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.

Disk Drive Frequencies

This morning in the newspaper Irish Independent on Saturday, I read the article entitled “Laptops can be crashed by Janet Jackson song, Microsoft says”, by Josie Ensor on pages 24 to 25. This prompted me to recount an experience I had and wrote about some years ago:

“Monday 8th March 2010. Yesterday I was using the compact disk […] that I bought on 10th January 2009. On my computer, I used the […] software on the disc to record and analyse […] in order to measure the frequencies […]. Almost immediately after I recorded […] there was an error on my computer and I was forced to close the software and I restarted the computer. Then I recorded the set […] instead. When I listened to the recording through earphones I noticed a descending glissando note […]. When I analysed the spectrogram of the recording, I saw what appeared to be like a formant of linearly decreasing frequency […]. By choosing spectrogram controls in the software program and setting the contrast at a maximum and brightness quite low and increasing the “Analysis window length” to a maximum I was able to see what appeared to be overtones or higher formants of the first glissando note. The other frequencies also decreased in pitch linearly with respect to time but with larger slope of the oblique lines as the formant increased in pitch. I noted particular frequencies of the formants at the arbitrary time 1.161 seconds from after the beginning of the recording. At that time in the recording, the fundamental frequency of the glissando note was 1549 Hertz; the frequency of the second formant or first overtone was 2268 Hertz; the third was 3139 Hertz; the fourth was 3888 Hertz; the fifth was 4597 Hertz; and the sixth was 5367 Hertz. I examined the ratios of these frequencies and found that they were as ratios of simple whole numbers. Setting the frequency of the first formant or fundamental frequency as one unit, the frequencies of the notes or formants are: f1 = 1; f2 = 3/2; f3 = 2; f4 = 5/2; and f6 = 7/2. From these values, where f_n = (n + 1)/2, it can be seen that the sequence of the frequencies forms an arithmetic sequence. Since I had noted these simple rational frequencies at an arbitrary time, I reasoned that the ratios would be maintained at all times during the glissando. I took more data points of the frequencies at different times and plotted the results in the software […] to determine the equation of the fundamental frequency trendline. I found the linear regression equation Frequency (in Hertz) = -1382.5t + 3134.7 where t is the time in seconds from the beginning of the recording and the frequency is that of the fundamental formant. I plotted the trend lines for the other formants by multiplying the above equation by the ratio of that formant frequency to the fundamental frequency and superimposed this graph upon the picture of the spectrogram. I found a better fit to the upper formants by changing the intercept by translation from 3134.7 to 3157 without changing the slope very much from -1382.5 to -1383. The graph according to the equations fit the spectrogram perfectly. I do not know what the source of the glissando note was, although since I was making the recording while running the […] software directly from the compact disc drive of my computer, the source may have been the disc drive spinning.”

I then went on to speculate about the source of the observed ratios of the overtones, comparing them to musical intervals. I continued:

“Thus, if the frequencies that were produced simultaneously were of similar intensities, which they were not, then the chord produced would be a major chord in root position or uninverted, with the fundamental frequency as the root and base of the chord.”

I then went on to eliminate various types of sources of sound:

“[…] a vibrating string is supposed to produce overtones at whole number ratios to the fundamental frequency, […] about overtones that are produced in cylindrical pipes, both open and closed at one end, […] in a pipe that is open at both ends, […]. Since the frequency of a wave is inversely proportional to its wavelength by the velocity or speed of propagation of the wave through the medium as the constant of proportionality, […]. Thus, the ratios of harmonics produced by a pipe that is open at both ends would be the same as for those produced by a vibrating string. In contrast, for a pipe closed at one end, […]. Thus, the ratio of the nth harmonic frequency to the frequency of the first harmonic in a pipe that is closed at one end is 2n-1. These ratios increase arithmetically as odd whole numbers rather than half integer numbers, therefore the pipe closed at one end could not be the type of source of the glissando note that I recorded yesterday. The source could not be a vibrating string or pipe that is open at both ends either, since in both of those sources the ratios of harmonics to the frequency of the fundamental frequency increase arithmetically by a difference of whole numbers rather than half integer numbers, unless what I assumed to be the fundamental frequency of the glissando note is not really the fundamental note. This frequency was by far the most intense of the formants of the glissando note. However, the fundamental frequency is not necessarily the most intense harmonic. I have read, for example, that in the clarinet the note that is heard to be the one intended and played by the instrumentalist is a higher harmonic than the fundamental frequency of the first harmonic. However, even if the fundamental frequency is not the most intense harmonic of a composite note, we would still expect the fundamental frequency to appear with some intensity comparable to the higher harmonics. On the spectrogram of the glissando note, I could not find any evidence of a formant or harmonic below the one that I attributed to be the fundamental frequency. I plotted an oblique line of a formant with one arithmetic sequence frequency difference below the attributed fundamental so as to show where the possible expected hypothetical alternative fundamental frequency harmonic could be that could allow the glissando note to have been produced by a source like a vibrating string or an open pipe. Since there were no signals along the expected line for a lower fundamental frequency, either there is no lower harmonic of the glissando note, or for some strange reason the real fundamental frequency or first harmonic was not produced or is virtual.”

I then argued to eliminate causes for the glissando, debating speed of propagation, energy of the wave, temperature and cooling, density, and pressure of the medium, the action of pistons on the volume of cylindrical pipes and the ratios of frequencies in compartments closed at all ends, and changes in the mass or temperature by mixing of air by an air pump. Having eliminated these, I proceeding towards a conclusion:

“Tuesday 9th March 2010. […] The remaining possibility could be a change in the energy of the source of the wave, by change in the energy of vibration of a tremulous source. I think that the most likely explanation for the glissando would be a decrease of the angular speed of some rotor or wheel at a constant rate, until being stopped at the minimum speed by application of a brake, such that the frequency did not appear to reach zero. I think that the most likely wheel or rotor to be the source was a disc drive of the computer, especially the compact disc or digital video drive. Another less likely possibility is the fan of the computer that is used for cooling, as this would revolve and is not always turned on. […] I do not know what pattern of ratios or sort of overtones that [sic] a spinning disc could produce. […] it might be possible for a spinning disk to have fractional revolutions that produce overtones. […] I think that a more likely explanation would be that the disk wobbles or precesses at different frequencies to its frequency of revolution.”

I next debated precession, resonance, and coupling.

“Thursday 11th March 2010. […] Half-integer ratios between frequencies could be caused by resonance or by coupling. […] a spinning disk can have any frequency because it can rotate at any speed, and this explains how the glissando note could have arisen.”

Next, I described tidal coupling of planetary periods.

“Friday 12th March 2010. […] Perhaps then analogously, there could be coupling between different periods of precession or spin in a single rotating object, such as a disk. This coupling would give rise to fixed simple ratios of frequencies of the overtones. If the fundamental frequency were also fixed, then all the frequencies would be quantised. […]”

The arguments spanned some pages and were longer and more detailed than I would reproduce in full here. Later, I noted the following:

“Sunday 4th July 2010. Sometime after Saturday 13th March 2010 I looked at a book called “Genius, Richard Feynman and modern physics”, by author James Gleick because I wanted to read about the ancient scripts or writing system […]. I noticed the following which may be of relevance to what I wrote about between Monday 8th March 2010 and Saturday 13th March 2010: “A few days later he was eating in the student cafeteria when someone tossed a dinner plate into the air — a Cornell cafeteria plate with the university seal imprinted on one rim — and in the instant of its flight he experienced what he long afterward considered an epiphany. As the plate spun, it wobbled. Because of the insignia he could see that the spin and wobble were not quite in synchrony. Yet just in that instant it seemed to him — or was it his physicist’s intuition? — that the two rotations were related. […] he tried to work the problem out on paper. It was surprisingly complicated, but he used a Lagrangian, least-action approach and found a two-to-one ratio in the relationship of wobble and spin. […]” […]”

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.

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.

The Trouble With Base Thirty

Whereas the importance of a prime number in metrology is influenced by such objective features as the ease with which a division into that prime number of pieces can be constructed, determined by laws of mathematical geometry, the optimally efficient bases for certain metrological procedures including measurement and the specification of values to the greatest accuracy balanced with the minimum number of steps or symbols, and subjective or human cognitive capabilities such as limits of subitisation, the importance of a prime number as a factor can be quantified mathematically as the expected number of times that the prime number appears in a randomly selected number.

For example, the prime number two appears at least once in all of the even numbers or half of all whole numbers. Furthermore, the prime factor two appears again in numbers divisible by four or a quarter of all whole numbers, and yet more frequently in numbers divisible by eight, and so on. The sum of these expected frequencies is the converging limit of the infinite series

1/2 + 1/4 + 1/8 + … + 1/2^n

which is a geometric series of common ratio 1/2 between adjacent terms. The sum is given by the first term 1/2 divided by the subtraction of the common ratio 1/2 from unity, which works out to be one. This means that if numbers were selected at random, the expected average number of times the prime number two would appear as a factor would be once.

Similarly, the expected number of times the prime number three occurs in randomly selected numbers is given by the sum of the infinite series

1/3 + 1/3^2 + 1/3^3 + … 1/3^n

and this works out to be a half or 1/2, so the expected frequency of the prime number three in randomly selected numbers is a half.

In general, the sum of the infinite series for a prime number p is (1/p)/(1 – 1/p) = 1/(p – 1). So, the expected frequency of the prime number five would be 1/(5 – 1) = 1/4, a quarter. Likewise, the expected frequency for the prime number seven turns out to be a sixth. [The author completed this reasoning and derived this formula for this purpose of the quantification of the importance of prime numbers by the month June of the year 2017]

From these calculations, and specifying the importance of a prime number as being directly proportional to its expected frequency of occurance in randomly selected numbers, we see that the prime number two is twice as important as the prime number three, which in turn is twice as important as the prime number five.

To choose or design a base of numeration to have in its prime factorisation an appropriate balance of prime factors according to their importances so defined mathematically, we ought to select the prime factorisation to be a power of the form

[2^1 * 3^(1/2) * 5^(1/4) * 7^(1/6) * … * p^(1/(1-p))]^n,

choosing some value of n such that the exponents of the prime numbers are whole numbers.

Since the number of prime numbers is infinite and they cannot all be included as prime factors in a usable base, the infinite product must be truncated after the first few prime numbers, such that only the most important prime numbers in order are prioritised. Truncation after just the smallest prime number produces the binary-type bases of the form 2^n, such as two, four, and the square of four.

Truncation after the second smallest prime number three produces the number twelve or its powers as the base.

If the prime factor five is insisted upon, truncation to exclude the prime number seven and larger primes gives the number 2^4 * 3^2 * 5^1 = 720, which happens to be six factorial or the product of the first six whole numbers, as the smallest possible example that maintains the correct frequencies of the three smallest prime factors in accordance with their mathematically expected occurrances.

For a practical base of numeration, the number 720 is far too large. Those who desired the prime number five at any cost would therefore have to decrease the frequency in the prime factorisation of the smaller and objectively more important prime numbers two or three, while preferably still maintaining that the exponent of a less important prime number will not be greater than that of a more important prime number, resulting in such numbers as

2^3 * 3^2 * 5^1 = 360
2^4 * 3^1 * 5^1 = 240
2^2 * 3^2 * 5^1 = 180
2^3 * 3^1 * 5^1 = 120
2^2 * 3^1 * 5^1 = 60
2^1 * 3^1 * 5^1 = 30.

The number 360 was used as the number of angular degrees in a full circular turn. The number 60 was used as the base of the sexagesimal system of numeration, now retained in the numbers of minutes in an hour, seconds in a minute, and for subdivisions of angular degrees. Others have emphasised that these numbers as bases remain too large for convenient computations from the viewpoint of the limitations of the long-term human memory for multiplication tables. However, the inconvenience of large numbers as bases of numeration can also be argued from less subjective or more objective reasons of mathematics and consequently metrology.

Consider the smallest of the above numbers 30 as a base to include the prime factor five. Since this number has a disproportionately large exponent of the prime factor five, the smaller and more important prime factors two and three are under-represented. Consequently, numbers as the denominators of unit fractions which are expected to contain more of the prime number two which occur more often would be represented by the same number of digital value places as much less frequent unit fractions having much larger denominators as powers of the less important prime number five. This is because the number of digital value places in the digital form of a fraction written in a base B is determined by the prime factors of that base B and the prime factors in the denominator of the fraction in its simplest form. So, in the case of base thirty, the fraction 1/8, an eighth, where eight is two to the power of three, which is expected to appear frequently compared to fractions with larger denominators, would have three digital value places, because the power of the prime number two in the base thirty is only one. In base thirty, the digital form of a fraction such as 1/5^3, which is decimally 1/125, or 1/3^3, which is decimally 1/27, would also have three digits after the fractional point. However, we should have preferred for rather the numbers that appear more frequently to be represented by fewer digital value places or significant figures at the expense of those that are less frequent. In contrast, a base such as dozenal having a larger and more proportionate exponent of the smallest prime numbers would represent the fraction 1/8 in digital form by fewer than three digital value places.

This mathematical reasoning becomes relevant to the representation of numbers and their measurement in metrology. In a base such as thirty, where the number of figures after the fractional point required to represent the fraction 1/8 digitally is three, the amount of precision by obligation to attain this number of significant figures is phenomenal, because thirty to the power of three is twenty-seven thousand. A requirement to reach such precision or one part in so many thousands in order to express a fraction so simple and frequent as an eighth or one part in eight is unwarranted and heavily burdensome. A result of such excessive precision is many irrelevant steps or degrees of the scale at each value position or place in the digital representation or on the ruler or implement of measurement. [This reasoning by the author on the base thirty was done on Tuesday 18th December 2018]

A base such as the square of four, being a binary power, would represent a commonplace fraction such as an eighth using fewer digits. However, a base with only the prime number two in its factorisation would then not represent the frequent ternary fractions with a judicious amount of concision. Only the base twelve and its square represent the fractions of the most important prime numbers in the optimal way, until the number 720 which is too large as a base for the reason that it would demand too much precision at each positional digit even for the representation of the simplest and most commonly used fractions.

In order to discriminate a unit fraction such as an eighth, 1/8, at its level of precision it is required to be discernable as distinct from a seventh, 1/7, and a ninth, 1/9. The difference to be discriminated or resolved between 1/8 and 1/7 is their difference 1/56. The resolution needed to separate 1/8 from 1/9 is their difference 1/72. The average of these differences 1/56 and 1/72 is 1/63, which is near 1/64. In general, for resolution of a unit fraction 1/n, it must be distinguished from 1/(n – 1) and 1/(n + 1), by differences of 1/(n(n – 1)) and 1/(n(n + 1)). The average of n(n – 1) and n(n + 1) is n^2, so the required precision is about 1/n^2. This means that if the fraction to be resolved is 1/n and the base is n, then there is a requirement to go to the next digital value place after those needed to just specify the fraction to ensure that there is a following zero.

Fractions whose denominators have prime factors that are not factors of the base of numeration by which those fractions are represented do not have terminating digital forms. Their number of significant figures would be endless. Such non-terminating digital expressions are not convenient either for mathematical calculations where their necessary truncation would cause the accumulation of rounding errors or for metrological measurement where they would require an unattainable infinite number of steps in procedure. Numbers such as a third in decimal as 0.333… are of this type, where the digital representation never ends but does repeat. The number of digits between repetitions depends on whether the denominator of the fraction shares a factor with one of the numbers beside or adjacent to the base or a power of the base. For a fraction to have fewer digits before they repeat is slightly better than having a longer stretch of digits. However, non-terminating digital representations remain a nuisance no matter how short the cycle of repetition because of the errors and inexactness that they introduce into calculations and recorded values, such that their use is preferably avoided altogether. In the metrological sense, satisfactorily representating fractions with long periods of repetition, such as a seventh has in decimal, does not practically occur, especially for large bases, because the necessary precision is physically unrealistic. This means that, for example, in ordinary circumstances it is not practically feasible for a quantity to be measured with the decimal base in order to recognise whether it was intended to be exactly a seventh.

From the viewpoint of metrology, there is an inconvenience in a base having too many different kinds of prime factor. Before the metric system, haphazard agglomerations of units of measurement gave rise to irregular and unpredictable multiples of units to make up to the next named unit in the same type of measureable quantity. Irregularities such as these make calculations with them quite inconvenient. For example, the number of feet in a mile has an erratic prime factor, and there is no predictable way to extend the system of units to larger or smaller extremes or incorporate this into a sensible pattern. Having three different prime factors as the number thirty has would produce a similar effect of irregularity within its subdivisions and multiples between its powers. Subdivisions and multiples between powers become practically necessary where the base is too large to be subitised.

While historically, powers of two were used for convenience in metrology such as for denominations of weights and coins of currency, in the modern world with the advent of digital electronic computers and their storage of information in binary format, powers of base two have become even more important. In the decimal base to which society became accustomed, a fraction such as 1/2^6 = 1/64 having its digital decimal form as 0.015625 requires a ludicrous number of significant figures and consequent amount of precision. In contrast, the base twelve in representing the same fraction would require only half as many digits after the fractional point, because the power of two in the prime factorisation of the number twelve is twice that in the base ten. This means that base twelve is twice as suitable in the modern era than decimal for the representation of binary powers.

In summary, base twelve is the only practically sized base suitable for optimally representing fractions according to their mathematical distributions.