Transposition

A rule of inference used in prepositional logic. It is often abbreviated to 'Trans' for short.

(P→Q) = (¬Q →¬P)
If P, then Q is the same as If 'not Q' than 'not P'.
And, of course, vice versa.

Think about it. Think hard. And when that doesn't work, construct a Truth Table.

Oi. Okay, look. If P happens, then Q happens. That's the (P→Q) bit. So, if Q didn't happen, you know for a fact that P didn't happen, right? Cause if it did, you would have Q. So, not Q, then not P.

Back up to Rules of Inference
Review your Logic symbols

Transpositions are permutations of {1,2,...,n} (or elements of the Symmetric group) which
swap two elements and fix the others. They are usually denoted (a b). This stands for the permutation that swaps a
and b and fixes all else.

transpositions, gender-identity/role: in G-I/R, the interchange of masculine and feminine expectancies and stereotypes mentally and in behavior and appearance.

Dictionary of Sexology Project: Main Index

Music can be transposed by putting it into a different musical key.

The most straightforward way of doing this is to just change the pitch; this is called chromatic transposition, and it will work perfectly under the tuning system of equal temperament. This can be used to make existing music fit the range or tuning of an existing instrument.

In most Western music (both classical and popular) we can also perform diatonic transpotision. In this process, all notes are moved up or down in the scale by a fixed number of notes - where the scale is formed by the seven selected notes of the major or minor key.

Generally, this is not a meaningful thing to do: melodies and harmonies change, and may even become nonsensical. There is a common exception, however. Every major key is a minor key transposed up by a third; e.g. C major is A minor except that it starts on C instead of A. So all music in a major key can be put in a minor key by diatonically transposing down by a two notes on the scale, and vice versa.

However, this only works for notes on the scale. Diatonically transposing a piece in C major to A minor turns every C to A, every D to B, every E to C; but a D# has nowhere to turn to, as there is no note between B and C.

So the major/minor change only works for all music that is firmly based in the major resp. minor scale, and only for notes that are actually on the scale, and this is true for diatonic transposition in general.

The movement of a gene from one part of the genome to another.

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From the BioTech Dictionary at http://biotech.icmb.utexas.edu/. For further information see the BioTech homenode.

In group theory, "transposition" means almost the same thing it means in ordinary language: the swapping of two things.

More precisely, let G be a group acting (from the left) on a set X. A transposition is an element of G of order 2 that only moves two elements of X. In other words, if τ is in G, then τ is a transposition if for some x and y in X,

τx = y;   τy = x

and τz = z for any other z in X.

The most important example of transpositions are the elements of the symmetric group on n letters, often denoted S_n. This is the group that consists of all permutations of n things ("letters"). In cycle notation, transpositions are the elements of S_n of the form (a b) for any letters a and b taken from {1, 2, ..., n}.

Transpositions are important because they can generate any permutation in S_n, and because Cayley's theorem tells us that any group is isomorphic to a subgroup of some symmetric group. Thus, in a sense, transpositions can generate any group whatsoever (although this might be the most inefficient way to generate an arbitrary group). This fact is of theoretical interest and reveals the fundamental nature of transpositions.

That any permutation in S_n can be written as a product of transpositions, should be intuitively clear. In fact, to avoid overkill, we can take the n - 1 adjacent transpositions

(1 2), (2 3), (3 4), ..., (n-1 n)

as generators of S_n. I will not give a formal proof of this fact, as it is bound to be more confusing and notationally entangling than enlightening. Instead, let us prove this "by inspection", as the engineers say. An arbitrary permutation σ of S_n moves the n letters {1, 2, ..., n} in some fashion. Now, imagine that these moves are performed only by moving two adjacent elements, one at a time. These motions of adjacent elements are exactly the n-1 transpositions I contend to generate S_n. For example, if σ moves 2 to 5, this can be accomplished by three transpositions: (2 3) then (3 4), followed by (4 5). Now 2 is in its rightful place according to σ, although other elements have been moved around in some manner. So the way to write an arbitrary permutation as product of transpositions is to turn this heuristic into something more systematic. This is best explained through example.

Suppose we want to write

   σ=     (1 2 3 4 5)
          (3 2 5 1 4)

as a product of transpositions. (The notation means that σ takes 1 to 3, 2 is fixed, 3 goes to 5, etc.) We have to move 3 into slot 1, and this can be done by two transpositions, (2 3), (1 2). 3 goes to 2 and then goes to 1. Now we have the following arrangement of the five letters:

           1 2 3 4 5   <--- slot number
          (3 1 2 4 5)  <--- contents of slots

Now 3 is in the right place, but 2 should be fixed by σ. No problem, we can move 2 back into its place with the transposition (2 3) (switches slots 2 and 3), without touching letter 3 in slot 1 to obtain

          1 2 3 4 5
         (3 2 1 4 5)

and so on. So far, as a product of transpositions, σ is (2 3)(1 2)(2 3), with composition going from right to left. If we continue in this manner, we discover that as a product of transpositions

σ = (4 5)(3 4)(2 3)(1 2)(2 3).

It is worth pointing out that there is nothing unique about this decomposition of σ into transpositions. For example, since a transposition is its own inverse (to undo a swap, you must swap again), we can stick any number of equal transpositions anywhere in the decomposition of σ so long as they are adjacent, for example,

σ = (4 5)(3 4)(2 3)(1 2)(2 3)(4 5)(4 5)(2 3)(2 3).

There are other more imaginative ways of decomposing σ into transpositions. Lack of uniqueness nonetheless, what is unique about the decomposition of σ is the parity of the number of transpositions. This is an important result.

Theorem. The parity of the number of transpositions in a decomposition of an arbitrary permutation is always the same regardless of the decomposition.

This result requires proof and is rather remarkable. It is what allows us to define even and odd permutations, as those who are formed by evenly many or oddly many transpositions, respectively. From this result, it is also clear that the product of two even permutations is again even (since even + even = even), so the S_n has a subgroup consisting of all even permutations. This is known as the alternating group on n letters, and for n > 4 is a nonabelian simple group. This is one of the deep reasons why the quintic equations and higher have no general solutions by radicals.

Being able to transpose keyboard music at sight is a useful, if impossible, skill for any budding organist or repetiteur. One finds that, many a time, a choir or congregation is not able to sing a piece of music, usually a hymn, at the desired pitch; therefore downward transpositions of a semitone, tone or minor third or common. An (church) organist may also be instructed to transpose a hymn in order that the key may be closer to the key of the voluntary at the end of the service.

After intensive (i.e. tenuous) study of transposition at the keyboard, I have identified four main ways of accomplishing this task. They vary in difficulty, and, to some extent, in "correctness".

1. The simplest method to achieve a transposition of up or down one semitone is to just imagine sharps, flats, or naturals (as necessary) in front of all the notes. So for example, were a hymn in D major to be transposed down one semitone, the organist can just pretend that the key signature has 5 flats (D flat major), and also remember to play C's and F's natural when he comes across them (as they are sharps in the original key of D major). Drawbacks of this method are that it only works for small up-or-down-one-semitone transpositions. Benefits are that it is perhaps the easiest method described here.
2. The next method is in some ways a generalisation of the previous way; instead of imagining sharps and flats, the player imagines that the notes on the page have been physically moved up or down by the required interval. Drawbacks are that it gets too complicated beyond a transposition of 1 tone in either direction; and sharps, flats and naturals in the original key signature can too easily foul things up. Note that this is probably the default method of transposition for almost all new organists, and perhaps inexperienced organists in general.
3. The third method is also perhaps the most idiosyncratic. It is as follows: observe that, once you have memorised a melody, it is possible to sing it back to yourself in any key; regardless of the note you choose to start from, your brain finds it perfectly easy to reconstruct the melody in real time (this also applies to people with absolute pitch – it is the memorising that counts). Therefore, if you are transposing a hymn that is fairly familiar to you, it is possible, if a little contrived, to play through the hymn in your head in the new key, perhaps glancing at the music here and there for a memory aid for some of the chords, and play the notes on the keyboard based on what notes you have in your head. A drawback of this method is that you are unlikely to have memorised perfectly all the chords of the hymn in addition to the complete melody, so this method perhaps only works for hymns that you know particularly well.
4. The final method is generally held to be the most "correct": as you play the hymn, you note the interval between each successive pair of notes, and move your fingers on the keyboard by that interval to the next note; the idea is that you ignore the absolute pitch value of the note, and so having started the hymn in the transposed key, everything thereafter is played at relative pitch. This method requires a lot of concentration and/or experience, but, once learned, is (at least in theory) just as easy for transpositions of a semitone as it is for, say, a diminished fifth.

It is expected that as you learn how to transpose, you tend to gravitate towards the more "correct" method (the fourth one in the list) – though it is important to note that due to the extreme difficulty in calculating the interval in every part in real-time and transferring that to the intervals on the keyboard, various other techniques must be employed: recognising the general shape of a chord and identifying it e.g. as "dominant, first inversion"; taking advantage of "free gifts" – notes that do not change in a particular part between two chords (sometimes not even two adjacent chords: experienced practitioners will be looking several chords ahead to find notes that do not change that they can use as "free gifts").

A final note: remember to choose a steady tempo and stick to it; don't dive in straight away with a transposition of a compound augmented fourth – have a go at small shifts on the order of one or two semitones (and start with method 1 above) – it's all good practice in the end.

Trans`po*si"tion (?), n. [F. transposition, from L. transponere, transpositum, to set over, remove, transfer; trans across, over + ponere to place. See Position.]

The act of transposing, or the state of being transposed.

Specifically: --

(a) Alg.

The bringing of any term of an equation from one side over to the other without destroying the equation.

(b) Gram.

A change of the natural order of words in a sentence; as, the Latin and Greek languages admit transposition, without inconvenience, to a much greater extent than the English.

(c) Mus.

A change of a composition into another key.

 

© Webster 1913.