Process of Elimination With Musical Keys

Question

This comes from a discussion I was having with my music teacher regarding the quickest method to confirm that you are playing in the right key starting from a given note.

This is complicated by enharmonic notes (e.g. G# = Ab) but, given their auditory equivalence, I have constructed the following key sets based on which degrees of the standard western 12 note scale they contain (C = 1):

• C|1,3,5,6,8,10,12
• G|1,3,5,7,8,10,12
• D|2,3,5,7,8,10,12
• A|2,3,5,7,9,10,12
• E|2,4,5,7,9,10,12
• B|2,4,5,7,9,11,12
• F|1,3,5,6,8,10,11
• Bb|1,3,4,6,8,10,11
• Eb|1,3,4,6,8,9,11
• Ab|1,2,4,6,8,9,11
• Db|1,2,4,6,7,9,11
• Gb|2,4,6,7,9,11,12

My queries are:

a) Starting from note 'x' and testing whether they sounded discordant (T/F) given what the backing band are playing, what is the maximum number of trials you would need in order to be certain you were in the right key?

b) I presume(?) that which 'x' you choose first is inconsequential and that it is the interval between x and y (choice 2) and then y and z (choice 3). If this is the case, what are the intervals that would take you down the optimum route to eliminate all the wrong keys?

Many thanks

Answer 1

Here is one optimal route. It takes a maximum of $4$ trials to be certain of a key. Most keys take all $4$ trials to separate, but in this route, Gb, D, A, and G take only $3$.

if 1: C,G,F,Bb,Eb,Ab,Db
  if 4: Bb,Eb,Ab,Db
    if 2: Ab,Db
      if 7: Db
      else: Ab
    else: Bb,Eb
      if 9: Eb
      else: Bb
  else: C,G,F
    if 6: C,F
      if 11: F
      else: C
    else: G
else: D,A,E,B,Gb
  if 4: E,B,Gb
    if 5: E,B
      if 10: E
      else: B
    else: Gb
  else: D,A
    if 8: D
    else: A