Western musical theory is based on a mis-conception: that there is a gap, called the wolf-tone, in the harmonic series, and that this must be eradicated!
What this gap actually reveals is that Nature is fractal: that the whole is reflected in the parts. And that the “cycle of fifths” is really a spiral of fifths – not a circle.
Building Scales from Harmonics
The strongest and loudest harmonic is the 2nd harmonic (also known as the octave). When we play a note, we hear the fundamental frequency plus a good dose of the octave.
The next strongest harmonic is the 3rd harmonic – beating three times faster than the fundamental. The note this generates we hear as the fifth note in the Western musical scale: Do-re-mi-fa-so. So, most musicians call this the “5th” interval – which is confusing, because it’s actually the 3rd harmonic – formed by a still “node” on a vibrating string or column of air, and causing a total of 3 lengths of the vibrating string to ring in unison, all three times faster than the original.
Music theorists maintain the position that Pythagoras constructed the musical scale by calculating 3rd harmonics on top of 3rd harmonics on top of 3rd harmonics. (And this is called the “cycle of fifths” for the reason outlined above).
And it’s interesting because 3rd harmonics of 3rd harmonics of 3rd harmonics is precisely how the Earth’s electromagnetic resonance seems to propagate as energetic harmonics just below human hearing.
fifths of fifths of fifths etc. until he’d derived all 12 notes of the diatonic scale. But to me, considering how easy it is to play the 2nd, 3rd, 5th and 7th harmonics on a single string – and how faint the “harmonic of a harmonic” is to hear – I don’t see why he would veer off into a theoretical approach when he had the mechanics for generating all 7 notes of the musical scale at his fingertips – literally – by touching his finger at various geometrical points along the length of a two strings tuned a harmonic fifth apart.
Leimma tell you a story
If you start with a fundamental frequency and multiply it by 3/2 you generate a musical fifth interval (e.g. B-flat to F), and if you repeat that multiplication by 3/2, 11 times you will make a complete round of the “cycle Of fifths”, bringing you back to your starting note and yielding all 12 notes of the western chromatic scale (as illustrated below).
But, in reality, the note you end up with is not exactly the note you started with.
For example, let’s start with a B-flat frequency of 0.9 Hz:
0.9 Hz x 3/2 =1.35 Hz, and F
And if we perform this harmonic multiplation a total of 12 times (and divide by the appropriate multiple of 2 to bring the frequency back down into a recognisable octave), we wind up with a B-flat frequency, not of the expected 460.8 Hz, but of 467.08681641 Hz:
The difference between what we were expecting, and what we end up with is referred to as the wolf-tone, or “leimma”, in ancient Greek.
It isn’t a whole note, it’s just a little chunk of dissonance – or so they thought.
467.08681641 Hz – 460.8 Hz = 6.28681641 Hz
So, harpsichord, piano and organ makers, tuners and composers, including Bach and Mozart, tried to account for the leimma by various schemes of “temperament” in which that gap was apportioned across the 12 intervals of the western diatonic scale – in such a way that the popular keys would suffer the least from the averaging, and the less played keys would suffer the most. Some of these “temperaments” are pleasant, ensuring perfectly resonant 3rd harmonics in one or two keys – and certain pieces of music were written specifically for these temperaments. In fact, Bach’s “Well Tempered Clavinet” was a series of explorations of multiple different temperaments.
Eventually, an equal apportioning of the leimma across all the intervals – known as “Equal Temperament” – was accepted as the standard way of tuning fixed pitch instruments (such as piano, organ and guitar) – because it enabled the same piece of music to be played in all keys, with equal harmonic dissonance.
The averaging that Equal Temperament (ET) introduces breaks the sympathetic vibration across the harmonic series. The whole self supporting interplay of harmonics is broken. The music “cancels out” its own harmonics and is essentially dissonant. Today we have food-like substances in our supermarkets – and we have music-like vibrations available for download – thanks to Equal Temperament. Welcome to modern times!
But the universe isn’t a watch-works as our 17th century predecessors thought; its more like a harmonic web of vibrating energy. Quantum physics tells us that matter is really an energy wave – photons, electrons, protons and neutrons all spinning and vibrating in harmony. Imagine if the vibration of these waves “didn’t quite add up” like the cycle of fifths doesn’t.
The leimma is fractal!
You know what else doesn’t quite add up? Fractals: the tiniest gap at the end of the fractal ends up being a holographic mirror of your starting point. Some scientists and authors are investigating the fractal nature of reality.
And here’s a little fractal insight I discovered for myself: Our gap above, 6.28681641 Hz, isn’t one of the Earth’s harmonics. The closest one is A-flat at 6.4 Hz.
What’s the difference between 6.28681641 Hz and 6.4 Hz:
6.4 – 6.28681641 = 0.11318359 Hz
This octaves up to 7.24374976 Hz – which still isn’t our B-flat frequency of 7.2 Hz precisely.
But this difference: 7.24374976 – 7.2 = 0.04374976 Hz,
And the octave of this is 2.8 Hz, which is the Earth F frequency:
So, the harmonic gap created by harmonics on top of harmonics on top of harmonics is itself the harmonic of a harmonic.
So, there is no gap – just a fractal fragment which itself is a harmonic microcosm of the note we started with. And in turn, that little harmonic fragment will generate its own harmonics – which will produce its own fractal fragment, and so on, and so on…
For some reason, music theorists didn’t examine the size of the gap between the last note of the cycle and the first. Instead, they felt this was a fundamental flaw in nature – a tear in the “music of the spheres” – and they attempted to make that gap disappear by adding it to all the other notes in the scale, breaking the fractal harmonic nature of Nature.
In fact, the “cycle of fifths” does not contradict the role of harmony in Nature; it reinforces it and perpetuates it – just as a fractal or a hologram perpetuates itself in the tiniest left over detail which contains the seed of the whole.
Did no-one else notice this before they went off any invented all the different “temperaments” of western music to try and eliminate this gap – including the abomination that is Equal Temperament? Beats me – the idea came to me at 2 o’clock in the morning in 2017, and it’s pretty simple mathematics to prove it out.
Foundations
OK – so musical harmony is fractal – the cycle-of-fifths is like a universal, harmonic fractal-generation engine. But if matter is vibration – all held in delicate interplay – is their a cosmic “first energy” which sets it all in motion? The starting frequency for the cycle-of-fifths, as it were?
Complete Cycle-of-Fifths lemmas:
This table investigates the full spiral of the “cycle-of-fifths”:
We go through 12 3rd-harmonics, 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 Earth’s harmonic series. 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.