The “cycle of fifths” is a way of taking a starting note/frequency and multiplying by 3 to produce the musical “fifth” harmonic – and to continue to move in fifths until you return to your starting note:
If I construct a cycle of fifths using one of our “magical frequencies” – and go around the cycle of fifths 11 times, musical theory says I should end up with an embarrassing “Lemma” – a gap between my starting note and the note I end up with.
So, let’s see what the Lemma is for our “magic” frequencies. Our starting frequency of 7.2 Hz when “octaved up” = a B-flat of 460.8 Hz.
460.8 x 3 (a fifth harmonic) x 11 should take us around the cycle of fifths 11 times to bring us back to B flat. What frequency do we actually get? 475.2 Hz.
What’s the size of the Lemma? 475.2 Hz – 460.8 Hz = 14.4 Hz
Remember what was our fundamental frequency for B flat? 7.2 Hz
What is the first octave of 7.2 Hz? 14.4 Hz !!
The Lemma from a cycle of fifths which starts at B-flat is itself a B-flat !!
So, the Lemma is not some useless, unexplainable gap. It is itself another harmonic of our B flat fundamental frequency and thus the cycle of fifths does not negate harmony. It REINFORCES and re-affirms it !
By the way, 460.8 x 3 x 12 (instead of x 11, as above) brings us to: 259.2 Hz. What note is that? Only EXACTLY a C – as we have calculated harmonically previously as the 9th of B flat.
In other words, there IS no Lemma! The cycle of fifths not only spirals around to yield a Lemma exactly equal to a low octave of itself. If you go one more fifth: that’s 12 trips around the cycle of fifths instead of 11, you end up exactly a full step up in the harmonic scale of B flat to a C !!
There IS no Lemma!!
Ok – let’s try the cycle of fifths for a C of 259.2 Hz:
259.2 x 3 x 11 = 8553.6 Hz. Divide by 2 a few times to bring it back to an octave we’re familiar with = 267.3 Hz.
What’s the difference between this frequency and our starting frequency: 267.3 – 259.2 Hz = 8.1 Hz. Multiply 8.1 Hz by 2 a few times to see what note it is in a range we recognize = 259.2 HZ !! The Lemma is the same note we started with: a C !!
The Lemma is itself a harmonic of the starting frequency!
Let’s try it for the other magical frequency which I had detected with my frequency generator to display the strange phenomena of being a “still point” that does not “beat” against the “background radiation”of sound. That’s an F of 10.8 Hz, which, as an octave we recognize, is 345.6 Hz
345.6 x 3 x 11 = 356.4 Hz (after you divide by 2 a few times to bring it to an octave we recognize)
So, what’s the Lemma?
356.4 – 345.6 = 10.8 Hz. What’s 10.8 Hz? Only, again, the magical frequency for an F that we had started with – a sub octave of our starting frequency of 345.6 Hz. The Lemma of an F is an F! Just as the Lemma of a Cycle of Fifths for B flat is a B flat! Just as the Lemma of a cycle of fifths starting with a C is itself a C !!
The Lemma is just a harmonic sub octave of the starting note!
Guess what? Is we go 12 cycles for an F instead of 11: 345.6 x 3 x 12 = 388.8 Hz. What note is that? Only EXACTLY a G as calculated harmonically as a 9th of F = 345.6 Hz. So, again, the cycle of fifths spirals right on top of the harmonic series, exactly coinciding with the next note in the harmonic diatonic scale, from an F to a G. No GAP!! Only exact matches to the harmonic scale of the starting note.
The same happens with a starting note of G, at 388.8 Hz: 388.8 x 3 x 11 = 400.95 (after you octave the result down a few times). The difference between the starting frequency and the ending frequency:
400.95 – 388.8 Hz = 12.15 Hz. Octave that up a few times = 388.8 Hz. Again, the Lemma is a sub octave of our starting frequency!!
The Lemma is not some embarrassing gap to be apportioned across an “equal temperament”. It is, in fact, a musical octave of the starting frequency and if you go one more hop around the cycle of fifths (at least for the fundamental frequencies for B flat and F) you don’t get ditched with some frequency in the middle of nowhere. What you get is the 9th harmonic of your starting note.
This demonstrable fact turns western musical theory on its ear. There is no Lemma, there is no justification for Equal Temperament.