What is transpositions: Definition and 19 Discussions

In music, transposition refers to the process or operation of moving a collection of notes (pitches or pitch classes) up or down in pitch by a constant interval.

The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
For example, one might transpose an entire piece of music into another key. Similarly, one might transpose a tone row or an unordered collection of pitches such as a chord so that it begins on another pitch.
The transposition of a set A by n semitones is designated by Tn(A), representing the addition (mod 12) of an integer n to each of the pitch class integers of the set A. Thus the set (A) consisting of 0–1–2 transposed by 5 semitones is 5–6–7 (T5(A)) since 0 + 5 = 5, 1 + 5 = 6, and 2 + 5 = 7.

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  1. nomadreid

    I Prove Even # Transpositions in Identity Permutation w/ Induction & Contradiction

    There is a proof that shows by induction (and by contradiction) that the identity permutation decomposes into an even number of transpositions. The proof is presented in the first comment here...
  2. nomadreid

    I Adjacent transpositions: question about definition

    Question: In defining adjacent transpositions in a permutation as swaps between neighbors, is one referring to the original set or to the last result before the transposition is applied? I clarify with an example. Suppose one assumes a beginning ordered set of <1,2,3> It is clear that (1,2)...
  3. B

    Permutations and Transpositions

    Homework Statement Attached are some screen shots of portion of the textbook I'm currently working through: Homework EquationsThe Attempt at a Solution My first question, why exactly can't ##\Delta## contains ##x_p - x_q## only once (note, switched from ##i,j## to ##p,q##)? As you can see...
  4. S

    MHB Proving Even # of Transpositions for Identical Permutations

    is there any easier way of proving that no matter how an identical permutation say (e) is written the number of transpositins is even. my work i tried let t_1...t_n be m transpositions then try to prove that e can be rewritten as a product of m-2transpositions. i had x be any numeral appearing...
  5. W

    Question on Permutations and Products of Transpositions.

    Hi all, I've answered a question but there's no answer for it, and if ye could tell me if I'm doing it right I'd appreciate it thanks :) Permutation: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 --------------------2 3 1 6 5 4 8 10 13 11 12 7 14 9 (i) Write it as a product of transpositions. I...
  6. M

    Expressing Transpositions as Products of Adjacent Transpositions

    Homework Statement Show that every transposition (i,j)(1≤i≤j≤n) in Sn is expressible as a product of adjacent transpositions. Also express the transposition (1,9) as a product of adjacent transpositions. Homework Equations none The Attempt at a Solution Really struggling to even...
  7. G

    Transpositions in Abstract Algebra

    Homework Statement Hi! There's a theorem 7.43 in p.221(Hungerford's abstract algebra) which states that every permutation in S_{n} is a product of transpositions. What I know about the concept of transposition is it is defined if there are at least two distinct elements. But, in the above...
  8. A

    Maximum number of transpositions required to make all permutations.

    Hi, Homework Statement Can all permutations of {A,B,C,D} be made by multiplications of transpositions (AB), (BC), (CD)? And by multiplications of transpostion (AB) and 4-cycle (ABCD)? What is the maximum number of multiplications needed in both cases? Homework Equations All...
  9. F

    Write as a Product of Transpositions

    Homework Statement Write the permutation P= 12345678 23156847  in cycle notation, and then write it as a product of transpositions Homework Equations The Attempt at a Solution I got the cycle notation to be (123)(45687), but i am now not sure now to write it as a product of...
  10. P

    Preserving cycle structure in transpositions

    Hi all, long time reader first time poster! Just need a hand on this problem I've been stuck on for a few days Homework Statement Let r=(a_1,a_2...a_k) be in S_n. Suppose that ß is in S_n. Show that: ßrß^-1=(ß(a_1), ß(a_2)...ß(a_k)). Homework Equations The Attempt at a...
  11. M

    Permutations and transpositions contradiction

    Homework Statement (1 2 3 4 5) (2 1 3 5 4) Write the bottom number as a product of transpositions Homework Equations The Attempt at a Solution 41352 41325 41235 42135 42315 24315 24351 21354 (2 4) (2 5) (2 3) (2 1)...
  12. M

    Problems with permutations and transpositions

    Homework Statement 1)Consider the permutation in S3 = ( 1 2 3 ) ( 1 2 3 ) NOTE: the two pairs of parenthesis are meant to be one pair that encases both rows Write as a product...
  13. E

    Show the Symmetric group is generated by the set of transpositions (12) (n-1 n).

    Homework Statement 5.1: Prove that S_n is generated by the set {(1 2), (3 4),...,(n-1 n)}Homework Equations None that I know ofThe Attempt at a Solution Any element in S_n can be written as a product of disjoint n-cycles. So now I need to show any n-cycle can be written as a product of...
  14. M

    Clearer Understanding of Permutation and Transpositions

    Let α (alpha) all in S_n be a cycle of length l. Prove that if α = τ_1 · · · τ_s, where τ_i are transpositions, then s geq l − 1. I'm trying to get a better understanding of how to begin proofs. I'm always a little lost when trying to solve them. I know that I want to somehow show that s is...
  15. M

    Can Any Cycle Be Expressed Using Fewer Transpositions Than Its Length?

    Homework Statement Let α (alpha) all in S_n be a cycle of length l. Prove that if α = τ_1 · · · τ_s, where τ_i are transpositions, then s geq l − 1.Homework Equations The Attempt at a Solution What I was actually looking for is where to start with this proof. I don't want the answer, just a...
  16. A

    Adjacent Transpositions of Permutations.

    Hi, Was wondering if anyone could explain to me what an adjacent transposition is (in relation to permutations, cycles etc). I know what a transposition is, eg the product of transpositions for (34785) would be (35)(38)(37)(34). I don't know what an adjacent transposition is though...
  17. S

    Max Depth for Submarine in Water: Practical Physics Transpositions

    Hello there. Can someone help me understand the following practical physics transpositions. I would like every last detail to be mentioned as I'm not really very sure on this at all. A submarine has a maximum allowable pressure of 588.6kPa. It is in water at its usual density of 1000kg/m^3...
  18. J

    Permutations and Transpositions problem help

    Can someone help me? I need to prove that for m>=2, m permutations can be written as at most m-1 transpositions. I can't figure this out for the life of me! thanks in advance :confused:
  19. W

    Permutations and Transpositions

    Hello, I am a little confused about an example. By definition, A cycle of m symbols CAN be written as a product of m - 1 transpositions. (x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn) Now Express the permutation (23) on S = {1,2,3,4,5} as a product of transpositions. (23) =...