Tag: base twelve

Logarithms of Primes in Bases

Properties of logarithms enable calculations to be conducted by simpler operations, provided the logarithms and antilogarithms are available for consultation from tables or by a mechanical slide rule or similar device if not by computation. A logarithm of a multiple of two quantities is the sum of their logarithms. This enables multiplication calculations to be reduced to addition. A logarithm of a fraction is the difference between the logarithms of its numerator and denominator, enabling division to be replaced by mere subtraction followed by conversion of the result to the power term by the antilogarithm or exponentiation. It follows from these rules that the logarithm of a composite whole number can be reduced to a sum of multiples of the logarithms of its prime factors, and indeed the logarithm of any rational number can be decomposed into addition and subtraction of multiples of the logarithms of the prime factors of the numerator and denominator of the fraction. Furthermore, logarithms can be used for calculating irrational numbers that result from algebraic operations on rational numbers. A rational number can be raised to any fractional power by applying a logarithm, which can then be manipulated by the rules of logarithms to provide the fraction multiplied by the logarithm of the rational number. For example, the rational number three to the power the fraction a half is the same as the irrational square root of three and can be calculated with the aid of logarithms by taking the logarithm of it to any base, resulting in half the logarithm of the prime number three to that base. That result is then converted from a logarithm to a power term by exponentiation to furnish the square root of three to an accuracy of the number of significant figures allowed by the logarithms and antilogarithms available.

While the rules of logarithms apply no matter what consistent positive real base they have, it is clear that logarithms would be especially useful if the base chosen were one such that the logarithms of the simplest prime numbers to that base were approximated peculiarly well by a limited number of significant figures of their logarithms. Unfortunately, logarithms are usually transcendental irrational numbers that cannot be exactly represented by terminating strings of numerals in positional notation. Nevertheless, sometimes the logarithm of a number such as a prime number can be approximated well enough to a few significant figures, which would make arithmetic with them involve fewer steps and be faster. For example, if the base of the logarithms is chosen to be close to a root of the prime number two whereby two is raised to the power of a unit fraction or reciprocal of a counting number, then raising that irrational base to the power of that counting number will result nearly in the prime number two with the consequence that the logarithm of two to that irrational root base will be nearly a whole number capable of being represented by a finite number of significant figures. The creation of such a root of two as a base is equivalent to a temperament of the musical octave into equal geometric steps, with the ratio between the frequencies of adjacent notes a step apart being the base of the logarithms. The task of finding a base of logarithms such that the logarithms of the smallest prime numbers are approximated very well by terminating numbers is hence equivalent to finding temperaments of the octave that align with the frequencies of the first prime numbers as harmonics.

A number of temperaments of the octave are already known to provide good coincidences with the first few harmonics. The most widespread is the equal temperament of the octave into twelve semitones, which corresponds well with the harmonics that are powers of the prime numbers two or three, and to lesser accuracy the prime number five. Other temperaments can approximate more of the small prime numbers. For example, temperament of the octave into six dozen equal step notes can provide good enough approximation to the first few smallest prime numbers two, three, five, seven, and eleven. Since the twelve semitones are a subset of these six dozen steps and six dozen is half the second power of twelve, temperament of the octave by a power of base twelve is particularly useful for music and would be good as a base of logarithms for mathematical calculations. While there is little wrong with temperament of the octave into six dozen steps in music, a slight annoyance could arise with this formulation of the base of logarithms mathematically because there are two bases involved: binary, the logarithm to the base of which gives the number of octaves, and dozenal for writing the logarithms in numerals and expressing the number of steps within the octaves, but just one base for the number of different numerals and for the base of the logarithms and all computation would rather be desired. Is there a way to have the same base for the base of the logarithms and for the number of different digits by which the logarithms are notated? The answer is almost.

If the problem is approached initially by finding base B such that B^(1/B) is nearly the frequency ratio of the step in the octave desired, then the same base can be used as the base of the logarithms and as the number of different numerals for notating numbers such that the logarithms will tell the number of those steps, but the logarithms will not necessarily round well at the octaves to powers or whole number multiples of that base B. It so happens that the temperament to six dozens steps per octave is approximated when the base B is the square or second power of twenty-six. A base for the logarithms can be constructed by sectioning the square of twenty-six into twenty-six steps geometrically. The logarithms of the smallest prime numbers two, three, and five to that step as base, which is equivalent to half the logarithms to the base twenty-six, will be particularly accurate to two significant figures when written in base twenty-six. The logarithm of the prime number seven to this base is not as accurate at just two significant figures. Thus, computations using logarithms of numbers containing the first three prime numbers would be unusually convenient in base twenty-six, which could be called base two dozen plus two or twenzy-two in another form of dozenal parlance. There are as many upper case letters in the modern English variety of the Roman alphabet to use as symbols for the numerals for this base.

More accurate results for the logarithms of also the next prime numbers seven and eleven can be had to two significant figures by base thrice eleven, using a third of the logarithm to that base. There are said to be as many runes in the English futhorc as the number of different numerals required for this base thrice eleven. However, apart from such a base being an odd one to use for general purposes in society, like base twenty-six it would not round to whole multiples for the number of steps at the octaves.

Is there a way to allow the numbers of temperament steps at the octaves to be round multiples of the base? This is possible if the step size for the base of the logarithms can be defined in terms of powers of the number of equal temperament steps per octave. To make that number of steps a convenient base to use, it is made to be a power of twelve, either twelve itself or its square. For a step near two to the power of a square twelfth, the square of thrice thirteen is close to the base in B^(1/B) required for the logarithms, but the logarithm of two is not as accurate to two significant figures and the notation would have to be in base twelve for the numbers of steps at the octaves to be rounded.

More conveniently, the base of the logarithms for the semitone is given nearly by the cubic dozenth root of twelve to the power of forty. When such logarithms are written in base twelve, there are few significant figures to a good degree of accuracy for the first few prime numbers. The prime number seven requires more significant figures for a not worse level of accuracy, but they are not too many to be useful in approximate calculations. These logarithms give approximations for the numbers of semitones for the harmonics, which are easy to remember when musical theory is understood. These logarithms are defined in terms of powers only of the base twelve and can be reduced to simple arithmetic using the base twelve logarithms.

It should be mentioned that the fortieth root of ten also nearly gives the semitone step, but expressed that way the base of the logarithms and the base of numeration to allow rounded numbers of semitones at the octaves for this to be useful in conjunction with musical theory are not the same. The common decimal logarithms of the smallest prime numbers two, three, and five are accurate enough to two significant figures if multiplied by forty or four. A thousand to the power of a hundredth partitions the octave into ten equal steps. Not many useful bases B have that property of having whole number values of n and m to make B^(n/B^m) partition the octave into nearly B steps. Binary bases, decimal, and dozenal do. The dozenal option that I have described has the extra benefits of giving accurate values for the logarithms of the first few prime numbers by a small number of significant figures and agreeing well with the most normal musical temperament.

Dozenal Numeral Ten

I have designed a glyph for the numeral ten to be used in base twelve numeration. Various characters have been used historically for the digit ten in base twelve numbers. One of the earliest used by a dozenal society is the Pitman turned digit two, based on the initial letter t of the word ten. Various conventions have been tried for the letter to be used for the numeral ten. One is the letter A, part of the system of transdecimal extensions to the Indo-Arabic digits used for hexadecimal numbers and called IBM computerese by dozenists. This scheme is considered to emphasise base ten as the ace of bases too much for dozenists by the first letter of the alphabet starting on ten. Another option is the letter J because it is the tenth letter of the alphabet. Yet another proposal is the letter d standing for dec or dek meaning ten. In recent years, some dozenists have begun annotating that numbers are to be read as written in base twelve by suffixing them with a subscript letter of the alphabet. This means that in order for dozenal numbers to remain distinct from decimal numbers, the decimal numbers need to be annotated with a literal suffix. However, these are not standard practice in formal mathematical literature, where bases are annotated when distinction between bases is necessary by digits in decimal format rather than by letters of the alphabet. The most popular numeral for ten among dozenists currently is the Pitman turned two ever since it entered Unicode. As such, this symbol can be interpreted as a numeral and not just a letter, bringing it into line with conventional mathematics.

My design is based on all of these literal characters for ten. Firstly, it is derived from the Pitman turned two by closing the curl in order for the character to have a distinct seven-segment modular element display. This fixed one of the disadvantages of the original unmodified Pitman turned two whereby it looked too similar to other numbers or characters, including the numbers two and seven, and the letter zed. However, closing the loop resulted in a figure that looked too similar to a style of the digit three having a horizontal top bar. To improve this issue, the next stage was to make the numeral look more like the tenth letter J of the alphabet while still resembling the Pitman turned two by a horizontal top bar and closing curve with the end meeting onto the partial stem. The present latest version modifies this further by making the join of the closed curve attach to the partial stem tangentially upwards. The resulting glyph can be written by hand quickly and effortlessly without lifting the implement from the page until the digit is complete. Additional bonuses are that it conveys a lower case letter d and double story letter a. Thus, my design proposal solves most of the conflicting usages of different letters for the digit ten by merging them all into one character. This character remains distinct enough from other alphanumeric symbols and glyphs to retain its unique meaning dedicated to the numeral ten for numeration using base twelve.

Further benefits of my design are that it has cues of the digits five and two that are the prime factors producing the number ten. As well as the turned digit two derived from the Pitman turned two, it contains a reversed digit five. Another effect is that a horizontal line and a closed curve suggest the numerals one and zero of the number ten in decimal format joined together and written vertically. This may increase subliminal identification in a transition from decimal to dozenal notation.

In summary, my design for the numeral ten for dozenal numbers satisfies the following inclusively:

  • The initial letter t of the word ten from the Pitman turned two;
  • The initial letter d of the morpheme dec or dek for ten;
  • The tenth letter J of the alphabet;
  • The letter a in lower case double story style from IBM “computerese”, often used for the numeral ten in hexadecimal numbers;
  • The decimal digits 1 and 0 forming the number ten in decimal format joined together and written vertically one under the other; and
  • The digits for the prime numbers two and five of which the number ten is composed as the product.

Dozenal Forum

There is a certain dozenal forum that currently has forty-three topics and 151 posts.

Posts include the topic of dozenal directions from Monday 21st September 2020 last year, where the main author on that forum proposed a system of dozenal compass directions whereby directions that are a twelfth of a turn from a cardinal direction are indicated by the word “of” between the names of two adjacent cardinal directions, with the direction after the word “of” being the angularly nearer of the cardinal directions. The abbreviation proposed was “XoY”, where X and Y are the initials of the cardinal directions. The system was extended to indication of quarter twelfths of a turn and preserves by incorporation the existing conventional scheme of binary divisions in compass directions.

Another topic in the forum is a proposal for a dozenal metrological system that preserves as much as possible existing units of measurement of the decimal metric international system reframed dozenally. The system allows existing decimal metric measurements and their units to be more easily converted mentally without the aid of calculational devices to dozenal than to any other dozenal system because it is so constructed that the conversions require only shifting decimal points or dozenal fractional points, apart from the conversion of numbers from base ten to base twelve. The most recent post under this topic was on Friday 12th February 2021 this year. The topic was started in the forum on Sunday 15th September 2019, but contains principles which were conceived and published earlier, for example the unit of time and the consequent unit of length leading to area and volume derived from it were communicated on the DozensOnline forum on Saturday 21st January 2017, and stated to have been conceived the previous year and can be read at the webpage address https://www.tapatalk.com/groups/dozensonline/all-your-base-are-belong-to-us-t1615.html#p40005757. The unit of mass was mentioned on Friday 28th April 2017 at https://www.tapatalk.com/groups/dozensonline/requirements-for-implementation-of-uncial-t1591.html#p40009356. The unit of temperature was mentioned on Monday 12th June 2017 at https://www.tapatalk.com/groups/dozensonline/requirements-for-implementation-of-uncial-t1591-s24.html#p40010490. There is a table for conversion from decimal metric to the dozenal system with hundreds of units of measurement.

Another topic on the dozenal forum, from Thursday 8th October 2020 last year, is on how the decimal metric millimetre can be retained as being dozenally incorporated into a system based on the troy weight and including ounces and a carat weight. It is possible for several consistently dozenal metrological systems to be employed to allow retention and peaceful co-existence of contemporary units of measurement. Another example of such a system is that containing the inch and foot, of which the yard is a multiple by a dozenal divisor or snapping point.

From Tuesday 29th September 2020 last year, there is a topic on how Old Irish metrics were apparently consistently dozenal.

In the Mathematics section of the forum, from Monday 13th April 2020 there is a topic on probabilities of prime and composite numbers related to their importance or rank by occurrence or frequency relevant to which of them should be chosen in the formation of a base of numeration. This concept is related to or an elaboration derivable from the post here entitled “The Trouble With Base Thirty” from Saturday 26th January 2019. Soon after this article was published here, several of its concepts or key persuasive arguments appeared by different authors on the DozensOnline forum without attribution. The concept of the importance rank of divisors appeared earlier on Thursday 8th June 2017 at https://www.tapatalk.com/groups/dozensonline/prime-numbers-in-metrology-t1713.html#p40010405.

Also in the Mathematics section of the forum, there is a topic on ratios of decimal and dozenal powers from Sunday 20th October 2019. The concept is relatable to the previous topic on dozenal rounding from Tuesday 30th May 2017 at Rounding, Surrogates, And Auxiliaries – Dozensonline https://www.tapatalk.com/groups/dozensonline/rounding-surrogates-and-auxiliaries-t1748.html. These topics are relevant to implementation of dozenism by conversion of decimal values to dozenal preferred values. They are contrary to the misconception of certain dissenters against dozenal that sizes converted from decimal could not be made to look elegant in dozenal. In fact, as was demonstrated, values converted to convenient dozenal numbers can be organised with excellent memorability and optics. The methods of conversion discussed demonstrated that interconversion between decimal and dozenal could be done mentally. There is some further discussion of roundness under the topic “Ripples and Awayness” in the Mathematics section on the dozenal forum from Monday 12th August 2019.

Another topic, from Thursday 19th Sep 2019, in the Mathematics section proves that fifths can be represented better in dozenal than thirds can be in decimal. This argument and conclusion is contrary to the mistaken beliefs of the antagonists infesting the DozensOnline forum. The argument also implies that octal encoding its square is better than the square of four as a base, again contrary to the sway on the DozensOnline forum. The topic incidentally contains a recurrence relation and summation defining the base of the natural logarithms dated from July 2010. A single function of a running variable there defines the numbers of the continued fraction of the base of the natural logarithms. It looks like original material to me despite lack of a reference for the fairly common knowledge of the continued fraction and associated notation, although continued fractions are not mentioned explicitly in the comment.

There is a topic from Saturday 7th September 2019 outlining a proof of the limitation of the number of regular four-dimensional figures bound by straight figures to six and mention of lack of fivefold symmetries in regular bricks. This is a recurrent theme on the dozenal forum, for example under “Angles in Crystals” in a comment of Friday 6th September 2019 under the topic “Comments on Angle Units” in the Metrology section. It is also stated earlier on Thursday 8th June 2017 at https://www.tapatalk.com/groups/dozensonline/prime-numbers-in-metrology-t1713.html#p40010405. Incidentally, there is a discussion of angles in a dodecahedral crystal in the topic “Pyritohedral crystal” in the Phaenomena section of the dozenal forum. In the Mathematics section of the dozenal forum, there is a topic on how simply angular fractions of a circle may be represented in rectangular co-ordinates. This demonstrates the greater simplicity of twelfths than tenths or even fifths of a circle. Furthermore, twelfths of a circle are not less simple than sixths or eighths, showing dozenal to be not inferior to senary or octal for the purpose of angular measure in the context of Euclidean constructability by straight unmarked ruler and compass. The unusually simple expressions for the double dozenths of a circle appeared earlier in an issue of the Duodecimal Bulletin, which was not cited.

Pyritohedral crystal Empty

In the Phaenomena section, there are also topics on global atmospheric air current zones and hexadactyly.

Another section of the Dozenal Forum is on Nomenclature. From Thursday 3rd October 2019 there is a topic on mnemonics which builds on the earlier post here from Tuesday 9th July 2019, “A Reply to Dozensonline, Number Bases, Dozenal, Request for Help with mnemonics for Dozenal”. From Monday 30th September 2019 on the dozenal forum there is a topic on Proto-Indo-European names for numerals. This is related to the earlier topic at https://www.tapatalk.com/groups/dozensonline/uncial-nomenclature-t1570.html on DozensOnline. The mnemonic solution and Proto-Indo-European nomenclature proposed are related.

As a proposal for common speech dozenal analogies of the decimal terms million, billion, trillion and milliard, billiard, trilliard used for example in finance, the nomenclature based on the suffix -lliad and Greek prefixes was described from Friday 9th August 2019 on the dozenal forum. For a more technical scientific style of nomenclature for powers of twelve to be used for example in units of measurement, from Saturday 28th September 2019 a system with the suffix -on or -ino and Greek prefixes was described. These systems incorporate the advantageous fourth power of twelve as base, as advocated here on Friday 22nd March 2019 in “A Reply to https://www.tapatalk.com/groups/dozensonline/duodecimal-myriad-system-t1970.html#p40018012”. Except the prefix “enkomi” for eleven seeming to be a bit long, in style and for international applicability, these proposals seem better than those on the DozensOnline forum. For example, -on or -ona is better than -qua, and -ino is better than -cia because there ought to be a vowel between the consonants in many languages. Derivation of the suffixes from the Latin word uncia was described to show that this is possible. However, in style it seems inferior to the earlier versions. A compatible version of the nomenclature for every power of twelve was proposed. In the nomenclatures, for example in the naming of geometrical figures, the consonant z rather than ch for twelve the dozen or zero was preferred.

Another section of the forum is on the design of numerals. There is discussed dozenal notation and design of numerals. On Saturday 17th August 2019 under the topic “Co-existing bases” was proposed the reversed semi-colon Unicode 204F as a dozenal fractional point marker. On Saturday 18th April 2020 last year under the topic “Font CSS” in the Forum Management section of the dozenal forum, the reversed comma was proposed for separation of groups of four numerical figures in dozenal numbers to match the reversed semi-colon as dozenal fractional point. These have been implemented on the dozenal forum so that they appear and anyone can type them with dozenal distinct against decimal numerals. On Friday 10th April 2020 last year, a character for the numeral eleven was proposed looking like a Gothic arch.

Lastly, there is a section for references on the forum and in the forum there are links to sites such as DozensOnline and an author of poorly competing schemes, such that if anyone monitors the source of internet traffic or referral webpages for those websites, he would know about the existence of the dozenal forum. Many ideas from here or the dozenal forum have been appearing soon afterwards by other authors on the DozensOnline forum.

Dozenification of Contemporary Weights

Previously (Jan 21 2017, 09:21 PM, DozensOnline, Applications, “All Your Base Are Belong To Us”; Apr 28 2017, 08:47 PM, “Requirements For Implementation Of Uncial”) was proposed how using division of the day by a power of the dozen, the gravity of the Earth, and the density of water can produce a unit of mass equal to the metric gram. All metric weights can thereby be converted into a coherent dozenal metrology by moving the decimal point to write the weight in grams, followed by changing the notation of the number from decimal to base twelve.

In choosing units of measurement for a dozenal metrological system, for minimum change and least disruption it is preferable to retain existing units wherever it is possible to incorporate these into a dozenal framework. Retention of the maximum number of existing units while allowing dozenal scales of magnitude would suggest that existing units of measurement still in use that already have dozenal ratios of their magnitudes should be chosen. Retention of existing measures acts as a celebration of the people’s choice of dozenal in the history of measurement and evidence for the benefits of twelve as a division. For this reason, I suggested retention of the inch as a unit of length, which is multiplied by twelve to produce the foot, and has been divided dozenally to the pica and point in typography or graphical design.

As mentioned previously (“Base Twelve in Measurement”, Knew’n’Tell, Ideas & Observations, Thought Views, WordPress, 17th Dec 2018), there are “for weight twelve ounces to a pound troy”. The troy system of weights was used in Britain for weighing of pharmaceuticals of medical prescriptions until it was made illegal since the year 1976 for such use as part of decimal metrication. The troy ounce is, however, still legal for use in the trade of investment precious metal bullion. Another pre-metric system of weights still used is the avoirdupois, which forms part of the American Customary and British Imperial systems of measurement. The troy and avoirdupois systems share as a common unit the weight called a grain, which is about 0.0648 grams, but differ in the multiples of this unit in the formation of larger named units. In the troy system, 480 grains form an ounce troy, so 480 * 12 = 5760 grains form a pound troy, 5760 * ~0.0648 g = ~373.25 g. In the avoirdupois system, on the other hand, seven thousand grains form a pound avoirdupois, 7000 * ~0.0648 g = ~453.6 g. But the avoirdupois pound has sixteen ounces avoirdupois rather than twelve, so the avoirdupois ounce is about 453.6 g / 16 = ~28.35 grams, nearly the same as an ounce troy of about ~373.25 g / 12 = ~31.1 grams.

For unification and dozenification of these current and related systems of weight, I propose that their ounces be dozenalised by being rounded to the same thirty grams, which dozenally is two-and-a-half dozen grams. This is the same as a metric ounce that already exists, so it is not a new unit. Additionally, I suggest that the multiples and divisions of this ounce to form the other named units of weight in the systems should as much as possible be the same as or similar to the ratios of units in the troy and avoirdupois systems, but should be changed only where such ratios involve awkward prime numbers in their factorisations, such as the prime number seven seen in the 7000 grains avoirdupois per pound avoirdupois, or where multiples or divisions are overly decimal (Conversion of decimally rounded numbers to dozenally rounded numbers was described earlier at DozensOnline, Number Bases, Dozenal, “Rounding, Surrogates, And Auxiliaries”, May-June 2017). This procedure allows the adjusted units to resemble in magnitude and relations the existing units, and is thereby along the lines of minimum change while at the same time being dozenalised. After conversion of the sizes of the units, it remains only for the numbers to be written dozenally using dozenal numerical notation of twelve phalangeal numerals.

The dozenified troy, which I call the trade weight system, would then be
0.0625 grams = 1 trade grain
1.5 grams = 1 trade pennyweight = 24 trade grains
30 grams = 1 trade ounce = 20 trade pennyweights = 480 trade grains
360 grams = 1 trade pound weight = 12 trade ounces = 240 trade pennyweights = 5760 trade grains

The dozenalised avoirdupois system or unified American Customary and British Imperial weight systems, which I call the system of civil weights, would become
0.0625 grams = 1 trade grain
1.875 grams = 1 civil dram = 30 trade grains
30 grams = 1 trade ounce = 16 civil drams = 480 trade grains
480 grams = 1 civil pound weight = 16 trade ounces = 256 civil drams = 7680 trade grains
5760 grams = 1 civil stone = 12 civil pounds weight
11520 grams = 1 civil quarter weight = 24 civil pounds weight
46080 grams = 1 civil “hundredweight” = 4 civil quarters weight = 96 civil pounds weight
829440 grams = 1 civil ton = 18 civil “hundredweights” = 144 civil stone = 1728 civil pounds weight

Common to both these trade and civil systems are the thirty gram metric ounce and the trade grain of a sixteenth of a gram. Sixteenths and other binary powers are very convenient for weights and are much simpler in dozenal than decimal, as mentioned before (“The Trouble With Base Thirty”, Knew’n’Tell, Ideas & Observations, Thought Views, WordPress, 26th Jan 2019). At first it might have seemed that the old grain being common to both the troy and avoirdupois systems should have been maintained unchanged. However, in effect that would be no change or no dozenalisation, as the value of the old grain would not be a convenient fraction of the dozenal basic unit of weight, and, with the old grain, only one unit would be common to both the troy and avoirdupois, whereas the sixteenth of a gram allows both the grain and ounce to be the same in both systems dozenalised.

Notice too that in the civil weight scheme, the number of civil pounds weight per civil ton is a cubic dozen, which is near the two thousand pounds per ton of the American Customary and is at the same time the dozenal analogue of the thousand or ten cubed of kilograms per tonne of the metric system.

A weight currently used for gemstones is the carat, which was formerly metricated to 0.205 grams and latterly downgraded to the more decimalised 0.200 grams. The carat is said to be derived from a 24th part of a solidus, where a solidus was a 72nd part of a Roman pound weight. Unifying the Roman pound weight or libra with the trade pound of 360 grams, this would lead to 360 g / 72 = 5 grams for the gem solidus, and 360 g / (24 * 72) = 360/12^3 = ~0.2083 grams for the gem carat weight. I therefore propose replacement of the metric carat by this dozenalised carat of 5/24 grams. However, in earlier times, the number of the gold solidus making up a Roman pound was sixty rather than 72, which would produce a gold solidus weight of 360 / 60 = 6 grams, and 360 g / (24 * 60) = 0.25 g or a convenient quarter of a gram for the gold carat weight. This value for the carat is necessary if the number of trade grains making up a carat weight is to be four. However, since the quarter gram value differs so much from the metric carat, I propose that the quarter gram carat be limited to being a unit of the trade system for precious metals, while the 5/24 gram carat would replace the metric carat used for gemstones.

Hence, the system of trade weights can be supplemented with these further weights, where:
5/24 grams = ~0.2083 grams = 1 gem carat = 1 square dozenth trade ounce = 1 cubic dozenth trade pound
0.25 grams = 1 gold carat weight = 1.2 gem carats = 4 trade grains
1.25 grams = 1 trade scruple = 5 gold carats weight = 6 gem carats = 20 trade grains = 1/4 gem solidus
5 grams = 1 gem solidus = 24 gem carats = 1/72 trade pounds weight
6 grams = 1 gold solidus = 24 gold carats weight = 96 trade grains
360 grams = 1 libra = 60 gold solidi = 1440 gold carats weight = 1728 gem carats

The Latin word siliqua can be used for a carat. The Greek word nomisma can be used for a solidus.

References:
https://en.wikipedia.org/wiki/Troy_weight
https://en.wikipedia.org/wiki/Apothecaries%27_system
https://en.wikipedia.org/wiki/Avoirdupois_system
https://en.wikipedia.org/wiki/Ounce
https://en.wikipedia.org/wiki/Carat_(mass)
https://en.wikipedia.org/wiki/Solidus_(coin)