In the early 1990s I had mentioned that the key that music is played in and the tempo might well be related because tempo can be thought of as a very slow note. To understand this, Let us for simplicity assume that a piece of music is in the key of C and that the frequency of middle C is 256 Hz (cycles per second). Then as we go down by an octave at a time, successive Cs will have frequencies in Hz of 128, 64, 32, 16, 8, 4, 2, etc. Now 2 cycles per second is the same as 120 cycles per minute. It might be the case that a piece of music is intended to be produced at a tempo of 120 beats per minute. In that case we may say that the tempo is in the same key as the music.
With harmonics theory I expect that after every two or three ratios of 2 (that is, moving down an octave) there might be a ratio of 3. So in that case the tempo would be 80 or 160 bpm (beats per minute).
My own investigations of some great composers such as Mozart and Beethoven found that they very often did match up key and tempo. But not all composers do. One problem is that the tempo is often not specified accurately enough for our purpose. However I found that Billy Joel specified the tempo accurately and of course we have his recordings to check. He also matched tempo to key. By match I mean that the tempo is either a number of ratios of 2 slower than the key note frequency, or possibly with one ratio of 3 also. On reflection it is possible that a second ratio of 3 should also be allowed.
Not long after I first mentioned this on internet, someone (I forget who. If it was you, please remind me) wrote to me saying something like “someone else has been thinking your thoughts”. They supplied a link to a web site by Stephen Jay and indeed he had been thinking very similar thoughts. He had gone a little further in identifying chords in the commonly used rhythms. Unfortunately after a while his web site disappeared.
A couple of days ago I was thinking about this and remembered the Internet Archive or Wayback Machine which lets you see the internet as it was years ago. Putting in the old link I found the web page existed until the late 1990s. But then I did a search and found that Stephen Jay was mentioned in Wikipedia and had a Web Site which included the Original Article.
Stephen Jay. Read his article on The Theory of Harmonic Rhythm.