Vowel Cube Formant Chart

Diagram of articulatory cube of vowels projected onto a formant frequency chart.

Characteristics of vowels, such as those of the International Phonetic Alphabet, are typically plotted on a flat chart by their first and second formant frequencies or by qualities such as front, central, back, close, mid and open along the axes. The two dimensional charts with articulatory features as axes typically show the vowels in a diagram with vowels having one of the articulatory features in common being joined by a straight line, usually such that the space of vowels is bound by a quadrilateral or a roughly triangular shape. Sometimes the distinctions between the vowels are displayed in a three dimensional plot with the acoustic properties of the first, second, and third formants as axes or by articulatory features with the three axes as frontness, height, and roundness. While the plots in two dimensions by the first two formant frequencies are often compared to those in which height and frontness are the two axes, comparison of articulatory and acoustic plots in three dimensions are not so usual as a method of teaching phonetics.

On the Base Dozen Forum there is presented a portrayal of vowels in a three dimensional plot by articulatory features of high versus low, back versus front, and rounded versus unrounded as the three axes. The space of vowels is let be a cube with extreme vowels at its corners. For example, the most high and rounded back vowel is one of the corners, and the lowest and furthest back unrounded vowel is another corner. Another corner is allocated to the most high and front unrounded vowel. These are three of the standard cardinal vowels proposed by the phonetician Daniel Jones. In the three dimensional space that is here proposed, these three cardinal vowels form the vertices of an equilateral triangle in the cube. To the other corners are assigned vowels according to the resulting articulatory dimensions of the axes aligned with the edges of the cube. Apart from vowels at the corners of the cube, other vowels of intermediate qualities are plotted between the corners, either along the edges or on the faces of the cube. At the centre of the cube is a schwa.

To compare this cube of vowels by their articulatory features to their acoustic properties on a formant chart, the cube is transformed by rotational orientation and scaling before it is projected onto the planar formant chart. The result is that the positions of the projected vowels agree extraordinarily well to their positions by their measured formant frequencies. Using the outlines of the edges of the projected cube, it is possible to predict the effects of changes in articulation on the acoustic qualities of the vowel. This should make this proposal more useful for understanding what a vowel will sound like in relation to known vowels than an ordinary chart in which unrounded and rounded vowels having the same features in other respects are merely placed beside each other in pairs without indicating how their formants differ, although it is expected that rounding applied to front vowels or unrounding applied to back vowels will centre them acoustically.

References:

https://dozenal.forumotion.com/t87-vowel-cube

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.