Mnemonics

There has been an appeal for a system of mnemonics by which dozenal figures could be remembered using words (SenaryThe12th, Dozens Online, https://www.tapatalk.com/groups/dozensonline/request-for-help-with-mnemonics-for-dozenal-t2020.html).

SenaryThe12th, 4:54 PM - Sep 30, #47, https://www.tapatalk.com/groups/dozensonline/so-how-difficult-are-the-multiplication-tables-of--t2045-s24.html wrote:Oh man, the holy grail for me would be a system of mnemonics which worked as well for dozenal as for mine does for senary.   If it works well for dozenal, it ipso facto would work for any smaller base, including 10, which would mean that the general public would find it useful, and would give yet another on-ramp to dozenal.   And it would therefore also ipso facto work for large bases like 60 and 120 using sub-base encoding.  

I mean I don't want to overstate my case by implying that it is IMPOSSIBLE to use mnemonics for mid-size bases like 12.  Just because i haven't been able to do it shouldn't discourage somebody else from trying.   I'm sure that with some creativity it could be done.   Heck, we are already sooooo close... its already possible to use the senary mnemonics for base 12 if you do a 2*6 or 3*4 subbase encoding, but that seems to be a bit much guilding of the lily.  Its within reach of somebody on this board, and it would be just an awsomely wonderful thing.

While a mnemonic system might help for remembering the numerical characters of individual numbers, for mental calculations it is unlikely that the accuracy of more than three significant figures for any given number would be required. In calculations with such numbers, trying to devise mnemonics for intermediate numbers in steps of the calculation would probably slow down and interfere with the computation, so it may be better to avoid mnemonics and simply remember the numerical figures until the calculation is finished. In any case, the use of mnemonics would be for the most basic arithmetic of addition, subtraction, multiplication, and division, common in finance, but would not be of much use in the kinds of mathematics that are more likely to be needed to be remembered, such as steps in derivations of formulae that involve variables represented by letters instead of given numbers, and all the steps in proofs of theorems and the references and citations to previously proven theorems invoked. It is very unlikely that all the steps of a computation involving basic arithmetic would need to be remembered for much length of time in the long term, as only the initial information and the answer are relevant, and the result if it is forgotten can be acquired and verified again through repeating the calculation, the steps of which should not be difficult to rediscover on account of them being only basic arithmetic. If something is hard to remember, it's probably because it isn't worth learning in the first place.

The first and only time that I used a mnemonic of letters for representing a number and for which I am aware of a record was before two thirds of an hour past eight o'clock on Wednesday the third day of January of last year 2018, and the number was in dozenal.

The proposals of others for mnemonics by phones assigned to the numbers have mainly involved representing each number by consonants, between which arbitrary vowels are chosen to make words representing strings of digits. The obstruent consonantal sounds were mainly grouped by place of articulation, while sonorant consonants such as nasals or liquids formed separate groups. Thus is the following list of groups for consideration:

OBSTRUENTS

• p,b [bilabial]
  
• f,v [labiodental]
  
• th,dh [interdental/dental fricative]
  
• t,d [dentalveolar plosive]
  
• s,z [alveolar fricative sibilant]
  
• sh,zh [postalveolar fricative sibilant]
  
• tc,dj [palatal affricate]
  
• k,g,q,x(,h) [velar/dorsal]
  SONORANTS
  
• m [labial nasal]
  
• n [dentalveolar nasal]
  
• l [lateral liquid]
  
• r [rhotic liquid]
  
• ng [velar nasal]
  

For the formation of mnemonics, it is better to have several options of consonant representing the same numerical figure so that there will be choice and variation of available words with the right permutation of consonants for the sequences of numerical characters. Some phones are rare in words or do not occur in all positions in the phonology of a language, so cannot be the sole phone by which a numerical figure could be represented. These phones, if they are to be included in the system of mnemonics, ought to be grouped with other preferably similar phones representing the same numerical figure.

For example, in English, h and q are not distinct finally, and a phonic or diphonic x does not occur initially. While these conditions might not be crucial for mnemonics since they can be followed or preceded by a vowel, nevertheless, if these consonants are being used for a number, they ought to be grouped with the nearest phones in quality to allow an adequate choice of consonants and increase the probability of being able to find suitable mnemonic words for random sequences of numbers.

The velar nasal ng does not occur initially without a preceding vowel, making it rare in certain positions. Besides that, it is not always clear whether ng should be analyzed as an allophone of n. For example, the word finger might be interpreted as having one of the sequences n-g, ng-g, or n-ng-g. Nevertheless, the velar nasal consonant remains a phoneme contrasted from the alveolar nasal. The same might be said of the labial nasal m, to which n is modified in front of labial consonants, as in input -> imput, although most languages, including English, contrast m from n generally.

In some dialects of English, the (inter-)dental fricatives are not distinct in pronuciation from the (dent-)alveolar plosives. Thus, for mnemonics, these two kinds ought to be merged into one group. Another reason for this is that there are also cases where it is not always clear from the spelling whether the phone should be plosive or fricative, for example, Thomas, Thailand, and hypothenuse. It has been stated that the digraphical spellings where a consonant letter is followed by h of Greek etymology represented aspiration rather than frication.

When dentalveolar plosives are followed by a palatalising vocoid, the result by some speakers can be pronounced as a fricative. For example,

• tune -> tiune -> chune
  
• action -> akchun
  
• natural -> natchural
  
• dew -> jew
  

Thus, it would not always be possible to reconstruct what numbers were intended to be represented by these words, unless these fricatives should be in the same group as the dentalveolar plosives. Proposals of others for mnemonics failed to recognise this pitfall.

A similar phenomenon happens with the sibilants in the environment of palatalising vocoids. For example,

• sure -> shur
  
• assume -> asiume -> ashume
  
• fissure -> fisiur -> fishur
  

So, the alveolar and postalveolar sibilants ought to be placed in the same group as each other. Incidentally, I would consider the pronunciation of the word fissure as fisher to be a "spelling pronunciation", since through the historically common process of medial lenition the sibilant has already become voiced. In a dialect, especially American, the sibilant in the word assume might not be palatalised, but this might not be assumed for all other dialects.

Some proposals for mnemonics have included the consonantal letters y and w and high vocoid phones in groups for digits. They were suggested before much the same suggestion in the response of Oschkar (7:02 AM - Jul 19#4, https://www.tapatalk.com/groups/dozensonline/request-for-help-with-mnemonics-for-dozenal-t2020.html#p40018843). It is natural that they should have been contemplated, since they have the only remaining consonantal phones that were not yet included among the groups. However, a phone of the consonantal letter y suffers from the same problems of ambiguity from palatalisation with an adjacent consonant revealed in the preceding remarks. Furthermore, if y or w were to be included as consonants, then the choice of vowels to go around those and other consonants would be restricted to monophthongs and non-high vowels, potentially making the formation of words representing strings of numerical characters too improbable.  Therefore, it is probably best to leave these high vocoids among the choices for vowels.

So a revised list of groups could be

OBSTRUENTS

• p,b
  
• f,v
  
• th,dh,t,d,tc,dj
  
• s,z,sh,zh
  
• k,g,q,x,h
  SONORANTS
  
• m,n,ng
  
• l
  
• r
  

With this arrangement, there are only five groups of obstruents, and three of sonorants, for a total of eight consonantal groups, not enough for all numerical figures of dozenal, though enough for a base at least as small as octal. The bilabial plosives and labiodental fricatives might be placed into one group, and the labial nasal could be separated into a different group from the other nasals to give the same number of groups.

As a solution for the extension of these groups to reach twelve groups, instead of using the contrast between frication and plosion, the contrast between voicing and devoicing can be maintained. Since the labials are not very common in words, and to prevent one labial being the only phone representing some numerical figure, the bilabial plosives and labiodental fricatives should be merged for this to work well. Thus, the groups become:

OBSTRUENTS

• p,f
  
• b,v
  
• th,t,tc
  
• dh,d,dj
  
• s,sh
  
• z,zh
  
• k,q,x,h
  
• g
  SONORANTS
  
• m
  
• n,ng
  
• l
  
• r
  

This list satisfies the required number of groups for representing dozenal numerical figures. It is notable that extension of this number of consonantal groups to many more than these twelve would not be very feasible.

The order in which the groups be assigned to the dozen numerical characters does not matter too much as long as it is done consistently, and can be a personal affair. After all, mnemonics should be designed internally to be as memorable as possible to the individual relying on them. It would not be too difficult to choose some order for the groups that would be more memorable than some other order. Preferably, the numerical figures representing smaller numbers which occur more frequently in random numbers because of Benford's law ought to be assigned to the phones that appear more frequently in the language. The most frequent phones are usually coronal. The assignments of groups to numerical characters could be done using a statistical distribution of the frequencies of occurrence of the phonemes in the language.

For mnemonic representation of the dozenal numerical figures by words of some other language with a different set of phones than those of English, the natures or how many there are of the consonantal groups might be different.

A Natural Language Order
On the numbers in the Proto-Indo-European family of languages, dozenal.forumotion.com/t26-proto-indo-european-numbers, a sequence such as the following as words for the twelve cardinal numbers was derived:

• zi
  
• uni
  
• duo
  
• trei
  
• kuator
  
• penc
  
• siks
  
• sept
  
• okt
  
• nen
  
• dek
  
• lif
  

Each initial consonant of these words would belong to one of the mnemonic groups of consonants.

Two of the words, uni and okt, begin with a vowel instead of a consonant, and thus have no consonantal group.

Initial vowels:

• u: uni
  
• o: okt
  

The consonants d and s are both initials in more than one of the words, so their groups are overused, and the initials of two words must be assigned to other groups if redundancy is to be avoided.

Overused:

• d: duo, dec
  
• s: siks, sept
  

Despite these, all the twelve consonantal groups are represented except these four groups:

Unassigned:

• b,v
  
• g
  
• m
  
• r
  

The number of unrepresented groups matches the number of words with yet-to-be-assigned alternative initials, therefore the unrepresented groups can be assigned to those yet-to-be-assigned initials. Justification of the assignments is attempted through the phones in words of natural languages for the numbers.

However, r is not in the Indo-European word for six, and so may be given to the word for the number four instead, while the word for six gets the group that the word for the number four had. The most characteristic phone or diphone of six is probably x from the Greek word for the number six.

• uni -> m [from Greek mono]
  
• okt -> g [from English eight]
  
• duo -> b,v [from Latin binary, vigesimal]
  
• six -> h,x,k,q [from Greek hex]
  
• kuator -> r [from Proto-Indo-European, and English four]
  

As an initial of the word for two, d is more expected and recognisable as such in Indo-European languages. For this reason, as a word for two I am not as fond of using from Latin derivation bi in itself for a kind of twoness.

Behold, the groups for mnemonics thus become:

• z,zh
  
• m
  
• b,v
  
• th,t,tc
  
• r
  
• p,ph/f
  
• c/k,q,x,h
  
• s,sh
  
• g
  
• n,ng
  
• dh,d,dge/dzh/dj/j
  
• l
  

The order of the groups here is the same as one that I had typed on Monday the thirtieth day of last month September 2019, this week.

After the typing of the previous post had begun but before it was posted to this forum, another post, by one Andrew, on a proposal for dozenal mnemonics was released, before the first post of this topic, that has similarities to this one in that the syllables for the numbers begin with consonants derived or related to consonants from words for numbers that happen to be from the Indo-European family of languages https://www.tapatalk.com/groups/dozensonline/scat-singing-mnemonic-t2044.html#p40019610.

Vowels for Mnemonics
In August last year, I allocated each of the twelve numbers from zero to eleven in dozenal numeration to a different vowel or diphthong with the purpose that these vowels could be used from English words as mnemonics for numbers in base twelve. I arranged the vowels such that they would be associated with vowels of the Indo-European words for numbers. This would allow mnemonics to alternate between consonants and vowels, and in reverse could allow almost any word in English to be converted into a dozenal number.

At first at that time, I selected twelve vowels and diphthongs systematically from phonetic and phonological principles. I chose four "short" or lax monophthongal vowels (though one of these is actually usually pronounced with a longer duration than unaccented vowels in English) spaced maximally apart from the formant chart, and completed a table of twelve entries by two further rows of "long" or tense vowels derived from the four selected monophthongs by diphthongalisation with high fronting or backing vowels. Next, I changed the order of the twelve entries to the irregular sequence of phones in the Indo-European words for numbers. I also chose Anglicized spellings for these vowels and diphthongs.

When I was a school pupil, before being of High School age, I analysed the phonology of the English language to determine its phonemes. I did not have many sources of influence, as I did not have access to the internet in those days. Mostly, I was operating independently without prior knowledge or formal instruction on the phonology. I came to the conclusion of there being a dozen plus one vowels and diphthongs altogether in English. This number provides enough for representing the twelve numbers for mnemonics, and not too many more. My conclusion on this aspect of English phonology has not changed much since then, although I was later to become more likely to represent diphthongs digraphically. I was already a dozenist at that age, and sought to arrange the vowels and diphthongs into tables of twelve entries. I was not aware of any other dozenist in existence at that time, and was not under influence of alternative bases, which were not taught in school. The principle of positional notation is mostly known when it is taught at the age of about four years, and while decimal points are taught later, there is little point in teaching the principle of positional notation again at the age of seven, eleven, or any other age. I recall considering the inclusion of a diphthong missing from the phonology of English for completing a row of them.

The vowels and diphthongs chosen for the twelve numbers do not correspond exactly to those in my phonological analysis of English. Some phonemes are merged, two of them into one kind of high schwa. I represented this by the available letter Y. Another two are merged into a single low vowel. In an American dialect of the English language, the vocalic phonemes of the words cot and caught are merged. The following was the representation of the twelve numbers from zero to eleven as words by their mnemonic consonants followed by mnemonic vowels:

zi
mo
vy
tei
roa
fai
ke
sau
gu
noi
da
leu

The orthography for the vowel in the word for the number four is not perfect. The allocation of the vowels was constrained by the aim for them to be different in each number, causing deviations from the nearest Indo-European simulation. If these words are learnt, they can be used as the guiding principle in selecting mnemonic words from the English language for dozenal numbers. They could also be used for constructing artificial words for dozenal numbers according to the same principle.