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

Lemma let you in on a secret

If we construct a cycle of fifths starting with one of our “magical frequencies” – and go around the cycle of fifths 12 times, Western music theory says we should end up with an  awkward gap between the frequency of the note we start with and the frequency for the same note that we end up with.   This gap is called the Pythagorean Comma – or “lemma”. But, it turns out, that gap is not some random noise that suggests a lack of harmony in nature. It is in fact always a sub-octave of the frequency you started with.

Lemma for cycle of C

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 (388.8 Hz = 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
  • So, what note is 8.1 Hz?  Let’s take it up 5 octaves (x 32) to a frequency we recognize, and we find that 8.1 Hz is a sub-octave of 259.2 Hz – the C frequency we started with!

I haven’t seen this insight anywhere else – I explored it on a whim, but it renders Western musical theory since Bach as obsolete because the Lemma was the reason that today’s “Equal Temperament” was invented.  And what does Equal Temperament do?  It apportions this “lemma” (in this case, 8.1 Hz) across all the notes of the scale – so that they’re all equally “off” and therefore no key is worse affected by using notes from these adjusted frequencies from the cycle of fifths.  The trouble with this is that it is inherently un-musical.  It makes every note, except the first and 12th note, no longer harmonically aligned with that first note.  It’s dissonant.

So, we’ve found that the lemma is itself a harmonic component of our first note – so it should not be broken up into 11 equal pieces, (as it is with Equal Temperament, apportioned across the scale – like the Horcrux pieces of Voldemort’s soul!) but it should be left as-is to generate the endless, fractal, harmonics within harmonics which music should truly envelop.

If you think about it, if Nature didn’t have this fractal sub-vibration left over at the end of the cycle of fifths, the Universe would just resonate itself to pieces – some howling positive feedback of constantly reinforcing harmonic frequencies.  But a faint little sub-octave lemma breaks the feedback, while itself quietly whistling the note we started with – in the same way that with a fractal, just when we think we’ve got to a new part of the picture we discover we’re back at the point we started at.

In the example above, both the starting note and the lemma after 12 hops around the cycle-of-fifths are C notes! So we can see that this lemma is not a random gap – but a harmonic under-tone, 5 octaves below our starting frequency.

Lemma for cycle of F

Let’s try it for another starting frequency – an F of 345.6 Hz as listed in the “happy notes” table at the top of the page (it’s my web-site, I can call these things what I want!):

  • 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?  Our ending vibration for F (356.4 Hz) minus our starting vibration for F (345.6 Hz): 356.4 Hz – 345.6 = 10.8 Hz.
  • What’s 10.8 Hz?  You may recognize it from the home-page as one of the “still-point” frequencies I observed with my tone generator.  And 10.8 x 32 (that’s 5 octaves higher) is 345.6: our starting frequency!

Again, the difference between our starting frequency and our ending frequency in the cycle of fifths is our our starting note, 5 octaves lower.  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.

The Lemma is not some embarrassing gap or hole in Nature to be apportioned across an “equal temperament”.  The Lemma is, in fact, a musical sub-harmonic of the starting frequency.

This fact turns western musical theory completely on its ear (pun by accident).  The Lemma does not break the harmonic series – it fractalizes it:  Natural harmonics do not extend in a circle – neatly overlaying each other.  If they did, probably the universe would shake itself to bits.  Instead, the resonance is more subtle. The cycle-of-fifths spirals close to the harmonic notes of the starting frequency, and the difference is a sub-octave of the starting note.  Each fragment, each gap, bears the signature of the starting frequency, like each fragment of a hologram contains an image of the whole picture.

Equal Temperament was clearly devised without this understanding – and before the invention of electronic calculators, which might have helped.  What Equal Temperament achieves is to kill the harmonic nature of every note by moving it equally away from its harmonic core, when in fact, the natural physics are completely harmonious – but not in the 17th century, “the-universe-is-a-clock” kind of way, but in a fractal, resonant way whereby each fragment/lemma/gap generates its own harmonic series, with the same fundamental frequencies but fainter and less powerful.

Complete Cycle-of-Fifths lemmas:

Cycle of Fifths

OK, rather than just give you a few spot examples, this table investigates the full circle of the cycle-of-fifths – (a “ring-tabled lemma” (that was a 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

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

  1. 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.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s