and the one suggested by Tom Pendlebury, is defined as 12-4 of the hour, or, in dozenals, *0·0001Hr, and this he named the "TIM".

1 TIM (Tm) = 25/144 seconds (*0·21) = 0·17361111...s and this is the Fundamental Unit of TGM.

Our Mean Solar Day is the first reality of TGM.

NB: TGM units are abbreviated to two letters. There are also prefixes (similar to the metric system's centi-, milli and Kilo) which we will introduce later.

From Time to Space.

The metre and the foot started from origins quite independent of the unit of time - they are arbitrary. The TGM unit of length is derived from the TIM by the law of gravity.

Watch a diver leaving a high springboard. His upward velocity gradually falls off until at the top of his jump he starts to fall. Down he comes faster and faster until he enters the water. This changing velocity is called "acceleration due to (Earth's) gravity". Laboratory tests in vacuum (so no air resistance) show this acceleration to be the same for all things, large or small, heavy or light, feather or lump of lead. It is given the symbol "g".
Its value in our traditional systems is 32·1741 ft, or 9·80665 m, per second per second. It is fundamental to a vast number of dynamic calculations (though in many cases not obviously so).
In TGM it is made the UNIT of acceleration.


Using Tims instead of seconds, g is just under 30cm per Tim per Tim. About eleven and five eighth inches, a little short of a foot. This length is called the Gravity foot or GRAFUT, abbreviation Gf.

No-one invented it. It is a natural phenomenon that comes to light when reckoning in dozens and hours. Whatever unit of length be chosen, g would still have to be defined. So let it be the unit itself and have

g = 1 Grafut per Tim per Tim

For the base of a system of measures it must be very accurately defined; all other units depend on it. Gravity is slightly stronger at the poles than at the equator. This allows within very narrow limits a choice of standard best suited for the rest of the system.
(The choice of standard, and all other details not appropriate to a brief introduction such as this is in the TGM booklet.)

With the unit of length defined we may also now define the unit of square measure , the square Grafut, or SURF (Sf) and that of volume, the cubic Grafut, or VOLM (Vm). (The names of the units are those created by Tom Pendlebury). We also have the unit of acceleration, the GEE (G) defined as 1 Gf/Tm2, and the unit of velocity the VLOS (Vl) defined as 1 Gf/Tm.

Acceleration due to gravity is the second reality of TGM.

From Space to Matter and Force

The stuff that makes gravity, and mostly responds to it, is called matter. It occupies space, and so has volume. But the pull, its weight, depends not only on volume. For a given volume, lead is heavier than aluminium, and both are heavier than water. Wood on the other hand is lighter and floats. The weight depends on the quantity of matter. This is called mass.

Mass per unit volume is called density. It has been found most practical to compare densities to that of the commonest liquid, water. So our next main unit is the unit of mass:

1 MAZ (Mz) = the mass of 1 Vm of air-free water under a pressure of one standard atmosphere and at the temperature of maximum density (3·98°C).

Tom Pendlebury pointed out that though the Maz is large compared to the pound or kilogramme, this maintained the 1:1 ratio between basic units, leaving prefixes to display the sense of proportion. (These prefixes will be described after the main units have been described in this introduction. Greater detail is in the booklet, including such long strings of figures as are necessary to cope with the most stringent accuracy that anyone might require). In metric the gramme came from a cubic centimetre, a millionth of a cubic metre. The kilogramme (the basic unit for SI) starts with a built-in prefix meaning thousand, but has a water volume about a cubic decimetre, only a thousandth of a cubic metre. In complex calculations decimal place errors often creep in due to these irregularities.

The unit of Density, the DENZ (Dz) is 1 Mz/Vm, the SG of water.

1 Kg of water at maximum density occupies 1·000028 cubic decimetres, which was the definition of the "litre" until 1964. The CGPM then redefined "litre" as the "synonym for cubic decimetre" but its use "is discouraged for precision measurements". This irregularity, excluded from TGM causes slight variances between conversion figures derived from the kilogramme and others derived from the metre.

It will be appreciated that a complete metric system, whatever number base it is constructed for, will require units to be defined and created for every possible application. All such units, along with auxiliary units have been so defined, and all physical constants (velocity of light etc) defined in terms of the TGM units; any metric system meant to replace the decimal metric system must have all these units and definitions so that the system can be properly evaluated by those who are interested in using it.

Before leaving the topic of the units we will add the unit of force

1 MAG (Mg) = 1 Mz x 1G

the strength required to hold 1 Mz of anything from falling, the weight of 1 Mz.
Standing, sitting, lying, walking, jumping, lifting, carrying, holding, climbing, running upstairs, weight is with us through every moment of our lives. The only escape is to go into orbit; then we sense the abnormality or weightlessness.

the density of water is the third reality of TGM

and weight (our normal experience of force to mass ratio) is the fourth reality of TGM

TGM preserves the good points of the present rival systems, discarding their flaws and awkward quirks, and brings metrology more in step with natural laws and counting.

What′s in a name?

We can count without actual digits, but we can't count without words. To be able to describe the units and their multiples we need a dozenal vocabulary. If we count in twelves instead of in tens, do we keep the words "ten" and "eleven", or make up new ones, such as "dek" and "elv" to make it clear we are not using base ten? Whatever the advocates of the dozenal system may suggest, someone will not like the words they choose. The names and words that now follow are those created and proposed for use with dozenals by Tom Pendlebury. They will not please or suit everyone, but they do allow us to describe the units in dozenal terms.

Where we need to use symbols for ten and eleven here in text published on the Web we shall use T for ten and E for eleven. (This is a personal choice, made in the days when special characters had to be soldered into place on typewriter keys...) In booklets and documents written in PDF format, where it is possible to embed special fonts and characters we shall use the symbols shown on the clockface at the beginning of this TGM section. (Alternatives to these symbols are discussed elsewhere on this site.)

Spelling numbers in dozens

Every second number is a multiple of TWO; every third a multiple of THREE; every fourth a multiple of FOUR, and so on. Because two twos are FOUR, two threes are SIX, two fours are EIGHT, three threes are NINE and three fours are TWELVE it is a natural law that the numbers 2, 3, 4, 6, 8, 9 and twelve play the most dominant roles in calculation. They come through in spite of decimalisation:

2 x 0·2 = 0·4, 2 x 0·3 = 0·6, 2 x 0·4 = 0·8, 3 x 0·3 = 0·9 etc

The lowest common multiple of 0·1, 0·2, 0·3, 0·4 and 0·6 is 1·2, a dozen TENTHS; and if 0·5 is included, the LCM is 6·0, the HALF DOZEN. Whether counting in units, tenths, sixteenths, hundreds, millions or whatever, makes no difference. The dozens still dominate, though this is often not obvious, due to our writing numbers in tens.

Recognition of this truth has led many individuals, from different nations and generations, to the conclusion that calculation and measurement can be more simply expressed by counting, not in tens, but in DOZENS. Yet is we still write twelve as "12", which means 1 ten and 2 units, we are not counting in dozens but in tens. The full benefit can only be achieved by using "10" to mean 1 DOZEN and 0 units.

We count from one to nine as usual, and then have ten (T), eleven (E) and the dozen (*10); Tom Pendlebury suggested "elv" for eleven, and "zen" (also "onezen") for *10.
We can continue our count with *11, *12, *13 and so on, read as "onezen one", "onezen two", "onezen three" up to *20 "twozen", then *30 "threezen" until we reach *10 x *10, zen times zen, or *100. The usual name for this is one gross, and though one dozen times this has the name "great gross" (*1000), there are no other common names for higher powers of twelve. We need to create new words, and from them derive the prefixes, like centi-, milli- and kilo-, which we can attach to the units of any dozenal metric system we may devise. The names which Tom created for use with TGM follow.

The following table lists the new words and gives their values in dozenal and in decimal. Putting a nought on in dozenal multiplies by a dozen, just as, in tens spelling, putting a nought on a number multiplies its value by ten. This gives dozenal counterparts of hundreds, thousands and so on that look like them but stand for different values. The job of words is to evoke and distinguish ideas, so these different ideas are given different names - and on a more straightforward basis. They are coined to suggest the index of the order - in simple English, how many noughts to put.

Name Dozenal value Noughts Standard Form Decimal equivalent
Zen 10 1 101 12
Duna 100 = zen x zen 2 102 144
Trin 1 000 = 10x10x10 3 103 1 728
Quedra 10 000 4 104 20 736
Quen 100 000 5 105 248 832
Hes 1 000 000 6 106 2 985 984
Sev 10 000 000 7 107 35 831 808
Ak 100 000 000 8 108 429 981 696
Neen 1 000 000 000 9 109 5 159 780 352
Dex 10 000 000 000 T 10T 61 917 364 224
Lef 100 000 000 000 E 10E 743 008 370 688
Zennil 1000 000 000 000 10 1010 8 916 100 448 256

No-one can make a mental picture of a million (the Egyptian symbol for a million was a man with his arms raised in astonishment) it's just a whopping big lot, only more so. To keep tediously converting numbers from dozenal to decimal, just to "understand" them, does not really help. It is only the "spelling" that changes. Only the simplest of numbers and fractions, easily convertible, can be pictured in the mind.

Nevertheless we use numbers to tell us what we cannot figure out without their help. They mainly tell us which is bigger than which, by how much or by what factor. Both in dozenal and decimal "300" is greater than "10" by a factor of "30". But whereas in decimal 3 hundred is thirty times ten, in dozenal 3 gross is three dozen times a dozen - a rather larger ratio on a rather larger number.

Armed with the number names we can define multiplying prefixes, similar to "kilo-" and "mega-"; in TGM they end with the letter -a. Thus, for example, a dozen feet (Gf) can be called the Zenagrafut, *100 years a Dunayear.
Similar to "deci-", "centi-", "milli-" we have dividing prefixes ending in -i. One twelfth of an hour is thus a zenihour, and the Tim (Tm) which we defined earlier, can also be referred to as the Quedrihour.

To avoid any confusion with letters used for decimal prefixes the abbreviations are written as numerals, raised for multipliers and lowered for dividers - e.g. with the Maz (TGM unit of mass), abbreviated Mz:
1 2Mz (1 dunamaz) = 100 Maz; 1 3Mz (1 trinimaz) = 0·001 Maz.

No matter how complex a problem, the order of the magnitude is kept track of by adding and subtracting the prefixes, which are in fact exponents. Duna times duna gives quedra; quena x trina = aka; but quena times trini = duna. In multiplication the "-a's are added and the "-i's subtracted. For powers the prefixes are multiplied; the square of quena is dexa. For roots they are divided; the cube root of neeni is trini.

The names themselves may not suit all; but they follow a more logical pattern than the current practice in decimal in which a trillion multiplied by a quadrillion comes out to an octillion, which is the cube of a billion, and that billion is no longer the square of a million but the trillion is.... All very confusing, when the English billion is a million million and the American a thousand million (for which the Germans have the word milliard...)

At the risk of becoming too detailed we have described the words devised by Tom to show what is needed to create the structure for a dozenal metric system - and for dozenal numeration. Much more information will be found in the full booklet, of which the Webmaster has a few copies. And you are invited to join in the discussions about TGM and other systems on our Dozenal Forum.