That’s a reference to Max Bygraves – the English comedian who started his jokes this way. Lemma is also the Greek word for a gap, or peel. 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 for the first note 1/3 along its length, thereby multiplying the original frequency by 3.
The “cycle of fifths” is a way of taking a starting note/frequency and going to its fifth, and then going to the fifth of that, and so-on until you’ve made twelve of these “5ths” hops and you’re back at your starting note having traversed all 12 notes of the western chromatic scale:
Now, you can do the following investigation of the harmonic nature of the lemma using any starting frequency, but because we’ve established (in the home-page) that there may be a fundamental vibration which generates all other vibrations as harmonics, we’ll do this investigation using those frequencies. Here again for easy reference is the summary table of the harmonic notes we generated from the sub-audio “still-point” notes of B-flat and F that we described on the home-page:
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”.
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 go up 5 octaves (x 32) to a frequency we recognize, and 8.1 Hz is a sub-octave of 259.2 Hz – the C frequency we started with!
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.
I haven’t come across this line of investigation anywhere else. It just occurred to me one night that the lemma might be sub-harmonics, and it turned out to be the case.
Equal Temperament was clearly devised without this understanding. 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:
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” ha ha!)
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!)
- 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 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-harmonic 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 manageable, 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 lemmings jump over cliffs, I suppose – three lemmings on top of each other – no wonder there’s such over-population! – oh, wait, lemmas ;-).
Something else of interest is in cycles 10 and 12, where I’ve determined 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 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
- B-flat 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
- 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
- 3/4 x 360 = 270 Hz C-sharp
- 3/4 x 270 = 202.5 G-sharp
- x 3/2 201.6 Hz = 302.4 C-sharp
- x 3/2 302.4 = 453.6 B-flat
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 founded on resonance, rather than false mechanics and altered “temperaments”. Thanks, but I like my temperament just the way it is – naturally!