The “cycle of fifths” is actually a spiral, because when you start with a frequency and go to its third-harmonic, and its third harmonic, in-turn until you go around in third-harmonic leaps covering all 12 notes of the western scale, when you return to the note you started with, it turns out there’s a gap between your starting frequency and your ending frequency.
For example, you start with a C of 256 Hz. Its 3rd harmonic (perfect fifth) is a G of 384 Hz. And the 3rd harmonic of that G is a D of 288 Hz, and if you keep going, you end up back at C, but the C is not the 256 Hz we started with, but rather 259.49267578125 Hz. And the difference is: 259.49267578125 Hz – 256 Hz = 3.49267578125 Hz. And this gap frequency, octaved (doubled, repeatedly) is: 447.0625 Hz – some kind of A-like frequency!
This presents a problem for anyone believing that nature is based on fundamental frequencies and principles because, “where is the beginning?”, and how come Nature is seemingly dissonant?
We did experience a couple of fundamental frequencies which are resonant with the Earth: an F at 5.4 and 10.8 vibrations per second (Hz), and a B-flat at 7.2 Hz. As it happens, we also know that an Eb, at 9.6 Hz, is a third harmonic below the Bb and corresponds to the time it takes the Earth to rotate 1/72nd of a degree (3.3333 seconds, called a “helek” by the ancient Hebrews and Babylonians. See the Discovery page for how I found these frequencies, their behavior and even the fact that those numbers are regarded by ancient cultures as sacred.
But, we don’t know if E-flat is the “foundational frequency”, or if there’s a frequency below that which generates E-flat, or a frequency below that which generates that? Or, is it “turtles all the way down”?
The strongest harmonics are the octave (multiply or divide by 2, or touch your guitar string half way along its length to hear the octave harmonic); and the second strongest harmonic is the 3rd harmonic (multiply or divide by 3, or touch your guitar string 1/3rd along its length).
Bb is already a third-harmonic below F, so we know nature is using those harmonics – but how far down do we go? The third harmonic below Bb is Eb, then Ab, then C#, then F#, then B, then E. But, where do we stop? Which frequency is the fundamental – that we can say is the “harmonic root” of all the vibration we experience – before we start repeating ourselves with each 3rd harmonic, e.g. E goes down to A, down to D, to G, to C, to F, etc. And the problem being, that when you get back to the frequencies you started with using the cycle of 3rd harmonics, the frequency you start and end at are not octaves of each other, even though they both should be. There’s a gap because in going around the cycle you actually overshoot your mark. 3rd harmonics are a little aggressive, you might say.
Now, another interesting phenomena is that the next strongest harmonic is the 5th harmonic (AKA our musical major-third interval – do-re-mi, see I told you the intervals were a confusing way to think of it, which is why i refer to the harmonics instead). And where the 3rd harmonic generally overshoots the starting point in the cycle of 3rd harmonics, the 5th harmonic generally falls short.
So, these two harmonics give us something to work with: one overshoots, the other falls short. Perhaps in some combination we could devise a harmonic scale where the ending note and the starting note are octaves of each other, or close enough for our human bodies to perceive.
With this in mind, I’ve read with interest the work of Ernest Macclain, Musical Theory and Ancient Cosmology. In Plato’s Critias he refers to “Poseidon and his five pairs of twin sons”, and McClain interprets this to refer to a harmonic series where Poseidon “begets” the other frequencies. Which frequency is “Poseidon” then, is the question!
So, long story short, I’ve used McClains’s idea to try to derive which our fundamental frequency would be, and to also construct a harmonic scale that doesn’t make war with itself at both ends (e.g. the starting frequency being dissonant with the ending frequency).
Poseidon is of course the God of the deep, the ocean, the abyss. It doesn’t hurt that my hypnotised synesthesia characterised the note B as Black, the abyss, the void. So, I’m going to combine art and science here and you can all hate me for it!
So, just to go with the idea that “Poseidon” is B for a moment, then the “twins” from the circle above are:
- F# and C#
- Ab and Eb
- Bb and F
- C and G
- D and A
And (their “mother” presumably) would be E. Interestingly, my synesthesia for E also gave it void-like energy (gray or white).
Now, we discovered Bb and F together, so those seem like pairs; and I would say the others form good pairs as well based on the synesthesia colours and my own aesthetics. But besides the subjectivity of my synesthesia, is there some other further evidence that B is the fundamental frequency of the Abyss? Well, let’s look at the numbers of these derived harmonic frequencies themselves:
There is a body of thought, which John Michell probably led, or perhaps Ernest McClain, that the numbers themselves have meaning, regardless of whether they’re vibrations per second, cubits, feet, furlongs, degrees of earth’s rotation, etc. The theory is that the ancient metrology which agreed how many degrees in a circle, how many cubits around the equator, etc., were based on an understanding of an interdependency of rotation and distance in physics (a “unified field theorem” we are yet to re-discover) where the Babylonian 60-digit counting system was the key.
With this in mind, I constructed the following table, just to look at the numbers associated with each musical note, as we go down in third harmonics from Bb (7.2 or 0.9 Hz) to Eb (0.3 Hz) etc. going in 3rd harmonics:
(I’m concerned that table may not be readable (try to zoom in), so I may re-create it, or you can ask me for the original table in Excel). Assuming you can read it, starting with the first row of data we have:
- Vibrations per second
- Vibrations per helek (1 ancient helek = 3.333 recurring moderns seconds)
- Earth degrees of rotation per vibration (e.g. we know that Eb is 1 beat every 3.333 rec seconds which is the time it takes the earth to rotate 1/72nd of a degree
- Number of seconds per vibration (for the really low vibrations)
- Number of Halakim per vibration – why not?
Here we are using the lens of harmonic numbers to identify the frequency that is the root of our harmonic series. Certain numbers, at various “octaves” and ignoring decimal points, seem to be fundamental and “sacred”. So, by looking at the numbers only, not only for Hz but also for the corresponding degrees of earth’s rotation for that frequency etc., this may help us use the points where the “magic numbers” begin and end as a way to determine the harmonic fundamental frequency which drives them all. Simply by dividing by 3 from our E-flat frequency these number patterns emerged in the table above. I’ve colour-coded the number types to help spot the patterns, e.g.
- 1234567901234579 recurring
- 37037037 recurring
- 11111111 recurring
- 33333 recurring
- 1: 8, 16, 32, 64, 256
- 3: 6, 12, 24, 48, 96
- 9: 18, 36, 72
- 27: 54, 108, 216, 432
- 81: 162, 324
Conclusions of numeric examination:
Using 3rd harmonics, the harmonic number range in the table above seems to start with low B, but goes no lower; and seems to end with C or G and go no higher. (e.g. 0.0012345679 is part of low B and high G, and this is an interesting number as it encompasses all the digits in sequence, except 8, and is recurring, and generated simply as a sub-harmonic of E-flat).
Also, if we notice, Eb is 1/72nd of a degree. The third harmonic below that (Ab) would be 1 divided by 72 divided by 3 = 1/24 degrees. And as harmonics and octaves, 1/24 is equivalent to 1/12, 1/6, 1/3 degrees.
A third-harmonic below that (C#) would be 1/1 = 1 degree. A 3rd harmonic below that (F#) would be 3 degrees; and a 3rd harmonic below that (B) would be 9 degrees of the Earth’s rotation.
9 degrees is harmonically equivalent to 18, 36, 72 degrees, as octaves. And 360 degrees divided by 72 = 5. So, what is the frequency that is 5 times slower than the B? It’s a G with a frequency of 0.0037037 rec / 5 (or 10) = 0.00037037 recurring Hz. And, octaved up, that is a G of 388.36148148 rec. Hz.
So, a day, 360 degrees, is an extremely low G which correspond to a higher G of 388.36148148 rec. Hz. And the good news is that this is very close to the G of 388.8 Hz generated as the 5th harmonic from E flat at 307.2 Hz.
So, we can start our scale at B, knowing that it’s equivalent to the time it takes the Earth to turn 9 degrees, (and also 18, 36, and 72 degrees as sub-octaves.) And if we build our scale in this way, our G for a day, and our G of 388.8 Hz match up. We have symmetry – and minimal discernible dissonance, plus it all aligns with the rotation of the only clock we know is true – the Earth’s rotation.
“Shut up and play your guitar!” Alright, well this is not guitar but here’s some music I created using this scale in Apple Logic:
It turns out the differentiation between the black notes and the white notes is quite handy because with 7 white notes, and 5 black notes, we can arrange it so that most of those 3rd harmonic (white) notes get a pure 5th harmonic (black) note to give them that “major third” interval:
- Eb has G
- F has A
- Ab has C
- Bb has D
- C could have E
And here’s how I’ve arranged the notes on my Apple Logic keyboard, where I can transpose by minus-6, like this. This puts the notes that were generated with 5th harmonics (G, A, C, D, E) as black notes, where they can be accessed from their corresponding white notes to form perfect major chords for Eb, F, Ab, Bb, respectively). (In the smallest text below, you can see the cent adjustments for each note (compared to an equal temperament scale where A is 432 Hz)).
(Note: on the Cents adjustments, I originally used this utility to calculate them from the desired Hz frequency. http://www.sengpielaudio.com/calculator-centsratio.htm. What I’ve found though is that the theoretical adjustment doesn’t produce the exact frequency – so the cent adjustments you see above are what I ended up with after using the “Oscillator” in Apple Logic which handily allows you to play every note of the scale and tells you frequency it’s playing, based on the off-sets you had put in the File -> Project Settings -> Tuning. And remember, I use a master offset of -31.8 cents to make my A = 432 Hz first, instead of A = 440Hz which is the default. )
If minus 6 is too extreme of a keyboard transposition for you, I can understand, here’s a plus 1 transposition, so you still get the benefit of the 5th harmonics being on most of the black notes.
One thing I didn’t mention is that last year I bought John Michell’s final book “How The World Is Made” and on page 11 he talks about how the geometry of 5 and 10 pertains to life. And I thought, well, we don’t want a harmonic scale based purely on 3,6,9 that is so sterile it misses out our biological essence! There’s quite a nice video about the 5 and 6 in geometry and music, here by Jain 108:
And also in John Michell’s book he references a print by Albrecht Dürer which seeks to encompass the hexagon and the pentagon into a combined geometry.
Anyway, all this is meaningless if it doesn’t sound good, so, hopefully you clicked on my SoundCloud link above and are enjoying the frequencies, if not necessarily my musicality!
- Playing the white notes only – they all sound great together!
- Playing the black notes only – they all sound great together!
- Wait for it, yes – picking our black notes to go with the white notes for “major thirds” – also sounds great!
Having “major 3rds” in a chord that are harmonically aligned to the tonic of the chord has been a goal of tuning temperaments from Bach and Mozart to the microtonalists of the present day. The trouble is, that unless you have an instrument with more than 12 notes per octave, your D as a major-3rd in a Bb chord, can’t be the “perfect 5th” in a G chord as well, as that D is slightly too low, and it sounds bad with the G. But by playing chords where the tonic or root of the chord is one of the white notes shown above, those “major-3rds” (pure 5th harmonics) are sequestered as black notes and can therefore be played only to give harmonic colour to a chord, rather than to be the tonic or 5th of that chord.
The other challenge of course is, can you play a harmonic scale in any key with this harmonic scale? Usually the answer is “no”, you can generally play in about 3 related keys in a harmonic scale, and then things start to go awry. Frankly, I’ve been trying to play harmoniously, so I haven’t explored if there’s a discordant side to this, but this scale is in harmony with our rotating earth and it’s magnetism, and our “low” 3rd-harmonic E is essentially the same as our high 5th-harmonic E – so it sounds consonant and good.
Being that the modern western scale at 440Hz for A is too high, this is an interesting solution to the problem: we should all transpose our keyboards up a semi-tone, and tune down to these magic frequencies. This would be the true way to “raise our vibration” – by raising it and lowering it!
Long story short, this is the sweetest scale I’ve produced – or heard. It feels more natural in my body. When I play with this tuning, it seems like the birds congregate to sing happily outside my window! It’s a good feeling. Life is good.
And because the low E below our starter B (323.635 Hz) and the high E (derived as the 5th harmonic of C) as 324 Hz, are pretty much the same, the scale is cyclical: there is no “war at the ends of the scale” – so it doesn’t really matter if E is “high” based on 5th harmonics, or low as our starting point for 3rd harmonics – it all gels.
The key sacred numbers are accounted for, as well:
- The frequencies I found on my tone generator (7.2 Hz for Bb, and 5.4 and 10.8 Hz for F)
- E-flat as 9.6 Hz or 0.3 Hz as a measure of the earth’s rotation
- C# as the time it takes the earth to rotate 1 degree
- Ab with a frequency of 0.1 Hz or 10 seconds per vibration, which has been thought by some to be the frequency of heart/brain coherence (e.g. Greg Braden)
- Plus we have A as 432 Hz – which everyone loves, and D as 288 Hz – both related to the dimensions of the planet.
- Our B is also the frequency of 72 degrees of rotation of the Earth;
- And a day is a G, the 5th harmonic below that B! – although we’re keeping our G as the 5th harmonic of Eb because it is a harmonic, and it sounds sweeter with the rest of the scale.
Plus, playing it, I feel like a child again, where everything is in tune.
I feel that western music has been so bastardized in every way: (the wrong frequency for A, the fact that A shouldn’t even be the reference note (e.g. probably should be E or B as the harmonic foundation, as we’ve discovered); plus we’ve been lumped with equal temperament – pretending that harmonic propagation doesn’t need to start with a common root. So, I figure if want to flip the black notes and white notes so that it’s more playable, that seems like a good thing. That said, playing the now black notes (the 5th-harmonic derived notes) all sounds great as I mentioned – so you don’t have to do the transposition if you don’t want to although you’re more likely to get combinations of notes that don’t gel quite perfectly.
So, now we have harmonic propagation from our fundamental B “word” of 0.0037037037 recurring Hz, using both the 3rd harmonic and the 5th harmonic: the numbers of sacred geometry and sacred biology.
By the way, I really recommend John Michell’s books if you’re interested in sacred geometry. The correlation between the numbers of geometry, ancient metrology, and now music can probably only be explained by a correlation between mass, gravity, electro-magnetism rotation and frequency. If you think about it, we’ve found that the Eb frequency of 9.6 Hz which corresponds to the Earth’s magnetic field is also an octave of 0.3 Hz which is the time it takes the Earth to turn 1/72nd of a degree (3.3333 recurring of today’s seconds, referred to as a ‘Helek’ in ancient Babylon). I suspect that all “vibration”, whether at the macro level like our planet, or the quantum level like electrons, is all really just rotation as a function of mass (or energy, as Einstein pointed out).
So, “energy has rotation” is basically the formula. Some clever mathematician will come up with the “holy-grail” unified field theory at some point proving that mass, rotation speed, gravity, and electromagnetism are directly related mathematically, but it’s clear that all we’re really doing here is rediscovering truths about the fabric of our universe which were also known by the ancient Babylonians and Hebrews when they developed geometry and their units of measurement. Somehow they incorporated these numbers into how they measured the rotating earth, their concept of time, and how they measured distance – and how they blew their walls down with trumpets!
And in the interim, the loss of this knowledge which we’re now re-assembling, has been at the heart of the horrible way we’ve treated each other, ourselves and our planet for the past few thousand years. But now we’ve got this harmonic knowledge back (or pretty darn close to it, I would say!) and are once more coherent, so we can feel good – music can be restored – our bond with nature and the conscious cosmos can be strengthened – and we can live natural lives in harmony with our natural world and each other!
Happy resonating!