Adjacent transpositions: question about definition

In summary, the conversation discusses the definition of adjacent transpositions in a permutation. It is clarified that the transpositions refer to the original set and not the last result before the transposition is applied. The example of <1,2,3> is used to demonstrate that even after composing (2,3) and (1,2), (2,3) is still considered an adjacent transposition. This may go against intuition, but the numbers in the transposition are fixed and always mean the same thing. The conversation ends with a thank you for the clarification.
  • #1
nomadreid
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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) (2,3), and (1,3) are the adjacent transpositions for <1,2,3>
However, if I compose them (2,3)(1,2), I first apply the transposition (1,2) to <1,2,3> I now have <2,1,3> and now 2 and 3 are no longer neighbors. So is (2,3) still considered an adjacent transposition?
According the the definitions I find on the Internet, it appears that the answer is yes, but this goes contrary to the intuition of sapping neighbors at each step.
Thanks.
 
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  • #2
Initial statement: (1,3) - end points - adjacent. After trans. (2,3) - end points - not adjacent. Doesnt look right.
 
  • #3
mathman said:
Initial statement: (1,3) - end points - adjacent. After trans. (2,3) - end points - not adjacent. Doesnt look right.
Thanks, mathman. I am not sure which part doesn't look right to you. I picked a short example, but perhaps a longer example would be appropriate. (I can give the source if desired.)
adjacent.png
 
  • #4
nomadreid said:
So is (2,3) still considered an adjacent transposition?
According the the definitions I find on the Internet, it appears that the answer is yes, but this goes contrary to the intuition of swapping neighbors at each step.
Thanks.
The numbers in the transposition are fixed dummy variables. ##(2,3)## always means the same thing: swap the second and third elements in the permutation. It does not mean "swap the numbers 2 and 3, wherever they may be". You can look at ##(2, 3)## as the mapping:
$$(2, 3): (x, y, z) \rightarrow (x, z, y)$$
 
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  • #5
Super! Thanks, PeroK. That clears up the confusion. Question answered!.
 

What are adjacent transpositions?

Adjacent transpositions are a type of musical transformation where two adjacent notes in a melody or chord are switched. This results in a new melody or chord that is similar to the original, but with a slightly different sound.

How are adjacent transpositions different from other types of transpositions?

Adjacent transpositions specifically involve switching two notes that are next to each other, while other types of transpositions may involve moving notes to different intervals or keys.

What is the purpose of using adjacent transpositions in music?

Adjacent transpositions can add variation and interest to a musical piece. They can also be used to create tension or resolution in a melody or chord progression.

Can adjacent transpositions be used in any type of music?

Yes, adjacent transpositions can be used in any type of music that involves melodies or chords. They are commonly used in classical, jazz, and popular music.

How can adjacent transpositions be notated in sheet music?

Adjacent transpositions can be notated using a variety of symbols, such as arrows pointing up or down, or specific notation symbols for transposition. It is important to follow the specific notation guidelines of the musical piece or style being played.

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