The Cycle of Fifths – the lemma is fractal!

“Lemma” is the Greek word for a gap.  Let’s investigate:

As we discussed on the home page, a “fifth” is so-called because it is the 5th note of the western scale (do-re-mi-fa-so).  But it is actually generated by touching a string  1/3 along its length, thereby multiplying the original frequency by 3.

The “cycle of fifths” is simply a repeat of this, playing the fifth, then the fifth of the fifth, and the fifth of the fifth of the fifth and so on until you get back to your starting note.  It’s a sort of harmonic journey around the cosmos – and this is one way to find all 12 of the notes of the western diatonic scale:


The trouble is, that you don’t come back to the frequency you started with.  When you get back to that note, there’s a gap.  And since J.S. Bach’s Well Tempered Clavier, this gap was treated as proof that Nature was flawed, that the Music of the Spheres was a poetic fantasy, and that the best thing we could all do is commit genocide and destroy the planet – well, that’s what we did anyway.

Well, no more my friends!  Because it turns out that Nature is not stupid or callous but an intricate fractal, and the “lemma” at the end of the cycle of fifths is actually a sub-octave of the frequency you started with.  Every time.

For easy reference, here again is the summary table of the harmonic notes we generated from the sub-audio “still-point” notes of B-flat and F that we discovered on the home-page:

Harmonic Notes summary

Complete Cycle-of-Fifths lemmas:

Cycle of Fifths

This table investigates the full circle of the cycle-of-fifths – (a “ring-tabled lemma“! – ring-tailed lemur joke.)

We go through 12 fifths, to bring us from a starting note of B-flat, back to B-flat.

In columns-B and C, we can see that the first three fifths produce frequencies which exactly correspond to the “magical” harmonic series we worked out on the home-page. There are no lemmas in cycles 1 through 3.

Then, when we get to the fourth cycle, the frequency we get for a D (multiplying the starting frequency by 3, four times) gives us a frequency for D of 291.6 Hz.  But, the harmonic frequency we had determined (top table) for D is 288 Hz (the third harmonic of B-flat: 230.4 Hz x 5/4 = 288 Hz).

The difference/lemma (291.6 – 288 Hz = 3.6 Hz), shown in column-E .  And what note is 3.6 Hz?  (See column-G).  It turns out it is exactly a B-flat, 6 octaves below our starting note!

There are lemmas for cycles 4 through 7, and they all resolve exactly as sub-octaves of our “magical” musical scale – to F, C and G (themselves consecutive 5ths of each other, as it turns out)

When we get to the 8th cycle (at F-sharp), something new happens:  Here, we actually get a lemma on top of a lemma.  (No, this is not some breeding programme for Norwegian rodents! (Norwegian lemming joke).)

  • The first lemma is 9.05625 Hz (column-E).  Taking the same approach as before, we multiply the lemma frequency by 32 to bring it up 6 octaves into more familiar territory, making the lemma frequency 289.8 Hz.  This time, the gap frequency is not itself a harmonic of the B-flat harmonic series
  • But if we choose the closest harmonic frequency (which is D at 288 Hz) and subtract them (289.8 – 288 Hz), we get a difference of 1.8 Hz.  And what note is 1.8 Hz?  If we multiply this by 128 (7 octaves higher), we can see that 1.8 Hz is equivalent to 230.4 – our starting B-flat.  So, this lemma of the lemma is a sub-octave of B-flat.

In the table above, I’ve used green to indicate the original lemma, pink to indicate that lemma upped by octaves to make it more recognisable, and if there is a lemma on top of the lemma I have indicated those in red, and blue for a third lemma (this is why there are so many lemmings, I suppose – three lemmings on top of each other: shocking! )

Something else of interest is in cycles 10 and 12, where I’ve determined the lemma in two alternative ways: by subtracting the lemma from the note above it, or from the the note below it.  Either way, the difference comes out to be a sub-octave of our harmonic notes.  This almost has a moral aspect, like, “it’s not the path you take, all paths lead to God”.

The incredible thing is that going all around the cycle of fifths, this truth never changes: if there is a gap between the cycle-of-fifths frequency and the natural harmonic frequency as we had determined on the home-page, that gap is always a sub-harmonic of our starting note.

From this we can see that the lemma is not a break in the harmonic cycle; it is a fractal curlicue of harmonics – where the lemma itself is a faint relic of the harmonic series it is part of, and which continues the harmonic series – creating an endlessly repeating harmonic fabric.

Corrected Cycle of Fifths Approach

So, that’s really the bit of information I wanted to share.
We can get into it in more detail if we like:  If instead of just going fifth-to-fifth and observing the differences to our harmonic scale, if we instead correct the cycle-of-fifth frequencies as we go so they match our harmonic scale, it turns out we still see the same harmonic fractal lemma phenomena.
Again starting with B-flat (460.8 Hz), if we multiply by 3 we get the musical fifth (and divide the result by 2 as necessary to bring the resulting frequency back down to an octave we recognize):
  • 3/4 x 460.8 = 345.6 F
  • 3/4 x 345.6 = 259.2 C
  • 3/2 x 259.2 = 388.8 G
  • 3/4 x 388.8 = 291.6 D
This is the D lemma we discussed above.  The difference between this frequency and our harmonic series D described above is 291.6 – 288 Hz = 3.6 Hz.  And 3.6 x 128 is 460.8 Hz: our starting frequency (the gap is the the same note, 7-octaves below)!  
This time, we adjust the next note in the cycle to our magic harmonic of 288 Hz – to put us “back on track”:
  • 3/2 x 288 = 432 Hz A
  • 3/4 x 432 = 324 E
  • 3/2 x 324 = 486 B
  • 3/4 x 486 = 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.
What this suggests to me is that the real “building blocks” of universal harmony are these little low-frequency fragments – like 4.5 Hz, 1.8 Hz, and the 5.4 and 7.2 Hz sub-audio frequencies that I detected with my tone-generator.
Anyway, getting back on track with F = 360 Hz:
  • 3/4 x 360 = 270 Hz C-sharp
  • 3/4 x 270 = 202.5 G-sharp
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. Multiply that by 512 to go up 9 octaves so we recognise it = 460.8 Hz: the lemma is a B-flat again!
Getting back on track with G-sharp = 201.6 Hz:
  •  x 3/2 201.6 Hz = 302.4 C-sharp
  •  x 3/2 302.4 = 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.  7.2 Hz is one of the original frequencies I detected on the home-page, and if you octave 7.2 Hz up a few times (x 64) = 460.8. The difference, as we’ve come to expect, is itself a sub-octave of our starting frequency.
So, either way – going with the pure cycle-of-fifths traversal to 12 notes and then making an adjustment, or adjusting the notes as you go to match the natural harmonic series we developed, whatever lemmas you encounter occur at the same cycle numbers, and they always resolve to a sub-harmonic of the starting note – in this case, sub-harmonics of B-flat.

Unlike the 17th century view of the universe as a giant watch-works, the understanding we have today is of a quantum universe, 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” (wouldn’t you rather have harmony than “order”?!) – founded on resonance, rather than false mechanics and altered “temperaments”.  There is no need for Equal Temperament – it only serves to destroy the harmonic message of our music.

Progressive cycle of fifths

Using 3^12

Another we to think of the “cycle of 5ths” is that you take your first frequency and multiply by 3, to give you your first “perfect 5th”. You then do this 11 more times, so 3 x 3 x 3 etc, or 3 to the power of 12.

3^12 is 531,441

So, let’s take one of our frequencies, a low B-flat of 0.9 Hz and multiply it by 3^12 to see what frequency we wind up with when we get back to B-flat having gone around the cycle of 5ths 12 times:

531,441 * 0.9 Hz = 478,296.9 Hz

Bring it down a few octaves:
478,296.9 / 1,024 = 467.08681640625

Closest harmonic note: 460.8 Hz (an octave of our starting frequency of 0.9 Hz (0.9 Hz x 512 = 460.8 Hz)

So the gap is:
467.08681640625 – 460.8 = 6.28681640625 Hz

We have a G at 6hz (the 5th harmonic of our Eb of 9.6 Hz = 9.6 Hz x 5/8 = 6 Hz)

So, the gap between this nearby harmonic and the gap we have above is: 6.28681640625 – 6 Hz = 0.28681640625

Let’s octave this up so we can deal with it: 0.28681640625 Hz x 1024 = 293.7 Hz.

What’s the closest harmonic we have to 293.7 Hz? It’s 288 Hz (our D, as the 5th harmonic of Bb: 230.4 x 5/4 = 288).

So, what’s the gap between the remainder of 293.7 Hz and our D: 293.7 Hz – 288 Hz = 5.7 Hz.

What’s our closest harmonic to 5.7 Hz? Well we could go with 6 Hz again – to yield a difference of 0.3 Hz (6 Hz – 5.7 Hz = 0.3 Hz). 0.3 Hz as the final gap is a sub-octave of our E-flat (9.6 Hz / 32 = 0.3 Hz).

Or we could go lower, with D at 4.5 Hz (288 Hz / 64 = 4.5 Hz). So, in this case, the gap is 5.7 Hz – 4.5 Hz = 1.2 Hz. And again, 1.2 Hz is another sub-octave of Eb (9.6 Hz / 8 = 1.2 Hz).

Either way, in this example, we started with a Bb at 460.8 Hz, we went around the “cycle of fifths” 12 times by multiplying our starting frequency by 3 (to give us the “perfect fifth”), and we did that 12 times (x 3 x 3 x 3, etc.

We ended up with a frequency that was not an octave of the 0.9 Hz we started with, but when we broke down what that gap was, we found that it contained within it, several sub-harmonics of our starting frequency:

  1. E-flat (the third harmonic below our B-flat)
  2. G (the fifth harmonic of that E-flat)
  3. D (the fifth harmonic of the B-flat we started with)
  4. Another E-flat

Someone could write a computer programme to illustrate this for any frequency you start with – but I’m confident there are no exceptions: every gap at the end of going around the cycle of fifths 12 times will resolve to being comprised of a series of sub-harmonics of the frequency you started with! It’s the lemmings law, and that’s the way they like it!

So, the gap (let’s not call it the lemma any more because I’m getting increasing flack from actual music theorists who tell me that the word “lemma” refers to a specific pythagorean interval gap, and not any old gap, but that’s good enough for me). Anyway, the point is, the gap is not some non-harmonic breakdown of the natural order. It is, as I said, a fractal reflection of the frequency you started with, as sub-octaves, petering out.

If you think about it, if the cycle of fifths did exactly overlay to the frequency you started with, the universe would have resonated itself into oblivion in the first mili-moments after every big bang – or creation event – or whatever we want to think of it as. But this fractal-underpinning of harmonic propagation is much more subtle: it means that the harmonic fabric of nature extends into all things, but with the increasing subtlety of these tiny sub-harmonics of harmonics, rather than some bombastic, positive feedback, forever amplifying the frequency you started with. Reflection and complementarity is how nature works, apparently – and humans too!

6 thoughts on “The Cycle of Fifths – the lemma is fractal!

  1. Hey Jack. Jacob Bronowski called it the “lemma” in his 1970’s series, “The Ascent of Man” – so I just went with that. I see from your papers that i”lemma” does refer to a specific gap, so I’ll just call it “gap” going forward. There was an English comedian in the 70s called Max Bygraves, who used to say “lemme tell you a story” so I was going to use that as the title for that section. 🙂 Anyway, I’ll revise it.


  2. The Ascent of Man was a BBC TV series (and accompanying book, probably) here’s the episode: at 4:10

    OK- I just watched the whole thing, and he doesn’t mention “lemma” so I must have found it somewhere else! Apologies. “Comma it is, I guess.

    Interesting in the summary at the end of the video how he speaks of the evolution of mathematics in the 17th century with calculus from a static view to one in which motion and the moment in time are a factor , re-inserting the dynamic and vibrational nature of matter and bringing the whole thing full circle to what Pythagoras had been saying about harmonics.


  3. Your calculations are wrong.
    You should be multiplying by 3^11 (three to the power of 11). what you are currently doing is essentially multiplying by 33/32, which means you get a “lemma” which is always 1/32 of your original frequency, hence your notions of a suboctave. If you multiply by 3^11/(2^17) you get 1.35152435303, (0.67576217651 if you want to go down the octave again) which is the ratio of the discrepancy between the starting and the final frequency after going through the cycle of fifths.


    1. Woops, I realised my calculations were wrong upon thinking about how great the discrepancy I commented was (the ending frequency would be a fifth/fourth out). I should have multiplied by 3^12 (of course, I was thinking that there were 12 pieches in the cycle, but there are 13 pitches, the first note reappearing as the 13th though slightly out), So the calculation should be 3^12/(2^19), which give a ratio of 1.01364326477.


      1. OK – I’ve removed the naïve and incorrect calculation of x 3 x 11 (which basically just gives us the 11th harmonic of the 3rd harmonic), and replaced it with a calculation based on the 12th power to calculate the ending frequency you get once you go around the “cycle of 5ths” twelve times. All I can say is that inspiration speaks to us in the language we can understand – but sometimes we have to apply more rigor to turn that inspiration into an objective proof. Einstein didn’t actually ride on the back of a light-wave, but that vision helped him figure out relativity, apparently.


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