Lemma tell you a story

That’s a reference to Max Bygraves’ – the English comedian who started all his jokes this way.  Lemma is also the Greek word for a gap.  Let’s explore:

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 we construct a cycle of fifths using one of our “magical frequencies” – and go around the cycle of fifths 11 times, Western music theory says we should end up with an  awkward gap between the frequency we start with and the frequency we get after making 11 hops around the cycle of fifths to bring us back to the starting note.   This gap is called the “Lemma” – which means gap, or “peel” in Greek.

For example, if we begin the cycle of fifths with a C of 259.2 Hz:  259.2 x 3 gives us the first fifth (a G).  Multiply that by 11 to make a full circle of the cycle of fifths and bring us back to our starting point of a C and we end up with a frequency of 8,553.6 Hz. Divide that 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 ending frequency for C of 267.3 Hz and our starting frequency for C of 259.2 Hz?

  • 267.3 Hz minus 259.2 Hz = 8.1 Hz
  • Multiply 8.1 Hz by 2 to move it up to an octave we recognize and we have, you guessed it, 259.2 Hz – our starting frequency for a C!!!

So, in this case, the Lemma of 8.1 Hz is a sub-octave of the 259.2 Hz frequency that we started with – itself also a C!!

The Lemma is not a gap – it’s a harmonic sub-octave of the starting frequency.

Let’s try it for another frequency – an F of 10.8 Hz.  If we octave this up we get a starting frequency for F of 345.6 Hz

  • 345.6 x 3 x 11 = 356.4 Hz (after you divide by 2 a few time to bring it down to a recognizable octave).
  • So, what’s the Lemma?  The vibration for F that we get having gone around the cycle of fifths 12 times (356.4 Hz) minus the vibration for F that we started with, 356.4 Hz – 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.  You can do the maths: multiply 10.8 Hz by 32 = 345.6 Hz.

The Lemma after 12 hops around the cycle of fifths starting with an F, is an F!  Just as the Lemma of a Cycle of Fifths starting with B-flat is a B-flat, and the Lemma of a cycle of fifths starting with a C is itself a C!

It actually doesn’t matter what frequency you start with – when you go around the cycle of fifths 12 times to return to your starting note, the gap between your ending frequency and your starting frequency is always a sub-octave of your starting frequency.

And, even if we go one more hop, we get the 9th harmonic exactly.  E.g. starting with an F of 345.6 Hz: 345.6 x 3 x 12 = 388.8 Hz.  What note is 388.8?  Exactly a G as calculated harmonically as a 9th of F = 345.6 Hz (345.6 Hz x 9/8 = 388.8 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 sub-octave of the starting frequency.  This demonstrable fact turns western musical theory on its ear. There is no Lemma, there is no justification for Equal Temperament.

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.

More Fractals

OK, let’s see what happens at each step of the cycle-of-fifths, multiplying the base frequency by 3 to yield the musical fifth (and dividing the result by 2 as necessary to bring the resulting frequency back down to an octave we recognize):
  • B-flat 460.8 x 3 = 345.6 F
  • 345.6 x 3 = 259.2 C
  • 259.2 x 3 = 388.8 G
  • 388.8 x 3 = 291.6 D
But this D is not the same as the D calculated as a musical third harmonic of our starting note (345.6 x 5 = 288 Hz). What’s the difference between the “cycle of fifths” D and the D calculated as a third harmonic of our starting note of B-flat? 291.6 – 288 Hz = 3.6 Hz. 3.6 x 2 is 7.2 Hz which is also 460.8 Hz when “octaved up” a few times which is our starting frequency!  The Lemma is again a fractal harmonic of the original frequency!
Getting “back on track” with D = 288 Hz:
  • 288 x 3 = 432 Hz A
  • 432 x 3 = 324 E
  • 324 x 3 = 486 B
  • 486 x 3 = 364.5 F-sharp
Our harmonic calculation for F-sharp is 360 Hz. What’s the gap? 364.5 Hz – 360 Hz = 4.5 Hz. Octave 4.5 Hz up a few times and you get 144 Hz, which is a D: The gap is the exact harmonic third of our starting note of B-flat.
Again getting back on track with F = 360 Hz:
  • 360 x 3 = 270 Hz C-sharp
  • 270 x 3 = 202.5 G-sharp
Except our harmonic calculation of G-sharp as a 6th of B-flat is 201.6 Hz.  The difference: 202.5 – 201.6 = 0.9 Hz. Octave that up a few times = 460.8 Hz: the difference is a B-flat again!
Getting back on track with G-sharp = 201.6 Hz:
  • 201.6 Hz x 3 = 302.4 D-sharp
  • 302.4 x 3 = 453.6 B-flat
But our starting frequency for B-flat is 460.8 Hz, not 453.6 Hz. What’s the difference? 460.8 Hz – 453.6 Hz = 7.2 Hz.  Octave 7.2 Hz up a few times = 460.8. The difference is itself a sub-octave of our starting frequency.
The cycle of fifths is a harmonic cycle with recurring, fractal gaps which are themselves harmonics of the starting frequency.

The understanding we now have of a quantum universe is based on a more harmonic and holographic view of energy and information and the fractal nature of musical harmony which we have identified here seems to sit comfortably with that notion; suggesting the possibility of a new world harmony founded on resonance, rather than false mechanics and altered “temperaments”.  Thanks, but I like my temperament just the way it is – naturally!

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