Stop the presses. A while ago, I figured it would make an amazing fractal if we could feed the equation for harmonic propagation into a fractal programme – because as we know, when harmonics propagate, the gap between your starting frequency and your ending frequency is always a harmonic of your starting frequency.
Well, I found this video where he shows how fractals can be shown to make music. If someone could just help me figure out what the equation is to represent:
* a starting frequency, * multiplied by 3, * multiplied by 11 to get us back nearly to where we started, where the difference nearly matches an octave of our starting frequency
* and the difference between that gap frequency and an octave of the starting frequency again becomes the input to the next iteration of the equation.
0.9 Hz x 3 = 2.7 hz X 11 = 29.7 hz The ‘expected’ frequency is 28.8 hz (0.9 hz x 32, which is several octaves above our starting frequency).
And the difference between the starting octave and the ending frequency is 29.7 minus 28.8 = 0.9 hz, which in this case is our starting frequency, although sometimes it might be the 9th or the 5th harmonic, etc.
So, the gap is the part that makes it fractal because the gap is always a whole number harmonic of the starting frequency. It would be a really amazing fractal because it would visualise how vibrational harmonics actually propagate, while also emitting the sounds of this process!
Dear All, I’ve been going down a line of enquiry recently and wasn’t sure how to share it, but I think I’ve today reached an interesting insight. This entry is going to be a little more technical than some of the others, but hopefully I won’t lose you. Meanwhile, here’s a recording of me playing my keyboard tuned as discussed below, which you can listen to while you read (sounds better loud).
And here’s the same track – in Equal Temperament. Sounds more like the kind of music we’re used to, but somehow the beneficial, therapeutic effect is missing. See what you think:
As you know, this website is based on my discovery of some odd, resonant audio behavior which I identified using an iPhone tone generator connected to Bluetooth headphones (video of this phenomenon is on the home-page, here). Here, I discovered that the frequencies 5.4 Hz (F), 7.2 Hz (B-flat) and 10.8 Hz (F again) are the point where a sub-audible “interference warble” on either side of these frequencies stops beating.
7.2 Hz is a B-flat. While 5.4 Hz and 10.8 Hz are both the F note, an octave apart (5.4 Hz x 2 = 10.8 Hz), both the 3rd harmonic of B-flat (7.2 x 3 /2 = 10.8 Hz) – as it turns out. (It’s the “so” in do-re-mi-fa-so-la-ti-do.)
When I say these are the F and Bb notes, I mean in a new tuning based on these frequencies – rather than the 440 Hz, equal temperament standard we’re lumped with – and this site is dedicated to putting together a harmonic scale based on these “found” frequencies. So, so my initial assumption was that the lowest of these notes, the B-flat, would be the fundamental frequency from which we could derive a harmonic musical scale – and I’ve done that analysis on the homepage: calculating the 9th, 3rd, 5th, and 7th harmonic of these notes; and in turn the 9th, 3rd, 5th, and 7th of those harmonics, etc. until we’d found harmonic frequencies for each of the 12 notes required for the the western scale.
Well, then I discovered here on the EarthPulse web-site that an E-flat (a 3rd harmonic below our B-flat frequency (7.2 Hz / 3 x 2 = 9.6 Hz)), is the frequency of the Earth’s magnetic field, as measured by Robert Becker (who wrote the book, The Body Electric) by driving poles into the ground and measuring the frequency.
(I have bought one of these EarthPulse devices to experiment with for sleep – and interestingly, the default setting is this 9.6 Hz frequency, which is how by Googling this frequency I found out about EarthPulse in the first place. But the best advice came from Sadhguru who advised sleeping East/West, rather than North/South!).
So, it turns out that the Earth itself happens to generate a frequency precisely a 3rd harmonic below my frequencies – indicating that it is the Earth itself which is responsible for that “interference” frequency which you can witness in the video – (or the Earth is following a universal frequency).
I have also previously reported in this blog entry about the ancient Hebrew and Babylonian measure of time known as the “Helek” – equal to 3.3333 recurring of today’s seconds. A Helek was determined in the ancient metrology as the time it takes the Earth to rotate 1/72nd of a degree as it turns on its axis! (There’s that number again, 72. Our Bb is 7.2 Hz, by the way.) If you banged a drum once every Helek, that’s 1 beat every 3.3333 seconds = 1 / 3.3333 secs = 0.3 Hz. So, 0.3 Hz can be though of as the vibration of the Helek. And 0.3 Hz is a sub-octave of 9.6 Hz Eb (0.3 Hz x 32 = 9.6 Hz), the earth’s magnetic resonance.
So the Earth’s rotation, and it’s magnetic field both happen to resonate at a very low E-flat, the 3rd harmonic below the B-flat which I experienced. So, there is a series of 3rd harmonics being propagated naturally here from the Earth: E-flat, to B-flat to F. (Interesting, isn’t it that the resonance of the Earth’s magnetic field and the Earth’s rotation are based on that same E-flat building block of 9.6 Hz. There must be a physics formula, and if there isn’t, we’ve just discovered something!)
(It’s also worth noting here that the “magical” number of 316.8 which John Michell documents in the The Dimensions of Paradise as appearing in Revelations, Plato’s Republic, Glastonbury Abbey and Stonehenge, and in the dimensions of the Earth and Moon, turns out to be the 11th harmonic of our 7.2 Hz B-flat tone (7.2 Hz x 11 x 4 = 316.8 Hz). And the “solfeggio tone” of 528 Hz turns out to be the 11th harmonic of the G note that is a major-3rd harmonic of this E-flat frequency (9.6 Hz x 11 x 5 = 528 Hz!).) Click on the links here to read my investigation of both of these.
And you can find out more about the miraculous power of the 11th harmonic for destroying cancer in this video. (I have suggested to them that if they’re going to use an 11th harmonic, they should base it off these particular frequencies, but they haven’t gotten back to me on that!)
So, to build a harmonic scale from these “found” frequencies we can keep going up in 3rd harmonics (multiplying the frequency by 3) until we’ve got all 12 notes of the western musical scale, or we could also try going down in 3rd harmonics (divide by 3) to see if there are other fundamental tones below the ones that presented themselves to me (the Bb and F frequencies).
Our E-flat is based on the rotation of the Earth (1 vibration per Helek) and gives us 0.3 Hz. Divide this by three to go down a fifth to an A-flat (0.3 Hz / 3 = 0.1 Hz). It’s interesting to me how we’ve now gone from 3 to 1, in an almost biblical, Trinity sort of way. Is this what the Trinity was referring to, “first there was the word, and the word was [a vibration of 0.1 Hz]?! From this came the Two (octave at 0.2 Hz), and the Three, the Trinity at 0.3 Hz, E-flat?
A-flat: 0.1 Hz
E-flat: 0.3 Hz
B-flat: 0.9 Hz
F: 2.7 Hz
C: 8.1 Hz
If 0.1 Hz is the A-flat building block from which all of the universe is constructed, then I think its interesting that however odd and discordant any frequency is that you might be playing, it’s going to be based upon that 0.1 Hz building-block. In terms of the “Aleph” at the beginning of Hebrew creation, it’s interesting that this note is an A (and an A-flat at that) as fundamental as you can get. In other words, the universe is fractal, built from this 0.1 Hz building block – and its harmonics.
Like the four fundamental musical tones of creation described by JRR Tolkien in the Silmarillion, (which I just started reading – funny how the poetic mind tends to be 100 years ahead of the scientific mind) even when over-reaching ambition makes us think we’re the centre of the universe and all reality emanates from our frequency, we cannot break the fabric of the universe, made up as it is in 0.1 Hz increments. The harmonic truth always rings through. Every evil plan is somehow botched by the goodness inherent in the system: Rockefeller also gave us mobility, Bill Gates also gave us computers. Anyway, enough of the realms of literature and tyrants 🙂
An additional interesting aspect of A-flat at 0.1 Hz as the fundamental frequency is that its octaves correspond to the doubling we’re familiar with from the binary scale (1, 2, 4, 8, 16, 32, 64, 128, 256, 512) except that because we’re starting with a decimal, the frequencies for the octaves of A-flat are 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6, 51.2, 102.4, 204.8, 409.6 Hz, etc.
But keeping with the goal of delving ever downward to find that “fundamental of fundamental” (or holiest of holies – the quarks of sound, let’s say) I tried then going down a 3rd harmonic from our 0.1 Hz A-flat to C#. 0.1 Hz / 3 = 0.0333333333333333 recurring Hz. (Recurring numbers are always interesting, I think. And if as Nikola Tesla said, the universe is built of 3s, 6s, and 9s – this seems like a good sort of number to find at this point!) (The C# octave for this is 273.066666666666 recurring Hz, by the way)
And a 3rd harmonic below that C# is F#: 0.0333333333333333 Hz / 3 = 0.0111111111111111 recurring Hz. We’re back to “the one” again – (and this corresponds to an F# at 364.08888 recurring Hz.)
This is where the harmonic simplicity seems to stop, however: If we go one more 3rd harmonic down, from F# to a B, we get: 0.0111111111111111 / 3 = 0.0037037037037037 recurring Hz. We’re no longer dealing with simple 1s and 3s.
So, it’s possible that B, is the underlying frequency – the foundation of it all. “Darkness was upon the face of the deep” and all that. We know from the cycle-of-fifths, that we can go upwards or downwards to get to our destination. In this case, I think we’d go down – and it sounds better there, but I’ve positioned B in both places in the table below, if you’d like to experiment)
( Now, it did occur to me to look at this B frequency in terms of Heleks: I just mention this because I think it’s interesting. I Googled this number, 0.0037037037037037, to see if it’s significant. Interestingly, this “day calculator” tells us that (0.00370370370370 days happens to be 320 seconds. In Heleks, 320 seconds / 3.3333 seconds = 96 Heleks:
If one Helek is the time it takes the Earth to rotate 1/72 nd of a degree
So, 96 Heleks = 1.3333 recurring; that’s 1 and-a-third 3 degrees of Earth’s rotation = 0.00370370370370 Hz = B, corresponding to 485.5 Hz )
So, then starting with B (or F#) as the fundamental frequency from which all of Earth’s vibrations emanate, then our harmonic table of 3rd harmonics (“cycle of 5ths” as it’s known in musical circles) looks like this:
Above, so that you could tune your keyboard or guitar to these frequencies, I’ve added the tuning “offsets” in cents from 432 Hz Equal Temperament as well as offsets from 440 Hz Equal Temperament, For example, to adjust to get an F# of 364.089 Hz, as above, if your reference pitch in your software is A=432 Hz then you adjust by +3.755 Hz. Or if your reference pitch is A=440 Hz you’d adjust by minus 27.834 Hz, as shown above.)
Regarding the title of this blog entry, about black notes and white notes, here’s the thing I wanted to share with you all: if we make F# the starting point of our cycle-of-fifths, as shown in the table above, you notice that all the sharps (and flats) happen in the first four ‘hops’ of this cycle-of-fifths: (F# => C#, C# => G#/Ab, G# => D#/Eb, and D# => A#/Bb. All black notes to start with. And then when we go from A#/Bb => F, we get the first white note.
Looking at the piano keyboard then, where we have the black notes and the white notes, isn’t it really interesting that the black notes are the fundamental building blocks from which our harmonic series (and perhaps the universe) is constructed; while the white notes are the later tones, more mundane, perhaps?
When constructing a piece of music, you could dip into the flats/sharps when you want to make those deeper connections to creation, and use the white notes when you’re dealing with things more in our every day experience. I’ll tell you, for me, playing these notes with my keyboard, using Apple Logic adjusted to these frequencies, I really feel it. And with the mind-set that when you’re playing the white notes your dealing with the more exoteric, day-to-day side of things, and when you’re playing the black notes your dipping into the more esoteric, fundamental side of things – it really gives a sort of quantum way of thinking about what each note stands for and corresponds to in the “sonic geometry” of creation.
Well, I learned today (on Reddit) where the white notes and the black notes come from: It turns out that the medieval church organ keyboards were designed with only 8 keys, to only play the key of C-major (AKA mixolydian G, AKA only today’s white notes). The black-notes were added in later. When I was growing up, we had an 18th century piano in the house (tuned down so it wouldn’t break), and the first piece of music in the piano book was “I am C, middle-C, left-hand, right-hand middle-C!” – and I always wondered why it was “middle-C” – why not middle-D, which looks more like it’s in the middle?!. But one thing in all this I’ve noticed is that when we do arrange the cycle-of-5ths in the sequence as I have in the table above (where F-sharp is the fundamental from which all harmonics generate), C is indeed the middle key of all of that (6 5ths below C to get back to our starting frequency at F-sharp, and 6 fifths above C to return us to F-sharp (cycle-of-fifths – the lemma is fractal!). Maybe that’s why the Church favored it. In my approach though, C is more a minor key than the “white-notes-only” C-major, .
The astute will notice, hey your scale doesn’t have A = 432 Hz, or D = 288 Hz. We thought you were a hippie!
So, let’s just explore that for a moment. I have noticed when exploring the octaves of my “found” frequencies for Bb and F is that they reflect the magic numbers of 432 and 288 and 72 – but behind the veil of a decimal point! For example, the B-flat octaves are: 0.9, 1.8, 3.6, 7.2, 14.4, 28.8 Hz. The magic numbers of 72, 144 and 288 are there, but as decimals.
Similarly with the octaves of F: 2.7, 5.4, 10.8, 21.6, 43.2 Hz. The magic “432” is there, along with 54 and 108, but as decimals.
Now, it’s a fact that the major-third interval can be found harmonically by multiplying a frequency by 10. So, 43.2 Hz as an F corresponds, when multiplied by 10, with an A of 432 Hz which is the major-third of F. And if you look at my home-page, those are the frequencies I gave for A (432 Hz), D (288 Hz), G (384 Hz), C# (270 Hz) and F# (360 Hz) – but these are all derived as major-third harmonics (of F, Bb, Eb, A and D, respectively), and they can make the overall scale less cohesive than when the frequencies are generated from perfect-fifths – in my perception.
In fact, using these 3rds-based frequencies basically renders half of the harmonic series incompatible with the other half. And here’s why: When we’re dealing with 5ths, 4ths and 9th harmonics, these frequencies are all multiples of 3: for 5ths we multiply by 3, for 4ths we divide by 3, and for 9ths we multiply 9. 3 and 9 (and 6) are all multiples of 3, and therefore any frequencies generated by using this multiplier will overlap and intersect when we go around the cycle-of-fifths (see my exploration of the cycle of fifths) and the infamous gap or “lemma”.
But, in terms of constructing a harmonic, musical scale that aligns with a fundamental frequency and reflects the power of that fundamental fully, I’ve come to the conclusion that it’s best to only use the perfect-5th (3rd harmonics). Because they are based on 3, and therefore the symmetry of the pattern is not complicated by deriving notes using major-thirds (based on 5 or 10 as the multiplier). (Frequencies multiplied by 5 and 3 only coincide as multipliers at multiples of 15. A bit like the Aztec calendar they don’t harmonise very often! – whereas 3s coincide often at 3, 6, 9, 12, 15, etc.)
I tend to think, like Nikola Tesla, that 3, 6 and 9 are the cosmic numbers. Actually, I believe the cosmic numbers are just 1, 2, and 3 because you can make 3, 6 and 9 from 1, 2 and 3, as well as every other number.
For example, let’s pick any old frequency. Let’s say 1.3 H. Harmonically, it would be 0.3 Hz (Eb) x 2 x 2 + 0.1 Hz (from the Ab). Meanwhile, in the Tarot, the number 5 is considered the human number – 5 senses, 5 limbs – it’s who we are, but as we know, we’re kind of out-of-kilter with the rest of creation – except at places where multiples of 5 and 3 coincide – such as 15, 45, 60, 90, 180, 360 (which might say something about geometry).
So, I’m thinking of harmonics derived from major-thirds as “satellite” frequencies – they compliment the music in a fractal sort of way, but cannot be used to generate other frequencies from. Therefore, I’ve released myself from the prejudice that my A has to be 432 Hz, and that my D has to be 288 Hz, etc. – even though harmonically those frequencies can be generated as major thirds from our “magic notes” of F at 10.8 Hz (10.8 Hz x 10 x 4 = 432 Hz), and Bb at 7.2 Hz (7.2 Hz x 10 x 4 = 288 Hz). With perfect-fifths, we still have the magical numbers of 432, 288, 72, 54 as decimals within the harmonic sequences or Bb and F (43.2, 28.8, 7.2, 5.4, as explained above) – plus, to my ears, it sounds better.
In fact, this is where I think most attempts like this go off the rails because they don’t understand about the decimal point, so you see scales where C is defined as 256 Hz (generated as the major-third of A-flat (0.1 Hz x 5 = 0.5 Hz), with octaves at 1, 2, 4, 8, 16, 32 … 256 Hz), instead of our C of 259.2 Hz generated as the perfect-fifth from our F of 10.8 Hz). But, we do have the number 25.6Hz, disguised as a decimal, as one of the octaves of Ab.
Nature is subtle, it understands decimal points! In fact, it’s almost as if the subtle roots of life emerge from the other side of the decimal point at A-flat = 0.1 Hz; and before that the origins of the C# and F# are tiny decimals, “lost in the mists of time”. Ha ha – perhaps that’s what that phrase really means!
It would be possible to construct a musical scale which included the major-3rds as well as the 5ths and even the 7ths, but we’d end up with the 11 notes each with their 3 harmonic “flavours”, so that’s 11 x 3 notes in an octave = 33 notes, which is probably the truth, but it makes making music on a keyboard or guitar, very, very ergonomically challenging! I had a guitar neck with 24 frets per-octave for a while, and in the end, I had to pull half of them out – the music actually ends up being more discordant because half the time you’ve accidentally played the wrong variant of the harmonic. Too much choice!
The good news is that most of the fundamental frequencies I had woven into a musical scale on the home-page and on the page on instrument design have not changed (what has changed are the F#, C#, D, A, and E) – and I have both tuning files in my Apple Logic, so I can go back to the old one if I want to. (I’ll add a link to the other one here, soon.) There is a different flavour between those two scales – but right now, I’m really liking this one, just based on the perfect-fifths.
And yes, using this harmonic scale, we can only play in certain keys – but if we really are playing “in the key of Earth” – then why would you want to play in keys not compatible with that?! Frankly, that doesn’t seem to be relevant because when I’m playing these notes, they feel so resonant with the fabric of reality – that I know it’s the right key!
Here’s a link to the Apple Logic project file with these offsets already in it. Hopefully it will download for you – and assuming you have Apple Logic. The other thing I’ve done is transpose the keyboard on each track so it’s playing 3 semi-tones higher. In this way, when it looks like I’m playing an F I’m really playing an Ab, a G is really a Bb, an A-minor is really a C-minor, etc. This is because I’m kind of new to the keyboard, and by transposing in this way I can play all the most sonorous chords and modes on the white notes (e.g. Ab Lydian, Bb Mixolydian, C-minor, Eb major, F Dorian). Also, remember, this isn’t a temperament – I haven’t adjusted the frequencies from the cycle of fifths – so some keys will sound great, like the ones I mentioned above; other keys won’t sound so good. After I worked out this harmonic series, I spent a day questioning all this and went back to my original mixed tuning using some major thirds, but in the end, this 5ths-based series just sounds more awesome!