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If two keys are played simultaneously (bitonality), I usually think of the tensions by the number of notes that are not included in both Keys:

C Major C D E F G A B

G Major G A B C D E F♯

Difference of one note = Little tension

But how should I think about 3 or more keys (Polytonality) simultaneously?

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One way to conceptualize it is to think of it in the same way, just from the other side. In other words, let's not count the number of different pitches, but let's count the number of pitches that are the same.

Between C and G major, there are 7 common tones (C D E G A B), which results in a high amount of uniformity and thus little tension. Seven is the highest number of common tones shared among two major keys.

Now, let's do the same thing with three keys. To C and G major, let's add A major. These three collections share four pitches (D E A B). In contrast, let's consider C, G, and A♭ major. These three collections only share two pitches (C G), and therefore will have less uniformity (=more tension) than the collection of C, G, and A major.

Note that the highest number of common tones between three major keys would be five (think C, G, and D major, which share D E G A B); this gives you a basis on which to compare polytonal relationships among major keys.

With all that said: whether this is the best way to measure tension in polytonal music is another question.


A Possible (but Incomplete) Systematic Approach

If you're looking for an easy way to compare the number of common tones, take the two key signatures the farthest apart, and determine their difference in accidentals. Subtract this number from 7, and that will be the number of common tones. among the keys.

For instance: between C, G, and F, the greatest difference in key signatures is 2 (G has one sharp, F has one flat, which is a difference of two accidentals). The common tones shared between these three keys will be 5, because 7-2=5. Between C, D, and D♭, the greatest key signature difference is between D and D♭, a difference of 7 accidentals (!); this means that there are precisely zero common tones between these three major keys.

The system breaks down with a large number of accidentals and if you allow enharmonic spellings as common tones; consider the common tones between C, G♭, and C♯ major. The largest difference between key signatures is 13 (6 flats to 7 sharps), yet these three keys actually yield one enharmonic common tone: F (which is spelled as E♯ in C♯ major). I'm not sure yet how to make sense of this within this system. In this case, I would suggest respelling C♯ as D♭; suddenly the key signature difference is 6 flats, which results in one common tone. But we can't always spell keys enharmonically...

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    "With all that said: whether this is the best way to measure tension in polytonal music is another question." Relevant to this point, "Density Degree Theory" by Orlando Legname: oneonta.edu/faculty/legnamo/theorist/density/density.html This takes Hindemith's ideas of tonality and consonance/dissonance and expands them out to several octaves; it may be worth looking at for figuring out just how dissonant a tension may be when far away from the "bottom" key. But that's a judgement call, really. – LSM07 Apr 27 '18 at 3:12
  • Thank you Richard. I'm very glad you're around to answer all of my questions. By the way, I couldn't find a "Poly-tonality" tag to add to this question. If you are able, would you mind adding it to the list of available tags? My thanks, again. – user47327 Apr 27 '18 at 3:54

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