Simple Linear Transformations of the Octave

A few weeks ago, a colleague in the music department sent me a page he had written describing some remarkable counting properties of the twelve pitches of the musical octave, with the circle of fourths and circle of fifths most remarkable.   Here is a generalization of his ideas.  There is some cool group theory on Z₁₂,  the set of integers 0,1,2,3,4,5,6,7,8,9,10,11 lurking here, but that will wait for a future post.  In the meantime, suppose we enumerate the twelve chromatic tones of a musical octave as in the following table:

C  C#  D  D#  E  F  F#  G  G#  A  A#  B  C
0   1   2   3   4   5   6    7   8   9   10   11   0

Notice that we will count modulo12  (mod 12) so that the C at the octave,which would have been numbered 12, becomes 0 instead.  Dave Benson refers to this as "clock arithmetic" in his comprehensive work Music: A Mathematical Offering.

The Chromatic Scale
We could generate the ascending chromatic scale with a very simple formula:

f(x)= x +1 (mod 12).

Notice that from any starting pitch x, every pitch of the chromatic scale can be expressed by n iterations of the function f.  That is,

 f (f (...f (x)))= x + n for n =1, 2, 3,....


Because the chromatic scales contains all twelve pitches, we can start with any choice of x  and eventually reach every tone in the octave. Another way to say this is that it takes twelve iterations of the generating function f to return to the starting pitch (modulo the octave). Notice the shift term “1” in f(x)is relatively prime to 12.

The Whole Tone Scales are generated by f (x)= x +2 (mod 12): 

0, 2, 4, 6, 8, 10, 0
and
1, 3, 5, 7, 9, 11, 1.

The shift term “2” in the generating function divides 12 evenly: 12÷2 =6.  Notice that six iterations of the generating function are required to return to the starting pitch, and that we need two distinct whole tone scales to reach all twelve tones.

The Diminished Seventh Arpeggios
are generated by f (x)= x +3 (mod 12):

0, 3, 6, 9, 0
1, 4, 7, 10, 1
and
2, 5, 8, 11, 2

The shift term “3” in the generating function divides 12 evenly: 12 ÷ 3 =4. Notice that four iterations of the generating function are required to return to the starting pitch, and that we need three distinct diminished seventh arpeggios to reach all twelve tones.

The Augmented Arpeggios are generated by f (x)= x +4 (mod 12):

0, 4, 8, 0
1, 5, 9, 1
2, 6, 10, 2
and
3, 7, 11, 3.

The shift term “4” in the generating function divides 12 evenly: 12 ÷ 4 =3. Notice that three iterations of the generating function are required to return to the starting pitch,and that we need four distinct augmented arpeggios to reach all twelve tones.

The Circle of Fourths is generated by f (x)= x +5 (mod 12): 

0, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 0.

The shift term “5” in the generating function is relatively prime with 12.
Notice that twelve iterations of the generating function are required to return
to the starting pitch, and that we need only one distinct circle of fourths to
reach all twelve tones.

The The Tritone Arpeggios are generated by f (x)= x +6 (mod 12):

0, 6, 0
1, 7, 1
 2, 8, 2
3, 9, 3
4, 10, 4
and
5, 11, 5.

The shift term “6” in the generating function divides 12 evenly: 12 ÷ 6 =2. Notice that two iterations of the generating function are required to return to the starting pitch,and that we need six distinct tritone arpeggios to reach all twelve tones.

The Circle of Fifths
is generated by f (x)= x +7 (mod 12):

0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 0.

The shift term “7” in the generating function is relatively prime with 12.
Notice that twelve iterations of the generating function are required to return
to the starting pitch, and that we need only one distinct circle of fifths to
reach all twelve tones. Because x +7 = x - 5 (mod 12) we see that the
ascending circle of fifths is equivalent to the descending circle of fourths.

We could continue with generating functions f (x)= x +k
(mod 12), for k =8, 9, 10, 11. However,we know that because x +8 = x - 4 (mod 12), we would simply generate the descending augmented arpeggios.
Likewise x + 9=x - 3 (mod12) generates the descending diminished seventh arpeggios, x +10 = x - 2 (mod 12) generates the descending whole tone scale, and x +11 = x - 1 (mod 12) generates the descending chromatic scale.