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the concept of key

  • when we derive the diatonic scale starting on C from the chromatic scale, the result is the C diatonic scale, or the diatonic scale of the key of C
    • note the different meaning for the word 'key' from the meaning when we're talking about a keyboard instrument
      • this is the other term named 'key'
  • if we call C 1, and map the other 6 scale steps onto the white keys, using the gap pattern (2, 2, 1, 2, 2, 2, 1), we have named the pitches in the key of C: 
       ______________________________________________________________
   |  |/|  |/|  |  |/|  |/|  |/|  |  |/|  |/|  |  |/|  |/|  |/|  |
   |  |/|  |/|  |  |/|  |/|  |/|  |  |/|  |/|  |  |/|  |/|  |/|  |
   |  |/|  |/|  |  |/|  |/|  |/|  |  |/|  |/|  |  |/|  |/|  |/|  |
   |  |/|  |/|  |  |/|  |/|  |/|  |  |/|  |/|  |  |/|  |/|  |/|  |
   |   |    |   |   |    |    |   |   |    |   |   |    |    |   |
   | 1 | 2  | 3 | 4 | 5  | 6  | 7 | 1 | 2  | 3 | 4 | 5  | 6  | 7 |
   |   |    |   |   |    |    |   |   |    |   |   |    |    |   |
   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  • i.e., C=1, D=2, E=3, F=4, G=5, A=6, and B=7
  • but, as we saw earlier, there are 12 keys
  • C is only one of the 12 keys
  • and the C diatonic scale is only one of the 12 diatonic scales
  • the key of G
    • if we call the G key 1, we can't use white keys for all the other 6 pitches
    • we have to use a black key - F# - for scale step 7 in order to follow the pattern of the diatonic scale
    • the resulting scale is the G diatonic scale
    • keys are named for the name of the diatonic scale step 1
   _______________________________________________________________
   |  | |  | |  |  | |  | |  | |  |  | |  | |  |  | |  | |  | |  |
   |  | |  | |  |  | |  | |  | |  |  | |  | |  |  | |  | |  | |  |
   |  | |  | |  |  |7|  | |  | |  |  | |  | |  |  |7|  | |  | |  |
   |  | |  | |  |  | |  | |  | |  |  | |  | |  |  | |  | |  | |  |
   |   |    |   |   |    |    |   |   |    |   |   |    |    |   |
   | 4 | 5  | 6 |   | 1  | 2  | 3 | 4 | 5  | 6 |   | 1  | 2  | 3 |
   |   |    |   |   |    |    |   |   |    |   |   |    |    |   |
   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  • calling any of the steps in the equal-tempered system scale step 1, and using the gap pattern to select the other six scale steps results in eleven more keys: Db, D, Eb, E, etc.
    • e.g., the key of Eb contains these pitches: Eb, F, G, Ab, Bb, C, D
    • a key contains a set of 7 pitches, selected from the full set of 12
      • it also contains a set of chords
  • any pitch in the equal-tempered system can be scale step 1
  • key: mapping the seven scale steps onto the equal-tempered system of tuning
  • this is why this tuning system - equal temperament - is used: it allows you to play in any key
    • playing in any key allows people playing fixed-pitch instruments to adjust to the limitations of the human voice (and other instruments)
  • there is a real price to be paid for using the equal temperament system rather than the just intonation system, though: you're always out of tune
  • you could, of course, tune a keyboard so that the key of C was just-tuned. But when you tried to play it in another key, you'd discover that the instrument was out of tune in that key.
  • an intelligent electronic keyboard would instantly tune itself in just intonation, having been told what key you're going to be playing in