Pendulum System

That Dozenal Transcends Metrology
There are and have been many collections of unrelated units of measurement and more structured systems of units of measurement that have been used at the same time and with one and the same base of numeration for representing the numbers without regard to the way units have been multiplied or aggregated into larger units or subdivided into smaller units of measurement. For example, there have been units of weight that increased as binary powers although their numbers were written decimally. For a number to be the default and only base of numeration used, a single unified, standardised, legally mandatory or obligatory, and conventional system of measurement is not necessary. While a fully global and international system of measurement would be desirable, the aim to design and introduce such a system of measurement should not be used as the main motivation for setting the default base of numeration. Just as having a single system of measurement throughout the world is advantageous, having a single base of numeration for all computations would be convenient. The argument for dozenal being as much as possible the exclusive base of numeration should be founded almost entirely in the advantages inherent from its mathematical and practical merits dissociated from any particular metrological proposal.

On Computational Conversion
It must be emphasised that having multiple bases of numeration being used concurrently in public interaction would be inconvenient and disruptive because of the computational conversion between their numbers that would be necessary. At the moment, some of the inconvenience of the use of more than one base is hinted at by the decimal scales being contary to the sexagesimal subdivisions of time and angle, though their numbers are all written with the same decimal digits. Thus, a complete replacement of decimal by dozenal should be desired. Lingering of decimal in a dozenalised society should be discouraged and eventually stamped out.

On Metrological Conversion
Since dozenal is to be the only base, there should not be permitted any contexts in which the decimal base persists. Thus, all measurements regardless of their original units should be converted to dozenal. To reduce the proliferation of units in the dozenal world, a scheme should be worked out to make these conversions towards a dozenal derivation as far as possible. To achieve this is proposed that all measurements in a quantity of the same kind and system of measurement should be converted to a single unit within that system which will then in turn be converted to a dozenal unit of measurement. For example, all measurements of the quantity length, whether they be parts of inches, feet, or yards, insofar as they are commensurate without the involvement of factors containing large prime numbers, from the Imperial or Customary systems could be converted first to inches, which then could be converted to be in terms of some unit of length that fits into a dozenally derived metrological system.

On Money
Replacement of decimal by dozenal may be begun with national currencies. Only denominations of notes and coins that are legal tender in a country are permitted to be used. Thus, making ultimately one base of numeration alone legal on denominations of national currency would not encroach upon commercial freedoms that are currently enjoyed. Furthermore, there would be no extra cost to changing the base of numeration in currency from decimal to dozenal, because the only cost is of replacing decimalised denominations by dozenalised denominations, which is nothing more than a matter of design and printing or minting, processes that are regularly renewed to counteract counterfeiting and replenish notes and coins withdrawn from circulation because of disrepair from wear and tear anyway. Other supposed costs such as of training and education or informing need not be entertained, because once there is only one system of legal tender, people will be able to suss out for themselves what to do and teach each other free of charge by word of mouth. All worthy independent news media would advertise the transition in the ordinary course of reporting. Enacting dozenal as the base for currency is merely a matter of being signed into law by elected representatives and heads of legislating government. This is really very easy to do once the decision has been made, as for the facile changing of laws generally where the populace do not object. There should only be one set of legal tender. If dozenal is to be allowed freely, decimal must by law be phased out.

On Numeration
Once the currency has been adopted and become familiar for some time, the mode of writing or printing numerals in documents regulated by the country may be impelled to be changed from decimal to dozenal. This may be done at no cost, as the emergence of freely available electronic converters and downloadable software to devices can be expected. Contents of textbooks are controlled by syllabuses or curriculums set by educational authorities. There would be no loss of freedom therefore from a demand to print all numbers in school books in dozenal notation. These books are reprinted very frequently, so there would be no extra cost to the printing or publishing house or consumer. Numerals extended to the full dozenal range can be supplied free to the printers on condition that they be used. If it is supposed that Indo-Arabic numerals from zero to nine be retained in expressing numbers dozenally, then the decimal numbers do not co-exist well and their use in decimal ought to be phased out to accommodate dozenal. This should be done through consumer protection, health and safety, and accountability legislation, to which in their current form citizens are already bound, among other acts.

On Priorities
By far, the most important agendas for dozenalists to start creating a dozenal world are probably dozenal currency as the only legal tender and dozenal notation. Nevertheless, many purporting to be dozenalists have devoted an inordinate amount of time and energy to less urgent matters, such that milestones of dozenal calendrical dates have come and gone while those dozenalists appeared to have not been ready with a contingency. They are still debating and arguing among themselves and against each other, and refuse as a matter of principle to decide on perhaps the single most important necessity for dozenal notation. And when some purporting to be dozenalists do come to an agreement, it often involves a strategy that makes being a dozenalist more difficult than it was before, which makes one wonder whether they are in fact scheming to hinder dozenalism. A tell-tale sign of those not committed to dozenalism is the advocacy of other bases than base twelve with invention in unnecessary detail of names for numbers and units in systems of those other bases not just in a way of investigation to prove the supremacy of base twelve, and ignoring the clear mathematical superiority of base twelve, while exaggerating alleged perks of other bases.

On Time
Many metrological systems proposed for dozenal have begun with the unit of time. Some have claimed that in a dozenal world, the multiplication and division of units of measurement, such as of time, would be consistently dozenal, such that larger and smaller units than the base unit would appear as dozenal powers of the base unit. This claim of consistency in the case of time cannot be true. There cannot be only one base unit of time, because the periods of time by which people reckon their lives are not commensurate with each other. The period of the day governs the light by which work and sleep are scheduled. The solar year governs the seasons by which the production of food is regulated. The lunar month governs the tides by which porting and sailing of ships in harbours are controlled. Yet the day does not divide exactly into the month or year, and the month does not divide by a whole number into the year. The number of times by which the day divides into these larger periods is not even a dozenal power. Hence, the supposition that the division of time can be strongly and consistently dozenal, or indeed any other base, is a fallacy.

One period of time, such as the day, in isolation from the other periods may be multiplied and divided dozenally. Preferentially, this would be done consistently at powers of the base twelve. However, the base ten is used in the represention of all numbers whether they be for divisions at decimal powers of the unit or otherwise. Hence, dozenal can instead be used as the only base of numerical notation for all divisions, whether they be at dozenal powers of the base unit of measurement or not. Ideally, it would be better if all subdivisions were as powers of the base twelve, as this would enable the easiest manner of notation and computation.

However, the concern is introduction of dozenalism in the world rather than in the fantasy of minds, and realistically the better way to introduce dozenal in time may be to retain as much as possible the current most common mechanisms of clocks, which divide the day into two halves and each half into twelve hours. There is then a way to be dozenal with time while keeping all major civic clocks that do not use a second hand unchanged except for the painting of their faces. One may relatively easily achieve such a dozenal clock hands-on by scrubbing off any tick marks between the hourly positions and writing new graduations between them. This makes the clock a measurement device for the dozenal unit of time that is fifty seconds in duration, the same as from completely consistent division at dozenal powers of the day.

A problem with the proposals of consistent division of the day by powers of twelve is that doing so would involve too many hands for the reading of the time. Hands must be distinguished by length, thickness, and design, but if there be too many hands, they would be liable to be less distinct from each other and would involve more mechanical complexity than desirable.

From a language point of view, to have a newly named or prefixed unit at each power of twelve is too frequent. By analogy, for the prefixes to units in the decimal metric system of measurement, prefixes appear at each power of the cube of the base ten, and not more frequently except for powers near the base unit, although even then a prefix for the square power is rarely used, such that the prefixes deca or deci are not often encountered, and the word decimetre is hardly seen outside of the context of discussing certain dozenal metrological proposals. Thus, for example, if there is a word, such as minette, for the dozenal temporal period of fifty seconds, then a word for a duration twelve times that or a twelfth of it should be discouraged. Instead, the named unit has derivative words at its multiples by powers of the square twelve or square twelfth, hence making a consistent division of scale by the square dozen base, rather than its subbase twelve.

Names or prefixes for units should only appear at intervals of the square power of the dozen. This being achieved on the clock face with a square dozen tick marks gives half as many hands, a simplification and an improvement.

Having as many as a square dozen tick marks on a watch face may be too many to be distinguished easily for reading the time. A lesser number such as half a square dozen of tick marks may be preferable. It is better to have as many tick marks as can be readily discriminated, as using fewer would waste space and energy.

Thus, the day can be divided into two dozen hours, each of which is divided into half a square dozen periods of fifty seconds, each of which would be divided into another half square dozen periods of the square of five sixths of a conventional second. An advantage of using this as the fundamental base unit of time is that its square and cubic powers approximate to simple rational portions of the conventional second, enabling easier mental rules of thumb for conversion between the units of the dozenal metrological proposal and the decimal metric, the most commonly used metrological system that should be eradicated by dozenal from the world.

On Length
Having selected a unit of time, some have derived with it a unit for length by using the acceleration due to gravity of the Earth set to some constant. However, the graviational acceleration on Earth is not constant, but varies with latitude and density of the geological formations underground. Nevertheless, it has been claimed that the deviation is small enough that such accelerational metrological systems are appropriate worldwide regardless of latitude. Using this logic, the gravitational acceleration at any particular latitude could be used as the constant for the purposes of the definition of units in the system of measurement.

In this proposal, the unit of length is derived ideally as the length of a pendulum with a period of the unit of time already selected for being a dozenal division of the day. The gravitation is chosen so as to make this resulting length of the pendulum exactly equal to a whole number of an existing unit of length in one of the legal standards.

Setting the unit of time to half a square dozenth of fifty seconds and the gravitation at \(2^{6} 3^{5} \pi^{2} /5^{6}\) ~= 9.8235 decimal metric metres per square second gives the unit of length \(L\) through the formula for the period \(T\) of the pendulum in terms of its length and gravitation \(g\):
$$T = 2\pi \sqrt{\frac{L}{g}}$$
Hence, the length of the pendulum becomes twelve centimetres. A twelfth of this as a decimal metric centimetre is constructed as the base unit of length.

From this length, units of area and volume by the second and third powers are constructed.

The combinations of the units of time and length determine the kinematic units in the system of measurement.

On Mass
The unit of mass is set as the amount of matter in a unit volume of water under specified conditions, equal to one gram.

The combinations of the unit of mass with the kinematic units determine the mechanical units of the system of measurement.

On Temperature
A theoretical consideration might involve Boltzmann's constant.

However, here a much more practical strategy will be proposed.

Temperature is measured by the height of liquid through a narrow bore in a tube of glass. The minimum height for water occurs at the temperature of its maximum density. The temperature of maximum density of water may be set as four degrees Celsius. This temperature in the dozenal scheme is set at zero. The size of the dozenal degree of temperature is set as being equivalent to the size of a degree in Celsius and Kelvin. Hence, the temperature, being a hundred degrees Celsius, of boiling of water at a specified pressure then becomes eight dozenal units of temperature. The conversion from degrees Celsius to the dozenal temperature involves only a subtraction of four degrees. In mathematics, the Greek letter delta is often used to indicate a difference or change in a quantity. In the Greek alphabetical order, the fourth letter is delta. Thus, the resulting dozenal unit of temperature may be called a delta degree, notated \(^{Δ}C\).

On Electromagnetics
Ideally, a natural system devised from theoretical physics employing something akin to Planck units and a Lorentz-Heaviside expression would be used for electromagnetic units.

Using a Lorentz-Heaviside type of system, a unit of charge may be derived from the physical constants and the kinematic and mechanical units.
$$Charge = \sqrt{ε_{0} L^{3} T^{-2} M}$$
Ignoring any unspecified constant of proportionality that may be hidden by the metric system of units, a unit for current may be derived as the flow of unit charge per unit time. This gives a unit of current near 2 * 10^-10 Amperes, allowing an easy rule-of-thumb for conversion from metric current to dozenal. That is, simply shift the decimal jot, divide by a factor of two, and write the number dozenally.

The most commonly used electrical units of measurement for ordinary people have been Amps and Volts. It is preferable that these more than any other electrical units should be easily convertible to a dozenal system of units.

On Molar Units
The Avogadro constant is little more than a conversion factor between two different units of mass, the gram and the atomic or Dalton. The atomic mass is set as a twelfth of the mass of a carbon-twelve isotope. One mole is then the mass number of an element in grams.

Luminous Intensity
It would seem that luminous intensity ought to be derived in terms of energy and spatial considerations.

Setting the base units is sufficient to define the system, as a physicist competent in these matters can simply put these base unit quantities into a spreadsheet and get the derived units as outputs.

On Unit Symbols
The symbols T, L, M used above were variables for quantities or numbers as well as their units.
In the specification of a particular amount of a unit, the number is followed by the unit in word form or by a letter symbol abbreviating the unit word form.
The letter symbol abbreviating the unit of time may be s.
The letter symbol abbreviating the unit of length, the centimetre, may be l.
The letter symbol abbreviating the unit of mass, the gram, may be g, as in decimal metric.
The letter symbol abbreviating the unit of temperature may be ^ΔC.
The letter symbol abbreviating the unit of charge may be c.
The letter symbol abbreviating the mole may be mol, as in the decimal metric system.

Reference
https://en.wikipedia.org/wiki/Lorentz%E2%80%93Heaviside_units

1. Transcending Metrology

What it should be remembered is that a measurement system gives one chance to explore different parts of the base.  Unless you have a purposeful access to large numbers in a base (as I did with symmetry groups), measurement units do this for you.

2. Priorities.

The metric conversion in Australia took just 10 years, because they got their priorities correct.  There is plenty to study here, but money, education should be up front.  Entertainment and weather follow, then the news.  One can start converting things like road-speeds etc at a middling point

3.  Time and Calendar.

The calendar ought be changed to a base-free form early on.  The preference here is for the Dee calenday, which provides a leap day 8 times in 33 years, and the corresponding lunar calendar 19 leap-days in 77 years.  Anything further than this, and one starts to run into things like what happened in France with their tutti-frutti calendar.

4.  The Pendulum.

This was tried for many years around 1780-1850.  It really doesn't give a ready unit of length greater than 1/10 mm.  The french tried it with their 'toise of peru', while the english considered it when their standards were destroyed in the 1834 fire.  You get much better results from using other rulers.

5.  Temperature.

I would be tempted for the Gas-law constant \( R = N_A k\).  This is how k is derived.  Putting R=1, would make PV=NRT the same as PD = uT, where P=pressure, D=density, u = gas-daltons, and T = temperature. 

6.  Molar Units. 

The moles count the number of molecules, and one can from this, derive the number of reactive sites.  It is a rationalision of five different units.  (g-mole, g-atom, g-equiv, g-ion and something else).  Molar and molal can be replaced by spig-mole (s.g. / daltons) and touch-mole (w.w fraction / daltons), the units would be adjusted from g/L to kg/L to be base-free.

The definition of the mole is of a dependent unit.  Its size depends on the gram.  It would be better to define the dalton as 1/12 of the rest-mass of a neutral \(^{12}C\) atom, and then define the M-mole as M/daltons.  I write this dimensionally as Mn. (mass-as-a-number), where 1/n = u (symbol for unified mass-unit).

7.  Electricity.

The system of setting \(Z_0=1\) sets \(\epsilon=\mu=1/c\), already brings the advantages of the HLU when c=1.  The trick here is not so much to ues just the base units, but to map the units onto powers of C, eg \( C^0 = T,\ C^1 = L,\ C^3 = Q,\ C^4 = M\)  Doing this maps all of the Planck units near C5, and the mass units are equal to the energy.  In other words, C lengths = 1 time.

wendy.krieger wrote:The system of setting \(Z_0=1\) sets \(\epsilon=\mu=1/c\), already brings the advantages of the HLU when c=1.

Perhaps the benefits of the Lorentz-Heaviside type of system are best seen when the speed of light is set to one length per time, so that the Maxwell equations become the same as in the Système International (SI). However, with the proposal for the units of time and length here, this did not happen.

The unit for charge in the SI form can be derived from the equation for Coulombic force \(F\), ignoring the directionality of the vector by taking an absolute magnitude,
$$|F| = \frac{1}{4\pi ε_{0}} \frac{q_{1}q_{2}}{l^2}$$
for point charges \(q_{1}, q_{2}\) at distance \(l\) apart, where \(ε_{0}\) is the permittivity of free space. Since the force magnitude \(|F|\) is defined as mass \(m\) times the magnitude of the acceleration \(|a|\) in units of lengths per square times, \(|F| = m|a|\) has units \(MLT^{-2}\). Letting the two point charges be equal each to \(Q\), and rearranging the equation to solve for this, a formula for the unit of charge becomes
$$Q = \sqrt{4\pi ε_{0} ML^{3}T^{-2}}$$
If the constant term \(4\pi ε_{0}\) is kept on the same side as the charge \(Q\), then they are incorporated into the unit of charge, and the result is the electrostatic unit (ESU) of charge, the franklin, which in SI may be calculated from \(\sqrt{4\pi  ε_{0}} (0.001\textrm{ kilograms})^{0.5} (0.01\textrm{ metres})^{1.5} (1\textrm{ seconds})^{-1}\).
The Lorentz-Heaviside is supposed to incorporate the vacuum permittivity \(ε_{0}\) into the charge, but not the spherical surface term \(4\pi\).

Reference
https://en.wikipedia.org/wiki/Centimetre–gram–second_system_of_units

The proposed set of units can be directly derived from SI, by setting \(\epsilon=\mu=1/c\).  Both HLU and SI are rationalised, as well as NR (the proposed units).

The general form of Maxwell's equations is

\( \nabla\cdot D = \rho / \beta  \qquad  \kappa\nabla\times H = \tau D + \rho v/\beta \)

and

\( \nabla\cdot B = 0 \qquad \kappa\nabla\times E = -\tau B \)

This gives for example  \(\epsilon \mu c^2 = \kappa^2\), and one is free to set any two of these, or set any relation that allows this equation to be satisfied. 

I wrote these equations in this form, that were included in the Wiki.  You will see my name on the talk page there.  I've read Heaviside.  I've read Lorentz.  I even corrected the wiki pages on these things.

In Gaussian and HLU, \(\kappa=c\)  It was set up that way deliberately.  In SI and NR, \(\kappa=1\). 

In Gaussian and other unrationalised systems, \(\beta = \frac 1{4\pi}\).  In HLU and SI and NR, \( \beta=1\).

In order to make \(\epsilon=\mu\) as does HLU, Gaussian and NR, you need to set \(Z_0=1\).

When one does this, maxwell's equations look like this:

\( \nabla\cdot D = \rho  \qquad  \nabla\times H = \tau D + \rho v \)

\( \nabla\cdot B = 0 \qquad \nabla\times E = -\tau B \)

This is the identical result in these systems.

The photon-equation \( czD = zH = cB = E \), which solves Snell's law for a single photon, where z is \(Z_0\), gives for any medium \(\epsilon = D/E = 1/ Z_0 c\) and \(\mu = B/H = Z_0/c\), will likewise give D=H=B=E when c=z=1. 

But setting \(Z_0=1\) overcomes the problem in both HLU and Gaussian, where there is c statC delivered by an abA in one second. 

It is of course, misleading to suppose that the \(4\pi\) jumps out of any particular value.  One can easily replace \( F,\ Q\) with \( 4\pi F,\ 4\pi Q\), and the \(4\pi\) will appear when these are cancelled out.

wendy.krieger wrote:NR (the proposed units).

What do these letters abbreviate?

wendy.krieger wrote:I wrote these equations in this form, that were included in the Wiki.  You will see my name on the talk page there.  I've read Heaviside.  I've read Lorentz.  I even corrected the wiki pages on these things.

In Gaussian and HLU, \(\kappa=c\)  It was set up that way deliberately.  In SI and NR, \(\kappa=1\).

In the Wiki talk page, Kodegadulo writes that \(q_{LH}=I_{LH}c_{0}t\). In the proposal was mentioned neglect of a constant hidden by SI in the definition of current from charge per time.

Phaethon, Sun 15 Sep 2019 - 15:22 wrote:Ignoring any unspecified constant of proportionality that may be hidden by the metric system of units, a unit for current may be derived as the flow of unit charge per unit time.

Would it nevertheless be possible to define the current such that the proportionality constant would be incorporated into it despite the rationalisation and the speed of light not being set to one unit of velocity? This may involve the defining of a magnetic constant.

Reference
https://en.wikipedia.org/wiki/Talk:Lorentz%E2%80%93Heaviside_units

There is a system of describing Gaussian units, using a prefix ab- for the electromagnetic and stat- for the electrostatic systems.  So, one might suppose that Gaussian = esu + emu.  Here https://dozenal.forumotion.com/t22-the-design-of-the-c-o-f-system#84 I give a definition of the base units of current (abampere) and charge (statcoulomb).  The amount of charge delivered by an abampere in 1 second, is {c} statcoulombs.

The idea of using prefixes suppose that G (gaussian) divides into E and M and other units. (eg N).  If one wants to comprise Gaussian as a complete composite of SI-like systems, one needs also suffixes to deal with the \( 4\pi\) factor.   That is, we suppose G = E+N+M, and U=I+R+Y.

There's something like http://www.os2fan2.com/files/physics.pdf which deals with this.  But in essence, one eliminates the cgs 'c' factor, by supposing current is charge / time.  This replaces the mixed prefixes of G with ab-, stat- or nen-.  Likewise, the mixed suffixes of an unrationalised system U is replaced by a fixed suffix  (-), (-ero) and (-ade) to represent the I, R, Y components.

The definition of charge and current in the first link, refer to defining units in the MI and EI subsystems.  Because these are different subsystems, the relation Q=It does not apply, it's Q=Ict.  That is, Kode is correct with 1 abamperero = c statampereros, where c is the speed of light.  These are the HLU units: rationalisation is about \(4\pi\), not about \(c\).

But these two subsystems do not contain \(\epsilon\) or \(\mu\).  These belong to ER and MR resp.  The effect of this is that the charge unit Q ~coulombs = \(\sqrt{4\pi}\) ~coulomberos. 

If you kept the same Gaussian charge and current units, and rationalised, GU -> GI.  Whence you have \(\epsilon=1/4\pi,\ \mu=4\pi\).  If the charge units are considered derived, then one can have GU (Gaussian) -> GR (Heaviside Lorentz).

The N prefix is something that i devised, is essentially \(\sqrt{ \mbox{ab }\times\mbox{ stat}} \).  It equates to setting \(\epsilon=\mu\).  Unlike the gaussian / hlu, this system has a single unit per quantity.  That is, the unit of charge is derived by the relation \( \sqrt{Z_0 ML^2/T}\).  The unit under the square root is that of Planck's constant.

If you take the MKS system, and define \(Z_0=1\), the unit of electricity is 19.4 V and 1/19.4 A.  If you set \(\mu=1\), then the units become 1.120 mV and 892.062 A.  If you set \(\epsilon=1\), you get 336066 V and 3 µC for the units.  All of these systems would function in the sense of \(\epsilon=\mu=1\) when c=1.  They all satisfy \( \epsilon\mu c^2 = 1\).  But for the emu and esu derived systems, you need to add a power of the base (like SI with \(10^{-7}\) does.  With NR, you don't need to, because the defined constant is already in the ordinary range.

I still have not figured out a worded definition for Z_0, since it is not an electrical resistance.  (The gaussian unit of resistance is in EI, not NR).  But it took some ninety years of the finest minds to write the definition for the ampere.  It involves a constant \(2\cdot 10^{-7}\) in it.  Using 'c' here would mean that the separate units of electricity and magnetism would disappear when mapped against exponents of c rather than the base.

wendy.krieger wrote:If you kept the same Gaussian charge and current units, and rationalised, GU -> GI.  Whence you have \(\epsilon=1/4\pi,\ \mu=4\pi\).  If the charge units are considered derived, then one can have GU (Gaussian) -> GR (Heaviside Lorentz).

wendy.krieger wrote:They all satisfy \( \epsilon\mu c^2 = 1\).

The magnitudes of the electric permittivity and magnetic permeability constants \(ε_{0}\) and \(μ_{0}\) could be set to \(1/4\pi c\) and \(4\pi/c\), to allow a current as charge per time without a proportionality constant of the speed of light.

By the way that the magnitude of the unit of current was defined in the pendulum system of the original post, its multiple by the ninth power of twelve would be very near one ampere, thus making this system very compatible with that current metric electrical unit, affecting its derived electrical units simply too. To convert from metric amperes to units of current in this system, simply write the number of amperes in dozenal format and append symbols for the ninth power of twelve and the unit of current.

Similarly, the unit of charge according to the formula times the sixth power of twelve is near four times ten to the power of minus four coulomb. This allows a convenient rule of thumb for conversion from metric charges in coulomb to this pendulum system. The procedure is to shift by four positions to the right the decimal point of the number for the charge in coulombs, divide by four, and append symbols for the sixth power of twelve and the unit of charge of the pendulum system.

The pendulum system proposed here is probably the best example of a "minimum change" metrological system. As it stands, this pendulum system could be more easily used with current instruments and the conversions of units could be more facile than for any other dozenal proposal. Its author is open to further discoveries on how a dozenal metrological system may be improved. At the moment, there appear to be no major problems with the pendulum system for ordinary use, and if nobody can offer a better proposal, this is the one that the author will proceed with in every day living until a better suggestion might arise.

Length
The unit of length is exactly the centimetre. This is a legally acceptable part of the metric system, allowing its use publically in business and with the most readily available rulers.

Volume
The same applies to its cube the millilitre which is used of grocery items. All that needs to be done to make these dozenal is the writing of the numbers in dozenal format.

Weight
The unit of mass is the gram, exactly the same as a unit in the metric system. It can be legally used, so there is no prohibition with it in trade. The gram is commonly used for grocery items, in commerce and industry. All that needs to be done is that the numbers of grams can be written in dozenal notation. The abbreviating letter g unit symbol can even be preserved.

Density
The unit of density is the gram per cubic centimetre, which is the density of water under some conditions. This measure of density is practically used in sciences such as geology and chemistry. Unlike specific gravity, the magnitude of which is the same, it is not considered to be a unitless quantity.

Temperature
The size of the unit of temperature is exactly the same as the degree Celsius and the degree kelvin. These are the sizes of unit for temperature used commonly by ordinary people for reporting weather temperatures, and also in thermodynamic science. From the start, thermometers would not need to be replaced. All that would need to be done is that the numbers be translated and written with the dozenal positional notation.

Molar Units
In this pendulum system, the molar unit is exactly the same as that used in the metric system for the science of chemistry. Its symbol mol can remain the same. There is no change here except the writing of the numbers in base twelve.

Angular Measure
Plane angles are the same as radians, but written dozenally as tau radians, where tau is the ratio of the circumference of a circle to its radius. Radians are the standard natural unit of angular measure used in higher mathematics. For plane angles up to a full circular turn, the number in front of the tau radians label would range from zero one. These numbers would most typically be written in positional format, such that fractional portions of a full turn or perigonal angle would appear in the positional figure places after a dozenal fractional zot. The plane angles could also be expressed as pergross tau radians, which differ from tau radians only in that the dozenal positional fractional zot is shifted and a symbol or word for pergross is appended between the number and the symbols for tau radians. Radians are not considered to be unitless quantities, and should be accompanied by a symbol, which is ordinarily rad.

A compact unit symbol for dozenal radians could be devised. The mathematical constant tau has the symbol τ. Some have used the abbreviation by letters pg for pergross. Combining these symbols, instead of full words, the pergross tau radians could be written as pg τ rad. There is little need or advantage to expressing the number as pergross instead of as a fractional number in positional form by figures after the dozenal zot, and hence a more common form for the expression of the angle could be expected to be τ rad. A further simplification of this unit could be designed.

Solid angles are measured in steradians, with the number being written dozenally as a portion or fraction of the full surface area of a sphere. The unit would therefore be double tau steradians. This unit could be abbreviated in some way, perhaps dτs. The steradian participates in some derived units, such as radiance and radiant or spectral intensity.

Basically, the angular units proposed for this dozenal system are the same as are already being used.

All of the above units in this pendulum system are equivalent to units in the metric system, and thus involve no change to their standards. They are all legally allowed units and could be used for dozenal immediately. Scales on instruments for measuring these quantities could be readily adapted to dozenal.

Photometric units
The metric unit of luminous intensity is the candela, one of the basic defining units of the system. Because the candela does not appear in too many derived units, even though it is an official unit of the updated metric system, it is envisaged that the candela could be retained for some time as one of the stranger historical units of measurement (https://en.wikipedia.org/wiki/List_of_unusual_units_of_measurement). In the metric system, it has undergone a number of redefinitions, one of which is:

https://en.wikipedia.org/wiki/Candela wrote:The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Such a definition could be readily adapted to a dozenalised expression by converting the numbers and units such as hertz and watt to be in terms of the dozenal pendulum units. The metric radiant intensity is a derived unit as watt per steradian. The luminous intensity must be defined for other wavelengths of light through a luminosity function, a model of which can only be determined by experiment and not fanciful definition, because it involves human perception of how intense light is at different wavelengths.

Time
The only major basic unit of this proposed pendulum system that differs from metric system units is the unit of time. Consequently, all the derived units from the unit of time in this dozenal pendulum system differ from the corresponding metric derived units, which is very many of the units indeed. However, many near simple ratios of the dozenal to metric units are designed by the raising of the dozenal unit of time in seconds to integer exponents near zero being close to simple ratios. Here, the dozenal unit of time was chosen to be the square of five sixths of a conventional second. Conversion of metric time to the dozenal time is therefore by a hundred or a square of ten seconds being equal to a square of twelve dozenal times. Conversion from metric time to this dozenal time therefore simply involves writing the metric time in terms of the metric basic unit of time which is the second, dividing by a hundred by shifting the decimal fractional point or jot to the left, and writing the resulting number dozenally. For example, the number of seconds in a day is two dozen times the square of sixty, which is 86400. Dividing this by a hundred gives 864.00. This number in dozenal is half a cubic dozen or six dozen dozen. To this number are then added two further dozenal zeroes before the dozenal fractional zot, to give six quartic dozen times. This dozenal pendulum unit of time also allows retention of analogue clocks, whereas other proposals for a unit of time could require complete replacement of mechanical clocks.

Electromagnetic units
The metric unit of current, the ampere, is so close to a simple dozenal power of the unit of current in this pendulum system that an accurate rule of thumb conversion between them could be used, allowing the ampere to be retained almost unchanged except for notation in dozenal form and a new symbol. As the metric electromagnetic units are derived from the ampere, these units should convert well to those of the pendulum system, with the exception only of some of those derived units that contain time.

The electromagnetic units in this proposed pendulum metrological system are based on the magnitude of a universal constant of nature, the free space permittivity, being set to one. In this respect, the resulting system resembles gram-centimetre Electrostatic Units. However, the units of the pendulum system by their equations being rationalised, alike to the Système international and Lorentz-Heaviside units, are an improvement over historical electrostatic units. An advantage that these pendulum units have on the Lorentz-Heaviside units is that in the pendulum system the current is defined as charge per time, like in the Système international, whereas in the Lorentz-Heaviside units there is a theoretically and practically troublesome constant of proportionality in the equation.

More research provides the possibility of another dozenal system of electromagnetic units with desirable properties that may be beneficial to theoretical and experimental sciences of the small and large beyond the human scale, such as particle physics and relativity at speeds approaching that of light.

Conclusion
Any dozenist who claims to support a minimum change stance for conversion of the world from decimal to dozenal ought to be proponents of this pendulum system, or answer with another minimum change proposal. It should be emphasised that other dozenal proposals involve more units that are not legal standards and therefore impose an immense barrier to their credibility for introduction into society, contrary to this pendulum system proposed here. For those who want to be a dozenist with measure in real life, thanks to this pendulum metrological system they can now be.

Of all the units of the pendulum system, only those derived from the unit of time were not the same as any units of the metric system. Since this proposed unit of time was not a decimal power of a metric unit of time, some might worry whether use of this unit of time could be allowed legally.

Yesterday, a possible legal loophole or way around this was thought of, which is to use hectoseconds as a basic unit of time for the purpose of the derived units. As pointed out in the previous post, the hectosecond is one gross of the unit of time that was used as the period of the pendulum. Using the hectosecond or square twelve times the period of that pendulum would enable units derived from integer powers of it to be simple powers of ten of metric units, and at the same time simple powers of twelve of the units of the pendulum system that was based on the square of five sixths of a second. If the hectosecond be used to derived the unit of current, it would be that current multiplied by twelve to the power of the sum of one and twelve rather than the ninth power of twelve that would be close to the ampere. It so happens that twelve to the power of the sum of one and twelve is an approximation to ten to the power of fourteen. This has the effect that the unit of current derived from the hectosecond, using the formula with the other variables as before, is close to a decimal power of the ampere. With the hectosecond, the same electrical units as before would be included but at different powers of twelve. The benefits to conversion from decimal metric of using the hectosecond would be more obvious for the mechanical and kinetic units and other units for them being derived from integer powers of the hectosecond and other basic units such as of length and mass.

It is not at all unusual to the decimal metric system for powers of the base to be used as a multiplying factor to the proportionality constant in an equation by which units of measure are derived. For example, the basic unit of length is the metre, but the unit of weight, meaning amount of substance, is derived from the mass of water in a volume of one litre that is the cube of a decimetre, not the cube of a metre. Similarly, a decimal power was used in the definition of the magnetic constant to bring the sizes of the derived units into the useful range. In the decimal metric system, it is normal to use different powers of the basic units depending on the size required and to convert between these easily by multiplying or dividing by powers of ten using the laws of indices. For example, while one person might measure in millimetres and another could measure in kilometres, it would be easy to use these measurements in the same calculations by conversion to metres.

Likewise, for the development of a dozenal metrological system, powers of the base twelve could be used between the units to consolidate the dozen as the base relating the units. Honestly, this is why nomenclatures for the powers of the base twelve have been proposed, as dozenal counterparts to the decimal prefixes.

With this solution, a dozenal metrological system based using mainly allowed units as a least change proposal is possible.

Decimal Système International to Dozenal Pendulum System Conversion Table
Up to October last year, I was creating a table for conversion of quantities expressed in the decimal Système International to the proposed dozenal Pendulum System. I have not done any further work on the table since then. I have presented the table as a HTML webpage, located at https://dozenal.forumotion.com/h1-si-to-pendulum-system-conversion-table, separate from this forum thread so as to allow updates and modifications to be made to it as they arise. The table as a work in progress is incomplete, though it is comprehensive enough to be practical for most purposes, and the method for deriving values not included in the table can be worked out from the information provided. The table so far has a dozen plus three columns, and more than two hundred rows, less than two square dozen.

From a theoretical point of view, some values or units might by expressed differently in the ultimate Pendulum system than how they have preliminarily been presented. For example, I do not consider angular and solid angular measure to be unitless, and thus units derived with those quantities, such as angular momentum, should be notated differently than how they are in the standard decimal system. Similarly, refractive index is also not considered to be unitless. I also believe that wave number ought to be qualified by the phaenomenon being counted.

The table demonstrates how in most cases the existing decimal units can be converted to the dozenal units relatively easily without non-trivial conversion factors, such that the only taxing computation would be conversion of the decimal numbers to dozenal numbers. Apart from that, conversion between the numbers for the different sizes of measured quantities of the two systems can be achieved simply by shifting the fractional point. No other dozenal system is known to provide such an advantage for conversion from decimal to dozenal units. Only the electromagnetic units do not have a terminating conversion factor, because of the magnitude value chosen for a constant of nature being set to unity. Nevertheless, from the "Ratio Interpretation" provided, it is possible to derive a rough rule of thumb that can be used for conversion of electromagnetic units to dozenal units. The exact values for the reference values by which the basic photometric unit is defined have not been set explicitly in dozenal terms at this stage, though it is possible to guess a suitably accurate conversion to satisfy one's purposes.

Some of the insight of Wendy Krieger has been contemplated in the selection of sizes for the base units such as to produce a system that overall has units of convenient sizes in the human range of experience. For this reason, various values for the size of the basic temporal unit were considered, and these are indicated under the column headed "n" in the table. The effect of the choice of this parameter "n" was assessed by the "average" and "geometric mean" computed at the end of the table.

At the moment, most formatting of text such as italics, bold, superscripts, and subscripts, have been removed for the simplification of the HTML code by which the table is expressed. The context can reveal where numbers following letters are exponents or indices. In the spreadsheet form, cells that were dubious or not considered final were highlighted in red, but this formatting has also been lost.

In the instructions for conversion under the column headed "To Convert SI to Dozenal", "dec" stands as an abbreviation for "decimal", and does not mean the digit ten.

The information on the derived units of the Système International is taken from various sources, such as Wikipedia.

Molar Units
The number of species in one mole by the Avogadro constant in decimal is not approximately a simple power of ten. However, in the Pendulum system, while the molar unit is exactly the same as that of the metric system, its number of species expressed in dozenal is approximately a simple power of twelve, namely twelve to the power of twice eleven. This property would make calculation of the number of species from the number of moles or vice versa a simple matter of shifting the dozenal fractional punctuation character a number of positional places in the number or changing the index or exponent in scientific notation. This could be a convenient rule of thumb for students.

The molar unit as merely a conversion factor between different units of mass within the metric system would be superfluous in a system of metrology based on fundamental units in which aggregated quantities appear as multiples of fundamental units. The molar unit would then be merely a shorthand replacing the name of the species being counted. Although the mole is listed as one of the base units from which other units are derived in the decimal metric system, it is not really a unit of any single physical quantity.

The only essential base units would hence become those of time, length, mass, temperature, and electric charge. I call the base units of these quantities in a dozenal system of metrology durz, lonz, gravz, thermz, and charz. For a system of unspecified base, the terminal letter zed would be omitted from these terms.

Standard Musical Pitches and Frequencies in the Pendulum System
The base unit of time in the Pendulum System being the square of five sixths of a decimal metric second and less than a second would lead to fewer vibrations per base unit of time for particular pitches of sound by that same ratio compared to the frequencies in hertz, which are counts per second. The quantity being counted, which could be vibrations of sound or of light or oscillations of any periodic process in principle, is omitted from the unit notation in the decimal metric system. In the Pendulum system, a frequency of two dozen dozen vibrations per durz would be 414.72 hertz decimally. If this is multiplied by the twelfth root of two of equal temperament, the result is just less than the standard concert pitch of 440 Hz, but if multipled by 16/15 or ①④/①③ of just intonation it is just a little bit more. Thus, twice a square dozen of vibrations per durz would be the A-flat a semitone below the concert pitch of A. Since two square dozen is such a divisible number, it would result in the pitches of other notes in just temperament in ratio to it being often simple numbers in this dozenal system, which would be convenient.

By simply converting the decimal number of hertz to a dozenal number, it is sometimes possible to get a good approximation to the number of dozenal vibrations per durz of the same frequency. For example the decimal number 440 in dozenal would be ③⓪⑧⁏ which is close to the number of vibrations per durz of 440 Hz. So, if you do not mind using decimal digits, then you could just change the unit symbol from Hz to per durz or d^-1 or /d after the number to convert frequencies approximately from decimal metric to the Pendulum system.

To properly do the conversion from hertz to per durz more accurately, multiply the decimal number of hertz by a hundred by shifting the decimal point to the right two positional places, write the number in base twelve, and then move the dozenal fractional point back two places to the left. You do not need to change your tuning forks, just how they are labelled to make them dozenal in the Pendulum system.

Musical Tempo in the Pendulum System
Beats Per Minette
Musical tempo is expressed in beats per minute. In a dozenal system, these tempi could be converted to beats per minette. The ratio of a minute to a minette is six to five, so for the same tempo there will be 6/5 times more beats per minute than beats per minette. To convert to beats per minette from beats per minute, multiply the number of beats per minute by five sixths. For example, a tempo of ⑧⓪⁏ beats per minute would become 80 beats per minette. In this case when the number is not too big, the conversion can be done by changing the base of the decimal number of beats per minute to base twelve and then pretending that the figures are decimal digits. These decimal digits could then be converted to a dozenal number. Alternatively, multiply the decimal number of beats per minute by ten by shifting the decimal point to the right one place, write the resulting number dozenally in dozenal figures, and then shift the dozenal fractional point back to the left one positional place in the dozenal number.

Beats Per Durz
Since there are six dozen durz per minette, to convert beats per minette to beats per durz, multiply the number of beats per minette by six dozen. This is the same as shifting the dozenal fractional punctuation mark two positional places to the right and then dividing by two, a simple operation to perform. A unit symbol for this could be bpd. These numbers would be likely to be conveniently rounded in dozenal if they were generated from divisible whole decimal numbers. Here, durz means the same as dyse, a double secondrie. They are both abbreviated monoliterally by their initial letter d.

Beats Per Secondrie
It would alternatively be possible to use beats per secondrie, where a secondrie is a square dozenth of a minette. The notation for this unit could be bps. It would be obtainable from beats per minette simply by shifting the dozenal fractional punctuation mark two positional places to the right. The numbers, though more likely to be whole or rounded, may be unnecessarily large or accurate in terms of the number of significant figures compared to beats per minette.

It is not necessary to change a metronome instrument to make it dozenal; as long as it is a mechanical pendulum metronome it is only necessary to change how the ruler scale along the metronome is labelled from decimal to dozenal units.