As William Blake, said, “to hold the universe in a grain of sand, and eternity in an hour!” Well, indeed Nature is fractal, and it seems the fundamental frequency of our harmonic scalar explorations just happens to be a Day. All the vibrations and music we make are sub-harmonics of that fundamental beat. The World turns, and we sing.
It makes sense – we live on a rotating ball of energy which emits an electro-magnetic and gravitation field, which I just happened to detect on a frequency generator in 2016 at the frequencies of 5.4 Hz (F), 10.8 Hz (F) and 7.2 Hz (B-flat); and it turns out that the ancient Hebrew measure of time (the Helek) was 3.333 recurring of today’s seconds, which is the time it takes the Earth to rotate 1/72nd of a degree, and just happens to be an E-flat, the 3rd harmonic below this same B-flat!
So, hiding in plain sight was the obvious notion that the Earth is the giant, rhythmic clock which defines our time, our harmony – the fabric of space-time which we, (and the birds!) furnish with music, conversation and thought. For every one of its rotations, we perform a myriad tiny tasks, with our little hearts trying to understand each other under the pressures of lies and dissonance passed down to us by History.
So, based on this notion that a Day might be the fundamental frequency from which we could construct a harmonic musical scale, I extrapolated the frequencies, by starting with the number of seconds in a day, (in the lower-left corner in the diagram above):
- 60 seconds x 60 minutes x 24-hours = 86,400 seconds per day (= 43,200 secs in 12 hours, for those who like to see the number 432)
- So, one beat per day = 1/86,400 seconds = 1.15740740 x 10-5recurring vibrations per second (Hz)
- Which, if you octave it up a lot (multiply by 2, many times) = 388.36148148 recurring Hz.
- This 388.36 Hz is close to the G of 388.8 Hz, calculated as a 3rd harmonic of C at 259.2 Hz – as shown at the top of the chart above.
Let’s take a quick recap of the harmonic and mathematical tools available to us:
- Octaves: The ear hears double the frequency of a note as an “octave”, e.g. perceptively the same note, only lower or higher. Multiply the frequency of your note by 2 to get an octave above, divide by 2 to get the octave below
- 3rd harmonics: Like it says on the tin, if you multiply the frequency by 3, you get the 3rd harmonic above. This corresponds with going Do-Re-Mi-Fa-So, and an example would be a G up to a D. Conversely, dividing by 3 gives the 3rd harmonic below, e.g. G down to a C
- 5th harmonics: And if you multiply your frequency by 5, you get the 5th harmonic, which corresponds to Do-Re-Mi, e.g. G to B above, and G down to E-flat, if you divide/go down.
So, these are our harmonic building blocks. Yes, harmonics form at the 7th, 9th, 11th harmonic – actually on all whole numbers, but they get progressively weaker, so the harmonics we hear primarily are the octaves, 3rd and 5th harmonics, and it’s this basic sonority that we seek in a musical scale: if these harmonics are clashing with one another, we’re going to feel that dissonance quite strongly.
So, back to the extrapolation of the musical scale from the musical Day.
We have our Day as equivalent to a G of 388.36168168 recurring Hz:
- Let’s see if we can build a harmonic scale from this which also coincides with the “phenomenal” frequencies we know (for E-flat, B-flat, and F, as discussed on the home page, and highlighted above. I refer to these as the “phenomenal frequencies” or “experiential” frequencies because I had experienced the unique “beating” phenomenon they exhibit at low frequencies.)
- Let’s see if a harmonic musical scale derived from the length of the Earth day actually sounds good! Does our existing cannon of Western music sound good using it (and hopefully Eastern music with it’s embracing of micro-tonality will also sound good, although beyond the scope of this page).
So, the 3rd harmonic is the strongest after the octave.
- The “3rd harmonic of a Day” = 24 hours / 3 = 8 hours;
- The frequency goes up as an inverse ratio – i.e. it’s a smaller portion of time (1-day going to 1/3rd of a day) so the frequency of that shorter period of time is 3 times higher, 3-times more frequent. So, multiply a G of 388.36 Hz by 3 gives us the 3rd-harmonic above (=1,165.08504504 Hz, which if we divide by 4 to go down a few octaves = D of 291.27111 Hz.
- Because in harmonics, dividing or multiplying by 2 gives you an octave, this D frequency for 8 hours can be octaved down to 4-hours, 2-hours, 1-hour, half-an-hour, 15 minutes, and still be this D.
Obviously, a Day corresponds to a complete turn of the Earth: 360 degrees. So, in the chart above I’ve also included the number of degrees, and the portion of the day in minutes which each musical harmonic represents.
In this way, we can find three, 3rd-harmonics going from G to D to A. I call these the “Geometric frequencies” because they’re all based on our rotating Earth.
Then, the other strong harmonic is the 5th-harmonic. Let’s see what happens when we take the 5th harmonic of “a Day”:
- The 5th harmonic of G is B (do-re-mi)
- As determined from a Day above, the frequency of 388.36168 Hz is a G, and the frequency for B then is 388.36 Hz x 5 = 485.452 Hz (after octaving it down)
- So? Well, as it happens, when we extrapolate downwards in 3rd harmonics from the “phenomenal frequencies” (from B-flat down through E-flat, to A-flat, etc., going down in 3rd harmonics (divide by 3) until we get to B, as shown in the centre column, above, this is the exact same frequency that we get for B based on those “phenomenal frequencies”.
So, there it is Ladies and Gentlemen – the “phenomenal” frequencies that I experienced with a low frequency tone-generator do in fact correspond directly and precisely to the rate of rotation of our Earth!
Conversely, the “geometric” frequencies generated from a Day exactly meet up with the harmonic series from the “phenomenal frequencies” (F, B-flat, E-flat). Seamlessly!
Coincidence? Well, no – clearly the frequencies we had experienced for F and B-flat were harmonics of the daily rotation of the Earth itself! It’s just that now, 6-years later, we’ve found the Geometric end of that harmonic equation!
So, that’s about it for this journey. We’ve found the musical scale. It seems to be a combination of 3rd and 5th harmonics based on the length of the Day. And as elsewhere noted, that E-flat frequency of 9.6 Hz also corresponds to a peak in the Earth’s magnetic field. So, there is some direct connection between Mass, Gravity, Rotation, Electro Magnetic frequency, and Musical frequency: The Earth is a piece of music.
So, the second question: Does this musical scale actually sound “as good as it sounds” and help achieve the miracles of healing and aesthetic pleasure we were hoping for when we set out on this quest? In my experience, yes. When playing these notes on the piano, I get a slightly dizzy sensation.
My proposal to you is to try this tuning yourself. I believe it can be transformative:
- If we create our binaural beats based on these frequencies, our brains will be attuned (I sleep with a 9.6 Hz magnetic resonator under the mattress, corresponding to E-flat, above. It brings on lucid dreams, and muscular relaxation!
- If our music and discourse are made against this fabric of sonority, we will be in-sync with the bird- and animal-song; we will hear what they are saying in the context of what it is – a hymn to the creative wonder that gave us life!
Of course, there are other harmonics of the Day, and harmonics-of-harmonics, but these ones above sound good together – I think. And, when I took this following recording of Beethoven’s Moonlight Sonata which I tuned on my computer to play only these exact frequencies, out into the fields, it seemed like the Birds and the Sheep were just singing along all the time: I just happened to show up with a piece of music which was based on the same musical scale they were singing to already!
I will admit that when I listen at home, there are parts that sound a little “off” – mostly around the E frequency. But when you combine it with the sounds of nature, somehow it fits. I suspect Ludwig didn’t choose the ideal key – but what I hear are the birds and the sheep singing to the scale that this piece of music is derived from.
Crazy? A day is a day. That may be the only thing left in our lives which cannot be re-construed, twisted and theoriticised to death. Take this piece of music out into the fields, and enjoy!
- One thing I’ve also shown in the diagram above is that, in the left column, you could keep going up from G beyond A – to E, and then on to B – but that B at 491.52 Hz is not the same as the B of 485.452 Hz in the centre-column, derived as a 5th-harmonic of our Day/G note. So, we make a decision that the harmonic the least number of “hops” from the primary frequency driving the harmonic engine (our Day/G-note) is the version of that note that we choose. I’ve highlighted the frequencies we choose with a blue box above, and the notes we are not choosing with a dotted-line box.
- Similarly, in the right-column, we can generate alternative frequencies for G, D, A, E and B. And I’ve tried to highlight in the same colour the multiple instances of a note – e.g. E in purple. This is fine for micro-tonal music – to have multiple slightly different harmonic frequencies for the same “note” – but I choose to keep the scale simple (with just 12 notes per octave) – and on the guitar, that’s what bends are for! Note, that these “upper” frequencies for G, D, A, E correspond to Zarlino’s frequencies in the 1500s, and the frequencies which enthusiasts for music based on A = 432 Hz generally espouse.
A Is Not 432 Hz
Ok – it can be, as shown top-right in the diagram above – calculated as the 5th harmonic of our “magic” F frequency. But, as you can see in the centre-column, our “phenomenal” frequencies for F, B-flat, and E-flat derive from a common root of B, which is in-turn derived as the 5th harmonic of a Day, as a G. So, A is not the “mother” frequency from which the others are propagated. G and B are. A at 432 Hz is more like the cherry on top – one of the most obscure and distant harmonics from our G-B driver frequencies.
Now, I know many will point out the significance of the number 432 (and its octave 864) in the dimensions of the Sun etc., but I put it to you that the number is correct, but the metrology is incorrect: we assume that 432 must relate to vibrations per second. But, for the ancient Hebrews and Babylonians who developed the sexagesimal counting system – the unit of time was the Helek – equal to 3.333 recurring of today’s seconds; and not the second.
So, does the number 432 appear in this harmonic scale? Yes, it does:
- Instead of 432 beats per second, 432 beats per Helek = 432 / 3.33333 seconds = 129.6 Hz. And what note is 129.6 Hz? It’s not an A, it’s an octave of C at 259.2 Hz
- Also, as it happens, there are 25,920 Heleks in a day (3.3333 seconds x 25,920 = 86,400 seconds (there’s that 432, 864 number again).
(John Michel is an excellent, readable source for information about the amazing coincidences of size, distance and proportion which mean that Pi can be derived from the ratio of the Earth to the Moon, how the Moon is sized and distanced so perfectly in relation to the sun that it precisely blocks it out in an eclipse, etc.; and amazing coincidences of how the same harmonic numbers we see here translate to the numbers of stades, furlongs and Egyptian feet necessary to span the Earth’s equator. Evidence of the harmonics of balance being foundational to our universe, and something never taught in today’s schools! )
In the paragraph above, we mentioned that there are 25,920 Heleks in a Day. There are also 25,920 years in the Great Year (the time it takes the precession of the Earth’s equinox to complete the tour of all 12 areas of the zodiac. So, the significance of 432 viewed with the ancient metrology of the Helek instead of the Second, shows itself more in the C-note as being of cosmic significance, than the A.
- Similarly for 288: Instead of 288 beats per second as a D, 288 beats per Helek = 288 / 3.33333 seconds = 86.4 Hz. And what note is 86.4 Hz? It’s not a D, it’s an octave of our magic F at 345.6 Hz
So, those numbers, 432 and 288 are significant, but in vibrations per Helek, not in vibrations per Second. It’s not for nothing that the size of the “enclosures” at Gobekli Tepe, and the distances reported for moving stone with sound correspond to the wavelengths of the frequencies for B-flat, F and C. They are “sacred” resonances because they align with the very tissue of our planet, and our conscious experience of it.
So, I would say that many well-intentioned New Agers are holding themselves back with this A=432 Hz mythology, and thereby preventing their getting to the heart of the natural harmonic scale. I know that I maintained that same prejudice for some time, which is perhaps why it took me 6 more years to figure out the rest of the harmonic musical scale.
So, you’ll notice in the chart above that A = 432 Hz, and D = 288 Hz are among the frequencies in dotted-line boxes – omitted from the scale; instead of pristine blue boxes included in the scale. They are harmonics of the great wheel; they are valid; but they’re not fundamental – and we already have frequencies for D (291.27111 Hz, corresponding to an Hour), and A (436.9 Hz, corresponding to 5 minutes.)
You can definitely make music using the A=432 Hz scale, I’ve created a piece using only Zarlino’s frequencies – and it sounds very nice. But, it somehow lacks the deep resonance that the scale based on the blue-boxes above, achieves – in my view.
When you look at the Earth frequencies in terms of Heleks, as I have noted them in the diagram above, you’ll see that these numbers 432, 288, 2592 show up again and again on different notes.
Guitars are tuned like the Day!
Basically, we’re saying in the diagram above that the magic notes extrapolated as 3rd harmonics of the Day are G, D, A; and the magic notes extrapolated from our phenomenal frequencies are B and E. It’s interesting that these five notes are how a guitar is tuned. Even more interesting is that we derived all of these as 3rd harmonics, except for the G to B, which is 5th harmonic (going horizontal on the diagram above); and on standard guitar tuning, G to B is the only string relation based on the 5th harmonic. The open notes on standard guitar tuning correspond Perhaps there is something fundamental in the way a guitar is tuned that has somehow been passed down through the ages.
- At C-sharp, it corresponds to both 1-minute, and 1-degree.
- B-flat corresponds to 1.11 seconds. There is an increasing number of people who find that most of the time when they look at a digital clock it is showing 11 minutes, 22 minutes, 33, 44, 55 minutes past the hour. Perhaps then this is a harmonic event making itself known.
60-based counting, vs 64-based counting:
The ancient Babylonians worked out our 60-based counting system. 5 x 12 is 60; 12 can be derived from 3, 4, 6, etc. In the guitar-string example it works well, because when you touch your finger at various whole-number divisions of the length of the string, you hear the harmonics that make up the fundamental sound of the string played open. And the strongest harmonics are divisions at 2, 3, 4, 5, 6, 7, 8, 9 of the string length.
But some have posited that the Ancients got it wrong: there are, for example, 64 codons in DNA, 64 hexagrams in the I-Ching, and, of course, 64 is a binary number: 2, 4, 8, 16, 32, 64. One extension to this idea is that C should be 256 Hz (because 64, 128, 256), instead of the cosmic 259.2 Hz that we see in the chart, above. (And this can be derived as the 5th harmonic of our Ab frequency (0.1 Hz x 5 = 0.5 Hz, which if you multiply by 2 gives you octaves of 1 Hz, 2, 4, 8, 16, 32, 64 Hz, etc.)
Notice though that, when viewed as vibrations per Helek, E-flat can be one vibration per Helek – or one vibration per 64 Heleks as it’s octave (1, 2, 4, 8, 16, 32, 64). E-flat, as one vibration per the Ancient unit of time provides that binary aspect to the sonic pattern, without having to distort C as 256 Hz.
As DNA has 64 codons, perhaps there is an interesting health opportunity with the use of E-flat as a healing frequency. As I mentioned, I use an electro-magnet under my mattress, set to E-flat (at 9.6 Hz) and I do seem to wake up refreshed. (The one I use, programmes in an hour of B-flat (14.4 Hz) at the end, so you are bought out of deep gamma. If I get up before that part of the programme, I’m pretty groggy!)
So, go to this section on Harmonic Instrument Design in order to play your music with these “Earth” harmonic frequencies.