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Topic: Ternary notation

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In the News (Mon 18 Jun 18)

Ternary digits are known as trits, analogous to bit.
Unbalanced ternary can be converted to balanced ternary notation by adding 1111..
Ternary is inefficient for human usage, just as binary is. Therefore, nonary[?] (base 9, each digit is two base-3 digits) or base 27[?] (each digit is 3 base-3 digits) is often used.
www.ebroadcast.com.au /lookup/encyclopedia/te/Ternary.html   (151 words)

 Cantor set - Wikipedia, the free encyclopedia
To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation.
Since there is a clear bijection between the ternary numbers consisting of only the digits 0 and 2 and the binary numbers consisting of the digits 0 and 1, it follows that the number of endpoints in the Cantor set is equal to the number of binary strings.
A closed set in which every point is an accumulation point is also called a perfect set in topology, while a closed subset of the interval with no interior points is nowhere dense in the interval.
en.wikipedia.org /wiki/Cantor_set   (1811 words)

 Cantor set
In this case we are removing everything between 0.01 and 0.02 in the first interval and between 0.21 and 0.22 in the second interval, or in other words, everything with a 1 in the second position after the point.
By the time you are done, the numbers which remain are those that can be represented in ternary (base 3) notation with no '1' in any position.
Therefore, the numbers in the Cantor set can be mapped onto the numbers in [0, 1] by replacing every 2 in the ternary expansion with a 1, and treating the result as a binary expansion.
www.ebroadcast.com.au /lookup/encyclopedia/ca/Cantor_set.html   (1003 words)

 Digital Image Archive of Medieval Music: Archive   (Site not responding. Last check: 2007-10-22)
Black chant notation on red 5-line staves; mensural values greater than one beat indicated by repeated notes on same pitch.
The polyphonic piece, in white mensural notation on a red 5-line staff, is found on the second front flyleaf of this 15th-century copy of the Old and New Testaments; the verso of this flyleaf contains chant.
1-4 and motets in Ars Nova notation on the concluding flyleaves.
www.diamm.ac.uk /apps/Archive.jsp?navToggle=2&archiveKey=23   (896 words)

 Data Modeling: Entity-Relationship Model   (Site not responding. Last check: 2007-10-22)
A ternary relationship involves three entities and is used when a binary relationship is inadequate.
Ternary or n-ary relationships are decomposed into two or more binary relationships.
The original notation used by Chen is widely used in academics texts and journals but rarely seen in either CASE tools or publications by non-academics.
www.utexas.edu /its/windows/database/datamodeling/dm/erintro.html   (1648 words)

 Discrete space - Wikipedia, the free encyclopedia
A product of countably infinitely many copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product.
Such a homeomorphism is given by ternary notation of numbers.
In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, which is a weak form of choice.
en.wikipedia.org /wiki/Discrete_space   (1101 words)

 Math Forum - Ask Dr. Math
This is called "balanced ternary notation," and is very interesting to play with; in particular, it makes negative numbers incredibly easy to handle.
You should be able to find a simple way to convert to this notation; I once found two rather different ways to do it.
In order to have only a single digit in each "ternary place," I had to rewrite the two-digit sums so that the Threes digit is added to the column on the left.
mathforum.org /library/drmath/view/60370.html   (756 words)

 Peter's personal rants. Today's special:A Rant Concerning the Number 3...
In decimal notation, the numeral 19 is shorthand for this expression:
Ternary also distinguishes between even and odd, but the signal is subtler: A ternary numeral represents an even number if the numeral has an even number of 1s.
In binary notation, the one-digit sequences 0 and 1 are obviously square free, and so are 01 and 10 (but not 00 or 11); then among sequences three bits long there are 010 and 101, but none of the other six possibilities is square free.
www.zzrose.com /pmrrant1.html   (4702 words)

 The Baffling Binary Gigue
Thirteen of Froberger's gigues are written in binary notation and, according to Ferguson, it is supposed that they were nevertheless to be performed in either 12/8 or a very quick 3/4.
Even though Froberger notated the gigue in binary, he may also have expected that contemporaries would be familiar with the style in which gigues were performed, regardless of the score.
First, historical evidence suggests that it was to be performed in ternary, and both notational and theoretical sources point towards the "tripleness" of the gigue in general.
www.scena.org /lsm/sm7-5/students-en.html   (1983 words)

 The Shwah Language: Logic   (Site not responding. Last check: 2007-10-22)
It's called ternary because there are three digits and each successive place represents another power of three, but instead of using the digits 0, 1, and 2 (which I would call trinary or base 3), ternary uses the digits 0 and 1 and the minus sign −, which represents −1.
In ternary, the numbers zero and one are represented by 0 and 1, as in binary and decimal (base 10) notation.
In decimal notation, negative numbers are preceded by a minus sign, which is kind of an 11th digit, since you can't write all the integers without it.
www.shwah.net /shwah6.htm   (3642 words)

 Mensuration in renaissance music - the rules
We'll write N for the longer note of the division (a breve when considering time, a semibreve for prolation), and n for the shorter one (thats is, a semibreve or a minim).
Following Apel, we will notate rest symbols and values between a pair of parenthesis: (N) et (n) for symbols seen on the document, (V) et (v) for their value once they've been interpretated.
In such a context dependant notation system, one must have some symbols keeping always the same value, and acting as milestones in the sentence, thus preventing context dependance to propagate too far away.
anaigeon.free.fr /e_mensur_regles.html   (3670 words)

 [No title]
It is a notation in which ;; everything seems easy.
Given a two-pan ;; balance, you are asked to weigh a coin known to have some integral ;; weight between 1 gram and 40 grams.
For instance, a coin of 35 grams?110 in signed ternary?will ;; balance on the scale when weights of 27 grams and 9 grams are ;; placed in the pan opposite the coin and a weight of 1 gram lies in ;; the same pan as the coin.
www.xs4all.nl /~hipster/lib/scheme/gauche/ternary.scm   (555 words)

 Coloration in renaissance music
Before going on, be sure to be aware of the principles of mensuration - specially the ternary one, of course - that I won't recall here, to make the story shorter.
The word color is used, depending on authors, either in a broad meaning, whatever may be the binary or ternary nature of the application level, or in a more precise one, in which color is said for binary levels only, and hemiola for ternary ones.
Under white notation, coloured really means fl, thus some pieces were entirely written in fl just to illustrate their meaning of relief or sadness.
www.medieval.org /emfaq/anaigeon/e_coloration.html   (1981 words)

 [No title]
However, as a matter of style, this book uses the notation () when it is desirable to emphasize the use of an empty list, and uses the notation nil when it is desirable to emphasize the use of the Boolean ``false''.
The notation 'nil (note the explicit quotation mark) is used to emphasize the use of a symbol.
A ratio is notated with / as a
www.ececs.uc.edu /~aralescu/403/LISP_STUFF/Lisp1.html   (6259 words)

 The 3n+1 problem in Binary and Ternary systems   (Site not responding. Last check: 2007-10-22)
For the statements of these two, a sequence x of ternary digits will be said to have the universal top property if for any ternary sequence y if there is an odd integer t represented by yx then t has the top property.
Any ternary sequence x either has the universal top property or there is ternary digit d such that the sequence dx has the universal top property.
It will be noticed, however, that any odd integer whose ternary representation ends in 00111, does go to a multiple of 3 after 4 applications of h, as can be seen by the appearance of 00111 at the bottom of Fig.
www.sju.edu /~rbanerji/rb/papers/paper1.htm   (4301 words)

 Microsoft Office Assistance: About ORM notation   (Site not responding. Last check: 2007-10-22)
Entity shape to represent an object type in ORM notation (that is, real-world objects).
Value types represent the strings, numbers, and so on that will actually be stored in a database.
Ternary predicate shape to indicate relationships among three entities.
office.microsoft.com /en-us/assistance/HP811500161033.aspx?mode=print   (480 words)

 [No title]
For example, #o105 means 105[8] (105 in octal notation); #x105 means 105[16] (105 in hexadecimal notation); #b1011 means 1011[2] (1011 in binary notation); #\L denotes a character object for the character L; and #(a b c) denotes a vector of three elements a, b, and c.
The double-quote and vertical-bar notations are similar but distinct: double quotes indicate a character string containing the sequence of characters, whereas vertical bars indicate a symbol whose name is the contained sequence of characters.
The default notation for structures is #S(structure-name slot-name-1 slot-value-1 slot-name-2 slot-value-2...) where #S indicates structure syntax, structure-name is the name (a symbol) of the structure type, each slot-name is the name (also a symbol) of a component, and each corresponding slot-value is the representation of the Lisp object in that slot.
www.barzilay.org /books/cltl/CLtL-text   (16319 words)

Postfix notation means that the operator appears after its operand.
All the binary operators use infix notation, which means that the operator appears between its operands.
The ternary operator is also infix; each component of the operator appears between operands.
java.sun.com /docs/books/tutorial/java/nutsandbolts/operators.html   (258 words)

The square notation is named by the editors as being in mode vii, the mixolydian mode, with the final g.
The hymn in modern, four-part notation, however, shows the piece with a final of f, in the key of two flats.
Palestrina's "Kyrie" from the Missa Aeterna Christi munera is an interesting catalyst for the discussion of ternary form within the basis of the original hymn tune and its structure, through Schenkerian analysis and the evidence of modal types within pre-Baroque music.
www.humanities.mcmaster.ca /~mus701/Antonella/pal.htm   (3596 words)

 Mensuration in renaissance music - introduction
Formerly some ratios were ternary, and this is more and more true when looking at older music, because these ternary ratios - which were said perfect - were the very foundation of the system.
Though the mensuration sign first specifies that a perfect note weights three smaller ones, this long note may be imperfected by a neighbouring smaller note, and then weights only two of them; the total values still amount to three, and this is called a perfection.
Ternary division of semibreves to minims is called prolatio major, and binary division is called prolatio minor.
www.medieval.org /emfaq/anaigeon/e_mensur_intro.html   (2106 words)

 "Fibonacci Arithmetic, Coding and Computers"   (Site not responding. Last check: 2007-10-22)
For the students who learn the computer theory and design the course "Fibonacci Arithmetic, Coding and Computers" is rather useful and interesting.
Conversion algorithms of numbers from one notation in other.
Number conversion from the "Tau-system to the ternary "golden" representation.
www.goldenmuseum.com /1704ArithmCode_engl.html   (73 words)

 [Fwd: minstrel: Verceppe]   (Site not responding. Last check: 2007-10-22)
Greetings Milady and the list, I will look through my files to see if I can identify this ballo for you, and to see if I can give you a reliable set of progressions (though I am not sure that music this early can be quantified with chord changes in a reliable way).
It is probably because there are hemiolas (emblematic of Italian dance music of the 16thC into the 17thC) such as a switch from a binary primary meter with a ternary secondary meter (eg 6/8 time), to a ternary primary meter with a binary secondary meter (eg 3/4 time).
It is customary to teach musicology students that because of an analogy to the perfection of the Trinity, ternary meters were referred to as perfect, binary as imperfect--you'll sometimes see that terminology.
www.pbm.com /pipermail/minstrel/1999/003924.html   (353 words)

 Ternary Fields   (Site not responding. Last check: 2007-10-22)
Following it, the basic axioms of a ternary field are displayed, the mode is defined and its existence proved.
The basic properties of a ternary field are also contemplated there.
The terminology and notation used in this paper have been introduced in the following articles [6] [4] [5] [1] [2]
mizar.uwb.edu.pl /JFM/Vol2/algstr_3.html   (126 words)

 KiXtart.org - official site: KiXgolf: Balanced Ternary Addition - Private Coding
Balanced ternary is a positional notation for expressing values in base-3, which uses the symbols -1, 0, and 1 as digits.
Inputs are two ternary notation numbers in the range of -255 and +255.
All provided ternary number combinations must be correctly added to have a valid BTA UDF and to have the KiXgolf score count.
www.kixtart.org /ubbthreads/showflat.php?Cat=&Board=UBB14&Number=133367   (1330 words)

 Weighing 12 coins, an Odd Ball puzzle
Here are two playful applications of numbers in the base 3 - also known as the ternary number system.
However, if the counterweights are allowed to be placed on either pan, then the answer is 40, using weights of 1, 3, 9, and 27 ounces.
Then I placed the 12 coins on the pans in such a way that, if any one of them turned out to be counterfeit and heavy, the outcome of the weighing would be a positive number in balanced ternary notation.
www.cut-the-knot.org /blue/weight1.shtml   (991 words)

 Cantor Set (PRIME)
When a number is written in base two it is said to be in binary notation, and when it is written in base three it is said to be in ternary notation.
But this must be as many as there are real numbers in the unit interval – for consider: we may represent all the real numbers between 0 and 1 in binary, and this is just every possible decimal with a 1 or a 0 in each decimal place.
The idea is evident: for every number whose ternary decimal expansion consists entirely of 0’s and 2’s, match it with the corresponding number whose binary decimal expansion has 0’s in the same place, and 1’s wherever the ternary number had 2’s.
www.mathacademy.com /pr/prime/articles/cantset   (1244 words)

 FreeSoftwareShop.org forum on Fri 4:21 Jul 28 2006
American Scientist has an article digging into the balanced ternary system; using trits instead of bits.
They are 'balanced' because they are arranged symmetrically around zero.
Note that negative values are easy to recognize because the leading trit is always negative.
www.freesoftwareshop.com /forum/article.php?sid=116   (139 words)

 [No title]   (Site not responding. Last check: 2007-10-22)
In balanced ternary, where 2-trits represent -1 values, this is the negation operation.
We cut off before 8 because octal notation is the smallest base used to facilitate human-to-machine communication, and this seems quite contrary to the basic principles behind INTERCAL.
The small data types hold 16 bits, 10 trits, 8 quarts, 6 quints, 6 sexts, or 5 septs, and the large types are always twice this size.
text.staticfree.info /intercal.txt   (2785 words)

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