Tillo
The Goal of Tillo is to make communication as easy as possible. To do this, the language must be simple to learn, and even simpler to speak and use. This language incorporates all forms of spoken and written communication – including mathematical operators, scientific symbols, and Units of measurement. Tillo reduces redundancy and confusion so that one can easily use it to work out mathematical and scientific problems – and explain them to other people.
Words are made to flow together – but are short and can be said quickly without misunderstanding. Spelling is systematic and uses no special characters to create every sound. The grammar is slightly different from any other language, but is simple and concise.
The vocabulary will be derived from the top most-spoken languages in the world. 35% of the vocabulary will be derived from romantic languages, 24% from Mandarin, 17% from Hindi, 8% from Arabic, 7% from Bengali, 6% from Japanese, and 3% from Russian. The words in Tillo will not be exactly the same as words in other languages, but will be distinguishably similar.
This spread of words from the most spoken languages worldwide makes Tillo easier to learn for those people speaking those languages. This is important for the language to be adopted by those groups of people. The percentages reflect the number of people that speak those languages. However, I tried making it so that at least *around* 10% of the vocabulary came from each language – therefore the percentages for romantic languages and mandarin Chinese are slightly damped, while the other percentages are slightly augmented. Vocabulary from the language Ido is used for romantic languages, but since Russian has far fewer words of European origin than romantic languages, it needs to have a little more vocabulary in the mix.
ALPHABET- same as English, roman lettering
SAYING LETTERS-
All consonants and semivowels (everything except a, e, i, o, q, r, u, x, y) are said as the pronunciation of the letter followed by an English “ee” sound (or tillo “yy” sound) after it. The vowels (other than “w” and “l”) are said as the pronunciation of the letter followed by an “m” sound.
a is said like “am” in English or the tillo-exact “am”
b is said like “bee” or the tillo-exact “byy”
c is said like “she” or the tillo-exact “cyy”
d is said like “dee” or the tillo-exact “dyy”
e is said like “em”, the tillo-exact “em”
f is said like “fee” or the tillo-exact “fyy”
g is said like the tillo-exact “gyy”
h is said like “he” or the tillo-exact “hyy”
i is said like “im”, the tillo-exact “im”
j is said like the tillo-exact “jyy”
k is said like “key” the tillo-exact “kyy”
l is said like “lee” or the tillo-exact “lyy”
m is said like “me” or the tillo-exact “myy”
n is said like “knee” or the tillo-exact “nyy”
o is said like the tillo-exact “om”
p is said like “pea” or the tillo-exact “pyy”
q is said like “ome” or the tillo-exact “qm”
r is said like “erm” or the tillo-exact “rm”
s is said like “see” or the tillo-exact “syy”
t is said like “tea” or the tillo-exact “tyy”
u is said like “um”, the tillo-exact “um”
v is said like “vee” or the tillo-exact “vyy”
w is said like “wee” or the tillo-exact “wyy”
x is said like the tillo-exact “xm”
y is said like “eem” the tillo-exact “yym”
z is said like “zee” or the tillo-exact “zyy”
Written Sounds -
Tillo consists of 26 letters which each have a unique and invariant sound. 9 vowels (a, e, i, o, q, r, u, x, y), 2 semivowels (l, w), 15 consonants, 3 dual-letter consonants, 4 dual-letter modifiers and 6 significant doubled-letters make up 35 distinct sounds that can be written in Tillo.
Doubled letters are pronounced as longer forms of the sound. Doubles letters are written as a succession of the same letter (e.g. “aaaaa”, “ghhhhh”, or “onhhhhhh”) The more letters there are in succession, the longer you hold the sound for. All doubled letters are meant to be longer forms of their base sound, but since sometimes it is hard to think of the correct way to do so, some are specifically clarified.
Dual letters are written as some consonant then an h (e.g. “nh”, “gh”, and “dh”). They are held by repeating the “h”
Consonants- everything is as in English, except:
Single Letter Consonants
Doubled Letter Consonants
Dual Letters Consonants
vh makes the nasal labiodental sound (kind of like a “v” sound but with your nasal passage open)
Vowels
Single Letter Vowels
Other sound modifiers
kh makes the previous sound into an implosive sound.
nh follows a vowel, making it a nasal vowel such as in the stereotypical french laugh (hqnh hqnh hqnh) or the way some people indicate no (opposite of mm hmm = onh onh). English doesn’t use it much, but French does. The French words “en” and “dans” are in tillo written “onh” and “donh” respectively. The French word “on” is in tillo written “qnh”. Vowels modified by “nh” are repeated by repeating the “h” (e.g. “donhhhhhhhs” is pronounced like the French “daaaaaans” (dans).
oh makes the previous sound uvular. This sound is pretty much never used in english. “kkoh” specifically is pronounced like the scratchy throat sound like in German and Hebrew (kind of like kk, but with throatiness) like the ch in the Scottish word loch, or the German Bach, the J in the Spanish Jose, or the “Kh” in the Arabic Khaled. As far as I can tell its pronounced the same way as the letter Ĥ in esparanto.
HOW TO MAKE CERTAIN SOUNDS
Tillo english or explanation of sound
dj - j (as in judge)
tc - ch (as in bench)
Some examples:
word - what it would be spelled like in Tillo.
oil - qyl out - awt air - er augment - ogment eat - yyt
pie – poy piece - pyys or pys europe - yrrup erupt - yrupt
euphemism – ywwfemizm go – gq judge – djudj
ago – ugq car - kor cat - kat care - ker state – steyt
book - bxk set - set see - syy were - wrr in - in wine –woyn sir - srr blood - blud cot - kot node - nqd
root - rwt look - lxk rut - rut mute - mywt
fur – fr church – tcrtc thimble – ttimbl measure - mejr
edge – edj door – dqr shook – cxk or – qr
purrer – pr’r ye – y’y or yiy
Comparison with the International Phonetic Alphabet:
IPA spelling Tillo spelling
Consonants (pulmonic)
Vowels
Consonants (Non-pulmonic)
Other symbols (??)
Parts of speach:
Nouns – In Tillo, nouns and adjectives are the same. For example, instead of saying “a green man” you would say “one that is a man and that is green” which would consist of two words (hypothetically “greeno mano”). Nouns can apposition to all the nouns in a noun phrase (a string of consecutive nouns), but a noun may especially modify the noun after it.
Normal Nouns – Normal nouns don’t take any argument, they’re just regular old nouns. These are words like “bird”, “blue”, “moose”, “wet”, “weird”.
Prepositional Nouns – These nouns take an object just like prepositions or verbs do in English. Because of the way these work, the word for “kill” is also be the word for “killer”. For example, “bob is killing you” could also be understood as “Bob is a killer of you” in tillo - “killing” would be a prepositional noun, with you as its object. “killing you” would be a noun phrase. Tillo equivalents for English prepositions are included in this category, and it will be a significant portion of the nouns.
Special Pronoun – The pronouns “ok” and “ot” (this and that) can specify which “him” “it” or “her” its talking about by placing the beginning letter, couple letters, or syllable of what its referring to before it. For example, If I said “Jill and the boy killed a squirrel with a bat by using ‘ot’ as a weapon to hit ‘ot’” , but I used the squirrel to hit the bat, which killed “it”; then you could say this “Jill and the boy killed a squirrel with a bat by using sot as a weapon to hit bot”. Thus “sot” would refer to squirrel, and “bot” would refer to the bat. If there are two different words that start with the same letter, one can simply use another letter to distinguish between them (eg. Alison and Amy fought: Alot won amot lost. You could also used the first syllable to do this: (eg. Jay and james died. jayot died first and jamot died second), however sometimes it would just be easier to say the first word anyway, as you can see.
Relative Pronouns – Relative pronouns would have the main relative pronoun start the phrase, then the rest of the phrase would somewhere contain a secondary relative pronoun that would define what its use in the phrase is. This would be unnecessary if what it relates to is the object of the new phrase.
- An example of relative pronouns is “The people are red, which is a color”. ‘which’ is a relative pronoun because it substitutes for ‘red’. Since the color (red) is the object of the phrase containing the relative pronoun, in tillo, there would be no need for a secondary relative pronoun. Another example: “The apple that I ate was intensely rotten” – in this case the object of the relative pronoun is *not* the object, a secondary pronoun is necessary. The general structure of a similar tillo sentence would be “Apple which I ate that, was intensely rotten” – which being the primary relative pronoun, and that being the secondary relative pronoun.
Verbs – a word that relates two noun phrases. There are 5 verbs in the language: simple, decisive, continuous, indecisive, generalitive.
Simple Verb – This verb denotes that an actions beginning and end occurred in whatever time (or time period) you are talking about. “I fed the cow”, “I feed the cow”, and “I will feed the cow” would all use the simple verb if translated into tillo.
Decisive Verb – This verb denotes that you are talking about an event that has ended in whatever time you are talking about. This is different from a “past tense” because the time (or time period) you are talking about could be the present or future as well as the past. “I had fed the cow”, “He finished feeding the cow”, and “I will have fed the cow” all would use the decisive verb if translated to tillo. (the last two sentences would have the same translations into tillo).
Continuous Verb – This verb denotes that you are talking about an event that is in the middle of happening at whatever time (or time period) you are talking about. “I was feeding the cow”, “I am feeding the cow”, and “I will be feeding the cow” all would use the continuous verb if translated into tillo.
Indecisive Verb – This verb denotes that you are talking about an event that has not yet begun at whatever time (or time period) you are talking about. “I was going to feed the cow”, “I am going to feed the cow”, and “I will be going to feed the cow” all would use the indecisive verb if translated into tillo.
Generalitive Verb – This verb denotes a general sense of doing or being something (in whatever time or time period you’re talking about). “I used to feed the cow”, “I do feed the cow”, and “I (generally) will be one that feeds the cow”.
NOTE: catenative verbs (verbs that use actions as their predicate) do not exist grammatically in tillo, mostly cause there are no verb infinitives (and because all verb phrases are essentially nouns themselves ); however, the same meaning can be produced by using the noun form meaning the act-of-(something) or using that-someone-does. (so to say “I want to run” is “I want running” or “I want that I run”)
Note about tense: The time and place and such that these particles are true at is given as modifiers and appositioned verb phrases. This means that to say “Yesterday I went to the store” you say “yesterday I go to the store” – to say “tomorrow I will run” you say “tomorrow I run”. To say “I had run” you say “in the past I [decisive verb] run” and to say “I will be going to run” you say “In the future [deI [indecisive verb] run”.
Conjunctions – Either connect two verb phrases together (e.g. “I can drive to the store or bike around town”, or create a compound verb phrase by joining two noun phrases (e.g. “I eat the cake and the pie”). Similar to (but different than) in English, in a list of more than two items, the conjunction can be omitted between all items except the first two and last two (usually pauses between items [commas] would be used to differentiate between items). NOTE: there are no conjunctions to join clauses; nouns (used like adverbs) will be in the place of this.
Ambiguous Conjunction – Conjunctions that have no indication of what they connect except for context. They can be used as Greater or lesser conjunctions, and can also be used as adverbial nouns. Ambiguous conjunctions should probably start with a vowel and end in a consonant – like greater conjunctions (perhaps they can start and end with a consonant).
Greater conjunction – conjunction joining two words or noun phrases that are NOT appositioned by the same words; (The rabbit kick and(GC) cabbage fly = the rabbit kicks and the cabbage flies (only one flys)). Greater conjunctions should start with a vowel and end in a consonant to provide audible contrast to the natural flow of the sentence (most words should start with a consonant and end in a vowel)
Lesser conjunction – conjunction joining two separate parts of a clause that are appositioned by the same words (The rabbit kick and(LC) cabbage fly = the kicking rabbit and cabbage fly (both fly)).
Expressions – Words that serve to describe someone’s attitude or mood, words that serve an interjectional meaning, or words and phrases that mean something other than what it means literally (whether it makes sense literally or not). In short, words that can’t be otherwise categorized.
interjections – expressions usually used alone that express emotion without specific meaning, such as “wow!” or “holy crap!”
Particles - words that indicate something not able to be connoted any other way. There is a particle that denotes the end of a verbal phrase. Particles that modify words are not spaced from them and placed before them, particles that modify phrases are separated from the words and placed where the desired function is supposed to start. Particles can also be used IN a word, but if makes it looked cluttered and mangled so its probably not a good idea. ?!‘/Ro’\yti’/o (pronounced like a stereotypical British "rightio" - tone goes up, then down, then back up)
Verbal particles - are pronounced and denote something in the sentence that is a menial task not directly related to either tone or the subject of the sentence.
Tonal Particles - are not pronounced but indicate a change in tone (tonal particles are made to be not able to be pronounced).
NOTE: about suffixes and prefixes, suffixes and prefixes will not be used in tillo, as it is just as easy to use a normal word instead of a prefix or suffix. Eg. instead of subfire which could mean a fire that is below, or a place that is below the fire, you could say “below fire” or “sub fire” with two words.
Punctuation: All punctuation are tonal particles.
The dash – indicates a very brief pause. May be used to set off an interrupting clause (eg “if we don’t succeed - and the critics say we won’t - the whole project is in jeopardy”). Also may be used in place of a comma, or colon to indicate a shorter pause. Can also serve as an equivalent of “through”, “to”, or “between” as an indicator of range (e.g. 104 through 670 is the same as 104 to 670 which is the same as 104-670).
The comma , indicates a casual short. Short pauses are commonly used for separating items in a list, separate words that might cause ambiguity (eg To mary, jane was someone special), and to set off items of increasing or decreasing specification (eg “Shreveport, Louisiana”; “Sunday, June 23, 1940”; etc), but a pause (a comma) is never a grammatical necessity.
A comma can also be used to end a subscript or superscript, to more clearly show what is and what isn’t a subscript – if necessary (e.g. H^2,0 is the same thing as H2O)
The Semi-Colon ; indicates a slightly long pause with the tone that the clause following the ; mark has something to do with the previous clause. Can be used to separate phrases that contain commas. Also, the semi-colon can be used to separate statements that are not directly related but related in some way, especially used in mathematical statements.
The Period . indicates a slightly long pause and an end-of-thought tone. The period is also used to indicate a decimal (e.g. 1.45) and to denote a Subscript (for example: M.r denotes and easily typed variable or symbol “M” with subscript “r” and is equivelant to Mr ; other examples: M.1 = M1 ) NOTE: this notation can’t be used to denote subscripts that come before the Variable or Symbol (eg. “ rM ” )
The mark ,,, indicates a trailing off or very long pause with a tone that you might continue. This might be used to indicate a long pause (as was said) or indicate an unfinished sentence. Also may indicate the omission of words in a sentence. (“and whom, by heaven I swear, I tender dearly” >> “and whom ,,, I tender dearly”)
The mark … indicates a trailing off with a tone with a finished tone. This may used to indicate an unfinished thought. Also may indicate the omission of sentences. (“The dog ate. The dog ran. The dog played. The dog died.” = “The dog ate... The dog died”)
The Colon : indicates that following words describe or elaborate on the previous clause or word. (dear sir: etc; Title: subtitle; three abstained: England, Russia, and france; etc). This can be used to separate related equations (for example: x=y^2[C]r : x/y^2 = [C]r : [C-](x/y^2)=r )
Parentheses ( ) enclose words to give the tone of an aside or whisper, to indicate a narrators note, or to group words together (Eg. “The rock (if you can call it a rock) is very light.”).
The Question mark ? is used to show a questioning tone. The ? mark is put before the word in the clause that you want to start the questioning on (do you ? want to go. ? can you go.). The questioning tone continues until the next conjunction, period, or semicolon, or until it is otherwise understood that the question has ended. (to say something like “OR, do you want to go?” with the “or” in a questioning tone, and the rest in a statement tone, you would write this: “?or; do you want to go.” NOTE: questions do not always need to use a ? mark; only if the TONE is questioning.
The Exclamation mark ! is used to show an excited, shouting, or emphatic tone. The ! mark is put before the word in the clause that you want to start the exclaiming on (! run you. Hello !milton [rising tone on milton]). The exclaiming tone continues until the next conjunction, period, or semicolon, or until is it otherwise understood that the exclamation has ended.
The exclamation mark is also used to indicate factorial, and is explained better in the Mathmatical Functions section.
Quotes would be used only for quotations, never for titles. quotes would always have anything you would like to include from someone's speach or from a writing. the things inside the quotes are taken as to be only one word and you must have a period and any other necessary punctuation or words after it. if you want to start a quote from the middle of a sentence, you would if you want to site the quote, you would put a ( sign and put any relevant information in. A period always goes on the end of a sentence even if there is a period in the quote or a quote at the end of the sentence. E.g. “quote”. Or “quote.”.
Underlining is equivalent to italicizing and are used only to indicate a lowering in tone. underlining or italicizing is and it doesn't matter if all the words are underlined with one line or with separate lines (like some extremely picky English teachers might insist on).
Capitol letters are used to indicate an emphasized tone. They can be used to indicate emphasis within a word (e.g. “EMphasis”). They are also used for abbreviations like “LAPD” (the LA police department) or “EG” (for “e.g.”) and name initials like BC for Bill Clinton, etc. Capitols are not used to start sentences or for the start of names.
The Apostrophe ' is used to make the pronunciation of a word clearer or easier to read and used to denote the missing letters. For example, if you have a word like sw'wwp (pronounced swoop) you need the apostrophe to tell if it is pronounced sw wwp (swoop) or sww wp (soup). Other examples of this: prr’rr (purrer, like a cat that purrs alot), y’yy (like olde English “ye”). An apostrophe does not indicate any pause or stop.
An apostrophe would also be used for large numbers eg. 4'345'354'455'000'000 (every three digits starting from the right of the decimal [or the 10s digit] and going higher). Can also be used for very long decimals (for example: 5.234’567’643’546’5 - every three digits starting from the left of the decimal)
Apostrophes are also used to construct contractions – used to denote the missing letters, digits, and other symbols (EG “don’t “). 1983 could be abbreviated to ’83, in the same way that English does.
Apostrophes can also be used as quotes inside quotes.
An other use for the apostrophe is used to denote superscript, it is placed between the Variable or Symbol and the superscript (for example: He’- is the symbol “He” with superscript “ – “ and is equivalent to “ He– “ .
NOTE 1: this notation can’t be used to denote superscripts that come before the Variable or Symbol (eg. “ +He ” )
NOTE 2: this notation shouldn’t be used to indicate a power. The carrot symbol (^)and the at symbol (@) are used for powers.
NOTE on accenting:. Accenting can also used freely as a way of stressing certain things about the word or sentence. If you want to speak normally then you shouldn’t stress any part of a word more than another. To accent as a rising tone, certain syllables would be CapitalIzed. To accent as a lowering of tone, certain syllables would be underlined or italisized.
Syllables are simply the vowel sounds of a word; the consonants have no connection to accenting or the syllable of one side or the other (vereeublee has the syllables at ‘a’ ‘ee’ ‘u’ and the second ‘ee’ but the consonant r does not connect to the syllable ‘e’ OR ‘ee’, because it doesn’t have anything to do with syllables). For example, tEnEycUs, hYWmUnhgUs, djOygAntIc, etc.
NOTE ABOUT COMPOUND WORDS – compound words don’t exist in Tillo as they do in English. To carry over the meanings of such words as coffee-table, one would use verbs such as a table for coffee or one would use apposition nouns like a word to mean “something having to do with coffee” or “something for coffee”. These types of constructions would be common in tillo.
Sentence Structure –
Noun phrase + verb + noun phrase
OR
Verb + Noun phrase
OR
Noun phrase + verb
Noun Phrase –
Noun
OR
Particles + Noun
OR
Noun + Noun Phrase
OR
Prepositional Noun + Noun Phrase
OR
Noun Phrase + Conjunction + Noun Phrase
Note: Since there are no passive verbs, simply omitting the subject will suffice. Or you can use the appropriate modifiers to make a verbs object change to a verbs subject.
Note: There are no indirect objects in this language, so for many things one may use an indirect object for, in tillo one would use the equivalent word for “to”.
How to say and write numbers works as follows: one would only add a post-positional after each of the thousands places, but you say no suffix for the 100s, 10s, or digit’s place. There are words for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, and 0. The words are related both by significant digit and by magnitude. For example, 5 might be “gq”, 6 might be “ro”, 50 might be “gyy”, 60 might be “ryy”, and 500 might be “ga”.
To write out fractions, one writes the numerator then the fraction verb, then the denominator (the number at the bottom of a ratio).
To make a number that is 1000 to the power of something (like million, billion, trillion, etc [1000^2, 1000^3, 1000^4, respectively]), you say thousand to-the-power-of X (to-the-power-of is one word). So million is thousand to-the-power-of 2.
- - - parents3
- parents 2nd parents 3rd parents
Children 2nd children 3rd children 4th children
children2 2nd children 2 3rd children2 etc
children3 2nd children 3 etc
children4 etc
etc
Tillo to english grammatical translation examples (using english words in both grammatic sets):
Slow fast-moving car – slow car moving fast
six or eight cylinder engine – “engine with/of six or(lesser conj’) eight cylinder”
green-headed, shifty-eyed man – “man with green head and shifty eyes” or “man that has green head and(lesser conj) shifty eye” or “Man Green of head, shifty of eye”
NOTE: Words in the language should flow together, so make the words so that most of the start with a consonant and end in a vowel (or at least end in a vowel). You can use deviations of this norm to mark special words (like conjunctions or something)
Scientific and Mathematical Abbreviations and symbols:
Abbreviations (such as ones on the periodic table) and symbols (such as unit symbols and mathematical signs) would be devised so that they can transcend disciplines and have no (or minimal) confusion and redundancy.
These will all be two letter abbreviations of element names consisting of one Upper case letter before one lower case letter. If ever there are too many elements, new elements can take three letter abbreviations with one Upper case letter before two lower case letters.
Unit symbols -
Note about the mole:
The mole is really just a number. There is no reason to have a “unit” of a mole because you can just say the correct (if very large) number. The only reason the mol exists is because it makes it easy to use weights on the periodic table. However, if an appropriate definition of weight is used, it makes weights easier to be used without the addition of the “mole”.
Scientific Variables -
These will all be single lower case letters indicating a specific type of measurement with an unknown numerical value. New variables are only created when more than 3 variables otherwise would need to be used to express it. Scientific Variables alphabetically correspond to the Unit Symbol relating to it – in other words, if you make a unit of measurement Capitalized, it turns it into a variable of that unit.
0 - this character denotes a *positive* infinitely small amount or infinitesimal amount. Tillo handles 0 in a different way than usual mathematical languages and interpretations. 0 can be thought of as the limit of x as x approaches 0 from the positive side. Using this definition, there is a clear positive 0 and -0 (negative 0). 1/0 is positive infinity and 1/-0 = -1/0 = negative infinity. 0/0 = 1, but one must be careful that the numerator 0 and denominator 0 are *exactly equal*. If you have an A and a B, but A!=B, then if A=0 and B = 5*A, then obviously A/B is 1/5, rather than 1. If you plan on canceling 0's, then make sure you didn't absorb another number or variable into a 0 (by multiplying it by 0). These properties do not hold for an "exact 0".
{0} - this represents an "exact zero", which has the special property of having no inverse. x/{0} is undefined for any x, including x = {0}. {0} would only be obtained when subtracting exactly equal quantities (e.g. x - x = {0} ). Using this definition, no "exact infinity" exists.
Mathematical Variables -
Variables such as one denoting it to be only in the set of integers, or only in the set of positive numbers – etc. Positive Integers, Integers (positive or negative), Real, Rational, Complex, Digits (a number that is a digit inside a number)
The language allows for subscripts by using the period ( . ) between the variable and its subscript (like in object oriented programming). The Apostrophe (‘) can be used for superscripts among its myriad of other uses.
Scientific Constants –
These will be abbreviations consisting of two lower case letters.
Math Signs -
These will all be non-alphabet symbols indicating a mathematical function (such as adding, subtracting, multiplying, etc)
+ - plus sign. This adds two numbers. In the case of a matrix, each corresponding cell is added to create one matrix in which each cell is the sum of the two corresponding cells from the matrixes that were added.
- - The opposite of addition, i.e. adds the opposite – it works in any case where the + sign can work. Also used
(+; -) - “plus or minus” is a way of shortening two similar equations into one. The symbol means that you can add OR subtract the two items in question to obtain two different expressions. (e.g. “x=3+4 and x=3-4” is equivalent to “x=3[+; -]4” )
/ - denotes multiplying by the inverse (i.e. division). If the denominator has no inverse, then this operation is undefined.
% - residual modulus (i.e. mod). A%B is equivalent to solving the system of equations: "x*B = A + y & y <= B" for y where x and y are both integers. Defining mod this way allows floating point and fractional numbers to be operands.
^ - carrot – this indicates a power (e.g. 5^2 = 25), but can also indicate iterations when applied to a function or operator (e.g. #^2(x^2) = 2(#x)^2 = the second derivative of x^2, or sin^2(x) = sin(sin(x)) , etc )
@ - Denotes “ *(10^4)^ ” – multiplies something by a power of 10^4 (older notation instead uses powers of ten such as "10^ " or “e” on simple calculators, however the latter brought up the discrepancy of the natural constant “e” ). This can also be used in conjunction with units of measurement (eg. m@-4 = pm = picometers = meters*10-15).
$^n [x] - Integrate x, n number of times
#x - derivative (infinitesimal amount) of x
#^n x - nth derivative of x (derive x, n number of times)
a.#x - partial derivative of x varying only a
[a; b; c].#x - partial derivative of x varying only a, b, and c.
** - Multiplies the entire previous expression by the following entire expression (is higher on the order of operations than the single * - stops at equal signs unless those are included inside parens) (e.g. “5+6**7” is equivalent to (5+6)*7). This is very useful for when you’re writing something and you don’t want to have to go back and modify parens, of course it must be used right. (NOTE: a SINGLE asterisk (e.g. *) can be substituted by a space, HOWEVER the double asterisk cannot be substituted by neither one nor two spaces.
// - like ** but with division
^^ - like ** and // but with powers.
.. - Denotes the subscript of the previous expression where the expression contains only two variables and one operator - either multiplication or division. This is used to create subscripts of units that have multiple basic units that make it up (e.g. velocity.a = va = m/s..a = ma/sa )
! - the factorial sign is coupled with any basic function to indicate the systematic reduction of one number while applying that basic function to each number. (must be immediately after the argument (no space)). Using this before an argument denotes "not" - i.e. the not operator.
*! n*! = n! = n * (n-1) * (n-2) * … * 1
- ( can also just be written without the * as “ ! ” )
+! n+! = n + (n-1) + (n-2) + … + 1 = n*(n+1)/2
NOTE: Matrices and vectors are never really different sizes – by default, the undefined members are 0. Consistent with this definition, scalars are defined as matrices who's first element is the scalar, and the rest of the elements are 0. Vectors are also matrices, who's undefined elements are 0 {e.g. (a,b,c) = (a,b,c,0,0,0) = ((a,b,c,0,0),(0,0,0,0,0)) } . The only object that does not follow this rule is the null set: "( )". The null set has no members, not even "0".
Variable - a variable is an unordered list. Can also be called a "set". Variables in many cases are infinitely large sets, i.e. sets with an infinite number of members.
Degenerate Variable - A set that only has one member. Can also be called a "degenerate set", a "constant vector", or simply a "constant". (note that this means the list cannot contain variables and cannot contain lists that contain variables.)
Vector - A vector is defined as an ordered list of scalars or vectors.
Degenerate vector - a vector who has one element in the first position, and 0s everywhere else. The element in the first position must also be a degenerate vector.
Degenerate member - a member of a vector that is a degenerate vector. Also called a "scalar".
Basic vector - a vector who's elements are all degenerate.
Matrix - A vector who’s elements are vectors.
Basic matrix - a vector who's elements are all basic vectors.
(a) - an ordered list consisting of the element "a". If "a" is degenerate, then "(a)" is also degenerate. This definition is here to distinguish it from (a;), which is an unordered list. "(a)" is equivalent to "(a,)" (for that matter it is also equivalent to "(a,,,)", "(a,,,,)", etc.
(a, b, c, etc) - any ordered list of scalars or vectors surrounded by parens is a vector - a matrix is a special case of vectors. When the variables inside are non-constant, the set has a number of elements greater than or equal to the variable (set) with the greatest number of elements. The commas indicate that the items in the list have a direct relationship - they are ordered. Older notation would be [w, x, y, z]
(a,>>, b) - This denotes an ordered list made from elements a, b, and every value in between. If a and b are integers, then all numbers between them are integers, same goes if they are real numbers, etc. The commas indicate that it is an ordered list
((a, b),(c, d),(a, c),etc) - given that the member vectors ((a,b),(c,d),etc) are non-degenerate, this is a matrix.
( (a,b,g),
(c,d,h), - matrices may be typed on more than
(a,d,f) ) one line for more clarity
( a b g
c d h - matrices may omit some parens and commas if they are
a d f ) formatted with distinct columns and rows, but must keep the outer parens, and any parens and commas that show any vector dimension higher than 2 (e.g. a list of a list is 2 dimensions, a list of a list of a list is 3D, a list of a list of a list of a list is 4D, ect)
{x}.N - denotes a diagonal matrix with N number of elements x down the diagonal of the matrix, starting from position one ( e.g. {1}.3 is a 3 by 3 identity matrix). Default N is the implied size of the matrix it is being added to (if it is being multiplied, the size can be considered to be either the same as in addition, or infinity - since the result will be the same for either case and any case in between). Note that x can be a scalar, vector, or matrix (also note that all those have matrix representations). The exception to this format is {0} and {00}. "0" is already a diagonal matrix. To make a diagonal matrix of {0} or {00}, simply replace "x" above with either "{0}" or "{00}" (i.e. "{{0}}" or "{{00}}" )
{1}.N - Consistent with the definition above - this denotes the “identity matrix”. N denotes the size of the identity matrix - i.e. the number of rows (= the number of columns = the number of non-zero elements).
{[y].M.N} - denotes a M by N matrix consisting only of elements y. Default M and N, are the implied size of the matrix it is being added to (if it is being multiplied, the size can be considered to be either the same as in addition, or infinity - since the result will be the same for either case and any case in between). Note that x can be a scalar, vector, or matrix (also note that all those have matrix representations).
A[num] - the num'th member of the ordered OR unordered lists A (the num'th member can also be a vector).
A[numA, numB, numC] - the numC'th member of the vector A[num1,num2]. Note that A[0] = A, much like A[1,0] = A[1] and A[2,0,0] = A[2,0] and A[2]. This is because 1 = (1,0) and 2 = (2,0) = (2,0,0).
{X} - cross - an operator denoting a cross product.
(e.g. A{X}B)
{G} - gradient
{GX} - curl
{D.n}[M] - determinant - defined to be the n-dimensional volume (or area for 2D things) of the n-dimensional parallelepiped (or parallelogram) described by a list of vectors that make up matrix M. If ".n" is omitted, the default is the implied width of the matrix (i.e. the number of columns). Note that if M is made up Note that {D.2} of a 3 by 3 matrix is ½ the surface area of the parallelepiped described by the 3 vectors. Similar properties hold for an {D.(n-1)} of a n by n matrix.
- tensor product
- wedge product
Starting NOTE: Functions are sets which are variables, therefore sets are functions. There is no difference.
Any operation or function performed on an unordered list is performed on *every* member of the list, unless of course it is one of the set operators (union ($), intersection (&), not (!), subset (-<), proper subset (=<), or the equivalent-sets operator (=<=) ). The set operators operate on whole sets.
Undefined elements of an unordered list are NULL, i.e. they don't exist. Because of this definition, a vector of sets has an equivalent set of vectors. (e.g. "((a; b), (c; d))" = "((a, c); (a, d); (b, c); (b, d))" )
(a;) - This is an unordered list of 1 element. The purpose of this notation is to distinguish it from (a) which is an ordered list. This is equivalent to "(a;;;)", "(a;;;;;;;)", etc.
(w; x; y; z) - This is an unordered list or set of 4 elements. A set can contain any number of elements, each element is separated by a ";" (semicolon). The semicolon denotes that the elements don't have any direct relationship. Older notation would be {w, x, y, z}.
(a;>>; b) - This denotes an Unordered list made from members a, b, and every value in between. If a and b are integers, then all numbers between them are integers, same goes if they are real numbers, etc. The semi-colons indicated that it is an unordered list.
f[x, y]= (x, 2 y x-4, -y) - This is a function, set or variable definition - functions and variables are sets, meaning that f is defined to be the set [y, x] . f[x, y] defines the name of the function (f) and [x, y] defines the input order (x then y). [x, 2 y x-4, -y] is the output. To be more specific, you may describe the set that contains the input variables: e.g. "f[x=<=(1;4;5;6), y=<=(2>>{00})] = (x, 2 y x, -y)".
f^y[x] - indicates a function being performed on itself y number of times (if y is negative, it indicates the inverse of the function being performed on itself y number of times). If y is a function, y must be surrounded by parentheses (e.g. f^(y[z])[x] )
f^-[x] - indicates an inverse function (is short for f^-1[x]
(*; /) - sets may also contain functions, and can be used in the way that plus-or-minus (+;-) is used.
& - "And" or "Intersection"
$ - "Or" or "Union"
! - "Not" (used immediately before the argument - using it after the argument is the factorial operator)
Function augmentation (denoting other properties)
Log[x] - Log base e of x (natural log)
Log.b[x] - Log base b of x
List.n{m}[x.n,] - This is a representation of an ordered list who's first element is x.(m[1]), second element is x.(m[2]), etc. For example, "List.n{1>>3}[x/n,]" = "(x, x/2, x/3)" and "List.(n, x){(1, 8, 2, 0), (2, 1)}[x^n,]" = "(2^1, 1^8, 0^2, 0^0)". The comma before the last bracket indicates that it is *ordered*
List.n{m}[x.n;] - This is a representation of an unordered list that contains all the members x.(m[1]), x.(m[2]), etc. For example, "List.n{1>>3}[x/n;]" = "(x; x/2; x/3)" and "List.(n, x){(1; 8; 2; 0), (2; 1)}[x^n;]" = "(2^1; 1^8; 0^2; 0^0)". The colon before the last bracket indicates that it is *unordered*
{+}[x] - Sums the elements of a vector "x". If "x" is a set, then each vector in the set will have its elements summed. Note that the individual vector elements of the vector x are *added* together (rather than summing their respective elements). For example: "{+}[( (3,4), (5,6) ) ]" = "(7,11)" and does *not* = "18".
(*)[x]
sir[x] - Is the sin function of x in radians
cor[x] - is the cos function of x in radians
tar[x] - is the tan function of x in radians
Note: since trig functions are still functions, their inverses can be written like inverses of normal functions (ie sir’-[x], cor’-[x], and tar’-[x] ). This is not the same as the reciprocal of a function, which would be written sir[x]^-1 or (sir[x])^-1 etc. Note that sir’2[x] is NOT equivalent to ( sir[x] )^2 .
Note about degrees:
Degrees don’t exist in Tillo. The old system of degrees is archaic and much less useful than the system of radians. Also, having two systems of angles makes the use of trigonometric functions confusing. Degrees are never used.
Note about Cosecant, Cotangent, and Secant:
Those words and functions will not exist in Tillo. They are all simply the reciprocal of each function, and those can be written 1/sir[x] , 1/cor[x], etc. The addition of these three functions makes trigonometric identity lists potentially more than twice as long, and make trigonometry harder to learn.
Note about trigonometric functions to a power:
To apply a power to a trigonometric function, you do NOT place the power immediately after the function name (e.g. sir^2[x] does *not* = (sir[x])^2 ). sir^2[x] = sir[sir[x]]. The correct way to apply a power is to put it after the entire function (e.g. "sir[x]^2").
: - Also isn’t an operator, but it is the standard way to separate directly related statements. It can mean “therefore”, “if, then”, “since the previous, it follows that …”, “examples are” or just “continuing that thought..”. It denotes that one equation, concept, or definition follows from or is related to another (e.g. in “F1 = F2 : F3 = F4” one came to the conclusion that F3 = F4 through analyzation of F1=F2 . It can also mean “if, then”: “a : b” means "if a, then b". Another example: “3x-7= y : 3x=y+7” ). Math functions are not the only things that the colon can be used with; in philosophy and scientific proofs, many times there are places where “therefore” is used – and that is a good place for the “->” symbol (e.g. “water is a clear liquid and object A is Solid : object A is not water” or “17+3x=A and 4*5x=A -> x=1” ). a -> b = "b follows from a" ) IT DOES NOT MEAN “EQUALS” OR “IS EQUIVALANT TO”.
; - Isn't an operator, but it is important to know that the semi-colon is the standard way to separate statements that aren't directly related. For example, if you're writing down some math where you write that one equations goes to another (using : ) but then you want to write an equation that is separate from that, you can use a semi-colon (e.g. x+2/(y+x) = 3 : (3-x)(y+x) = 2 : 3y -xy +3x -x^2 = 3 ; y = 2 s k ; s/k = t ; etc).
x >> y - Means “goes to” or “changes to”, or "from x to y". (This *cannot* mean "much greater than")
x->y - Mean "approaches" or "gets closer to"
= - equals, is equivalent to (older notation may be = )
a~=c - about equal to (e.g. F~=5.43). Note that the meaning of this depends greatly on context.
a~b=c - "a compared to b = c" means that a is much closer to c than b is (older notation would be |a-c| << |b-c| ). Note that the meaning of this depends greatly on context.
=? - tests if statements are equal (or asks if the statements are equal)
?= - same as =?
> - denotes that the value to the left of the character is GREATER THAN the value to the right
< - denotes that the value to the left of the character is LESS THAN the value to the right
NOTE about << and >> : In this language, >> does not have anything to do with the "greater than" function, and does NOT mean "much greater than". Also, the symbol << does not exist in Tillo.
+, -, {+;-}
Significant Digits -
Precision –
Musical Notation -
-END-
References
http://www.qualitycode.com/glosa/glosaenglish-cat.html
http://www.glosa.org/en/g18s.htm#S1
http://members.aol.com/idolinguo/bgrammar.html --- for information on ido. Useful for comparison
http://www.antimoon.com/sound/hotrock-br.mp3
http://www.interlingua.com/en/easy.htm
http://www.terra.es/personal5/transnauta/interlingua.htm
http://www.basiceng.com/basiceng.html
http://www.sil.org/computing/ipahelp/ipaconsv2.htm
http://www.ling.hf.ntnu.no/ipa/full/ipachart_other_fbmp3.html