Harmonics as seen by an electronics engineer

A friend of mine asked me to help him build an electronic door chime. The idea was to generate the different notes from a common high frequency by subdividing using electronic counters.

From an old physics book I had the information that the frequencies of a Pure Major Scale have the following proportions:

24  27  30  32  36  40  45  48

c   d   e   f   g   a   b   c'

If you look at these numbers you'll find that they only contain the prime factors 2, 3 and 5 or some powers of these. The lowest common multiple of these numbers is:

 32*27*5=4320

With an oscillator of 4.32 kHz one could generate the above frequencies (and a lot more) through dividing by factors that only contain powers of 2, 3 and 5. Now 24 Hz would be a rather low frequency for a door chime. If one multiplied all of the above numbers by e.g. 11 one would get a note a at the correct value of 440 Hz.

This is what we built. The 6 switches could be closed to bypass the divider. This increased the output frequency by the respective factor.
In the finished device the switches were replaced by electronic components that were controled by patterns of 6 bits read from a prom which stored the melody. All switches open gave an inaudible very low frequency that was used as pause. We used the version with the switches to test the principal. With a little practice it was possible to play tunes on this 6 key piano. I put down the pattern of switches to close or not as 2 digit octal numbers and used 3 fingers of one hand to operate switches 2-16 and the fingers of the other hand to work on switches 3-5.
Soon I noticed that the three switches 3,9,5 determined which note and the other three switches in which octave. A c always requires switch 3 to be closed and 9 and 5 open. Closing 9 and 5 with 3 open always gives a b. To get an overwiew of which switch produced which note I drew the following diagram:

                       switch 5 c(losed)   ----- a e b

                       switch 5 o(pen)     ----- f c g d

                                                 | | | |

                                      switch 3   o c o c

                                      switch 9   o o c c

Going to the right here corresponds to a multiplikation by 3 e.g. if you multiply the frequency of a c by 3 you arrive at a g. Dividing by 3 will take you from c to f. Going up from c to e corresponds to a multiplikation by 5. The factor 2 and its powers are ommitted here because they do not change the name of the note only the octave. One perhaps could envisage the factor 2 as belonging to a third axis that is perpendicular to the diagram.
The sequence f c g d continued with a e b seems familiar. It is the sequence of major scales having 1b, no signs, 1#, 2#s, 3#s ...
The interval between c and g is one fifth. It has a frequency ratio of     c / g = 2 / 3.
The interval between c and e is a big third. It has a frequency ratio of     c / e = 4 / 5.
If we ignore the octave then c - g could also be regarded as a fourth (going down) with a ratio of c / g = 4 / 3.
We see that intervals that blend nicely like fifth, fourth, and thirds correspond to neighbourhoods in the above diagram. But there is more to come.
Looking at diagonal neighbourhoods e.g. e-g (looks cryptic, doesn't it?) we find a small third with a ratio of e / g = 5 / 6.
The other diagonal c-b is a seventh with a ratio of   c / b = 8 / 15. This does not sound nice at all.
The represent the fact that one diagonal is a 'nearer' neighbourhood than the other the diagram needs a little modification. We shift the top row by half a position and get. A -   E -   B / \    /   \   /   \ F - C - G - D We find the seven notes of the c major scale arranged in a pattern of triangles. Each triangle encompasses 3 notes.
The 3 triangles pointing up: FAC, CEG, GBD are major chords. CEG being the c major chord and GBD as g major also called the dominant of c major. FAC is then as f major the subdominant to c major which itself is called the tonica.
In this diagram we see at one glance which notes belong to e.g. c major and what is the dominant or subdominant chord to this.
There is more. We find triangles pointing down like ACE. Now that is a minor chord in this case a minor (sorry). Like before we can see in the diagram that EBG i.e. e minor is the dominant to a minor. The subdominat to a minor on the other hand poses a problem unless we extend the diagram to the left which we do now.

To be continued...