Mandelbrot Music

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.


E.g.

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!

Can anyone help with the equation?!

Black Notes and White Notes

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
  • Then, 96 Heleks = 96 – 72 = 1 degree and 24 Heleks
  • 24 / 72 = 0.3333 recurring
  • 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.

Post-script:

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, .

Technical Note:

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 25.6 Hz, 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!

I would be interested to hear what you think.

Robert Johnson – how he may have tuned his guitar

Legend has it that Robert Johnson wandered down to the cross-roads one night, handed his guitar to the Devil – who tuned it for him and handed it back. And in this way, he went from a guitar player who his mentor, Son House described as “noising” to one of the most delicate, lyrical bluesmen recorded. When you hear Robert Johnson’s recordings from 1936/7, you can hear the fundamentals of jazz and rag-time; and old Scotty Moore’s guitar work in the 1950s on those first Elvis Sun Session recordings is essentially just an electrified Robert Johnson.

There’s two things with Robert Johnson’s guitar playing. One is that he plays pretty fast, and the other that it’s really difficult to play if you tune your guitar in the normal style (E, A, D, G, B, E). There’s another interesting thing though: the released tracks are generally a semi-tone sharp. If you visit the Wikipedia page which lists all his recording dates, you’ll see that in some cases there was a second “take” recorded. Now, his recorded songs were generally released as 78s by his record company, and what seems to have happened is that the versions that were released were all speeded up slightly, while the un-released takes where left alone. Here’s an analysis of the key of each of the songs as released in the wonderful “Centennial Collection” (which manages to get rid of all the hisses and pops – you see, computers can be useful for some things!). So, I’ve used this release as a reference, and simply noted down the released key of each recording (in parentheses) and in [square brackets] is what I reckon is the correct key:

Session: 1936, November 23rd:
Kind Hearted Woman (B) [Bb]
** 2nd take in (Bb)
Dust my Broom (E) [Eb]
Sweet Home Chicago (F#) [F]
Ramblin’ on my mind (F#) [F}
** in (F) 2nd take
When You Got A Good Friend (F#) [F]
Come On In My Kitchen (B) [Bb]
** in (Bb) 2nd take

What’s interesting about this session is that the second take is always a half step lower than the first. It’s possible that he just decided to down-tune on the second take of each song – but pretty unlikely! And there has been a rumor for years that his record company sped up the recordings for the release – whether by accident or to make the recordings more “exciting”. It seems clear to me based on the 2nd takes above, that the original recordings were a half-step lower. e.g. Kind Hearted Woman was released on 78 in the key of B, but the 2nd take on the Centennial Collection, which was not released on 78, is in Bb. Similarly, Ramblin’ On My Mind was released in F#, but the second take is in F.

As it happens, lots of people have attempted to slow Robert Johnson recordings down to the “corrected speed”. What I’ve done is create a YouTube playlist of all the Robert Johnson songs, but adjusted to what I think is the right key – per the tables above and below. Have a listen. One thing I think you’ll notice is that his voice just sounds more natural in these recordings, slowed down essentially just a semi-tone for the early sessions, and the final session recordings all left as-is.

So, if RJ is playing a low Eb, he’d have to be tuned down to Eb, even if he was in regular tuning. Now, anyone who plays guitar knows that the blues is a starting chord (the 1), a second chord (the 4th harmonic), and a turn-around chord (the 5th above, or the 4th below). So, with a regularly tuned guitar you can start on the A-string, then play a D-chord a 4th above, and then play the turn-around E-chord a 4th below – all using open strings. We guitarists try to keep it simple – using the open strings where possible. So, if Robert Johnson was playing in Eb concert tuning, then the easy open chords to play would be Ab (1), Db (IV) and Eb (V) – and while there’s quite a lot of Eb in his recordings – there’s only five songs in Ab. So I think this also suggests we may have been tuned to an Open-Eb chord. A lot of blues players in the Delta were tuned to open-D and open-E. I think Robert Johnson just tuned to Open-Eb like his mentor, Son House in this video of “Death Letter Blues”.

But there’s more: There’s an excellent documentary about Robert Johnson on Netflix called “Devil at the The Crossroads” (where would we be without these kinds of sensationalist titles for our media?!). The documentary is excellent because it includes first-hand accounts by people who knew him – including Son House who says, “yeah, he had a 7th string on his guitar”. A 7th string eh? Old Robert just tied another string on and voila, he’s a guitar genius? Well, you can’t just tie another string onto a six string guitar and turn the tuning peg and expect the two strings to go into tune. And if he had a 7th tuning peg on his guitar, you’d think other people would have commented on it – and when he lost his guitar in a fire, he wouldn’t have just been able to replace it – even if there was such a thing as a “luthier” in any of the towns RJ visited – who could have added a seventh notch to his nut, another string insert on his bridge, moved all the strings around a bit – and managed to conceal all this in the photo above! So, what I reckon is that Robert J tuned his guitar according to the harmonic series, in open tuning – but, here’s the thing: Instead of making the top string another Eb (the tonic), I think he tuned it to the 7th harmonic – i.e. a C# (a whole-tone below Eb), like this:

6th string (lowest): Eb (harmonic = 2)
5th string: Bb (harmonic = 3)
4th string: Eb (harmonic = 4)
3rd string: G (harmonic = 5)
2nd string: Bb (harmonic = 6)
1st string (highest): C# (harmonic = 7)

This is pure pythagorean tuning – meaning that he’s tuning each string according to the 2nd harmonic, 3rd harmonic, 4th harmonic, etc, as shown above. Similar to Keith Richards with his 5-string, open-G tuning (GDGBD – which, by the way is based on a 5-string banjo tuning, an instrument introduced into the delta by Africans). This Open-Eb tuning is the same sequence of harmonics, except it starts with an Eb, and because Robert Johnson has 6 strings instead of Keith’s five, the next harmonic in the series is the 7th harmonic.

So, I reckon our Robert’s “7th string” was just a string tuned to the 7th harmonic! If you play in open tuning, as I do, you do think of your strings as the tonic string, the 5th harmonic string, etc., and if you tune the top string to a harmonic 7th, you’re going to think of it as your “7th string”. When you tune your guitar like this, lo and behold, those tricky RJ suspended 7ths and octaves, and minor-3rd double-stopping become easy to do. Give it a try.

Did the devil teach Robert Johnson how to tune his guitar? To quote Bobby Johnson, “It must have been that old evil spirit, so deep down in the ground” – or perhaps he was just sensitive enough to feel the power of these keys resonating through the Earth, and intelligent enough to try a 7th harmonic for his 6th string – or his cousin gave him that tip when he was off re-learning guitar with him. I had wandered into this tuning several years ago based on the Pythagorean approach, and abandoned it after a few years for being too “weird” – but when I used it play along with RJ at the right speed, I realized that this was probably it. It’s certainly a more rational explanation for the “7th string” myth, and it does make it easier to sound like RJ. (Note: I know that Robert Lockwood Jr claimed he was taught to play guitar by RJ himself, and he seems to be in regular E tuning – but I’m sticking to my story. He was only a young boy when RJ was teaching him, after all.)

You’ll recall from my Home Page that Bb and F are the “magic” frequencies I found on a tone generator, which seem to resonate against some background sonic fabric of the Earth. What’s a beautiful thing is that, slowed down a semi-tone, these recordings are all now in comfortable keys for a guitar that is tuned to open Eb. Keys of Eb (open 6th string), F (2nd fret), G (4th fret), Ab (5th fret), Bb (7th fret and as a chord based on the 5th string) and C (9th fret). And we know he did use a capo, and the photo above shows it on the 2nd fret, so, if he’s tuned to Open-Eb, you’d expect to hear some songs with F chords (6th string, 2nd fret) and C chords (5th string, 2nd fret) – which we hear plenty of!

Here are the rest of Robert Johnson’s recordings, with the (key of the official recordings), [and my adjusted key in square brackets]. Again, the 2nd take is in one of the natural keys of the Eb harmonic series, and the rest are sped up, so a (B) was probably a [Bb]. What also interesting is that in the final session in June 1937, they are all in our keys – it seems that these final recordings were not sped up for release:

1936, November 23
Terraplane Blues (B) [Bb]
Phonograph Blues (B) [Bb}
** 2nd take in (Eb – an altogether different key/feel)
32-20 (Ab – perhaps not tampered with, or maybe in A)
They’re Red Hot (C) [B, but it’s OK because it’s rag-time and it hits the key chords]
Dead Shrimp Blues (B) [Bb]
Crossroads Blues (B) [Bb]
Walkin Blues (B) [Bb]
Last Fair Deal Gone Down (B) [Bb}
Preachin’ Blues (Up Jumped The Devil) (E) [Eb]
If I Had Possession Over Judgement Day (Bb) [perhaps not altered]

1936, November 27
SESSION IS SLIGHTLY MORE THAN A HALF STEP SHARP
Stones In My Passway (A) [Ab]
Steady Rollin’ Man (A) [Ab]
From four until late (C#) [C]

1937, June 20th
THIS FINAL SESSION, SEEMS RIGHT:
Hell Hound On My Trail (E) [May be sped up, Eb more likely]
Little Queen Of Spades (Bb)
**2nd take also in Bb
Malted Milk (Eb)
Drunken Hearted Man (Eb)
**2nd take also in Eb
Me And The Devil Blues (Bb)
** 2nd take also Bb
Stop Breakin’ Down Blues (Bb)
** 2nd take also Bb
Traveling Riverside Blues (Bb)
** 2nd take also Bb
Honeymoon Blues (Bb)
Love In Vain Blues (Ab)
** 2nd take also Ab
Milkcow’s Calf Blues (Bb)

So, there you have it, perhaps a couple of mysteries explained:
1. That Robert Johnson tuned his guitar to Eb.
2. That it was probably Open-Eb, not concert Eb
2. That his top (highest) string was tuned to the harmonic 7th of the Eb major chord – giving that haunting sound, and the facility for RJ to play delicate harmonies without flying about all over the fret-board. The simplest explanation is usually the right one.

Do have a listen to my hand-selected collection of “corrected speed” Robert Johnson recordings to hear the true nature of the man and his music – here on YouTube, and play along on guitar if you have one – and let me know what you think!

Using sound and wavelength to move rocks

Here’s a page from Scott Hill and Guy Lyon Playfair’s book, The Cycles of Heaven, (used without permission) which I bought in 1979 when I was 15. This account of Tibetan monks using the sound from drums and horns to move rocks up a sheer mountain face includes detailed measurements: the monks stand in a 90 degree arc at a distance of 63 meters from the stone to be moved, which is placed over a shallow cutting in the ground and is 250 meters from the cliff face behind.

On a whim this afternoon, I used this frequency calculator to figure out the frequency of the sound wavelength at 63 meters and 250 meters. It turns out, it’s 5.45 Hz for the 63 meter distance from monk to stone at 20 degrees centigrade and 5.26 Hz at 0 degrees centigrade; and 1.3728 Hz at 20 degrees centigrade for the 250 meters from stone to cliff – which if you multiply it by 2 a few times, turns out to be a sub-octave of 5.49 Hz (and 5.3 Hz at 0 degrees centigrade (freezing)). We don’t know at what time of the year this experiment was conducted, but Tibet is likely to be chilly!

So, the resonant wave between the monks and the stone to be moved is our 5.4 Hz magic “still-point” vibration for F which we documented on the home-page; and for the distance between the stone and the reflective cliff the frequency is also a sub-octave of this F.

The total wavelength from the monks to the reflective cliff behind is (63 meters plus 250 meters) = 313 meters. At 0 degrees centigrade, the wavelength is 1.0585 Hz (which if you octave it up (multiply by 2) a bunch of times = 270.97 Hz (our C-sharp is 270 Hz).

The relationship between C-sharp and F is a major 3rd. And from F at 5.4 Hz to C# is a major 3rd (x 5) of a major 3rd (A) (x 5) = 135 Hz x 2 = our C#.

So, there you have it folks: the secret to moving masonry with sound is to create a resonance around infrasonic vibrations of our F frequency, with a harmonic of an augmented 5th at the same time.

The theory of how this works put forward by Swedish aircraft designer Henry Kjellson, who recorded this event and drew the diagram, is that the sound creates a low pressure wave above the rock, and atmospheric pressure moves it up the cliff. The author recounts that he watched the monks move several pieces of stone in this way, although some broke on landing.

I don’t really get that surprised these days when I find that the F and B-flat frequencies which made themselves known to me, displaying remarkable properties that suggest they are fundamental to the fabric of the universe. But this seems like a bit of amazing lost knowledge which we may be able to explain and revive. How the Mayan temples and Egyptian pyramids were built might be related. Some of you may be thinking, “Jules, you’re just going too far – sound isn’t stronger than gravity”. But the notion of Tibetan monks levitating themselves and objects is almost a legend – something we’ve all heard of but which seems to have died out with the incursion of outside cultures. But we find one carefully documented and measured account, and find that the distances and the sound-waves are precisely the “magic” frequency for F which I’ve documented on the home-page as creating a resonant still point against the background resonance of our universe. So, it could just be another coincidence – but at some point the coincidences stack up to such a point that they become evidence.

Forty years on, I see that this book – actually the first non-fiction book I bought – seems to have been at the core of my interests all my life; and even the name (The Cycles of Heaven) is closely connected to the name of this website, the Harmonics of Nature – something I didn’t think about when I named it. In fact, I bought The Cycles Of Heaven with a book-token I had won at school. It was the only book in the store that “spoke” to me. The subconscious takes us on journeys we don’t realise we’re on, until we look back.

Music and the fabric of time

Well, I just came across this article where the author goes back to the ancient Hebrew divisions of time and overlays musical frequencies in keeping with these time divisions and – lo and behold – comes up with the exact same frequencies for the complete musical scale that I did, based on the phenomena of the “resonant still points” which I demonstrate on my homepage.

https://ethnographicsblog.wordpress.com/2019/04/19/a-horological-and-mathematical-defense-of-philosophical-pitch/

The Hebrew measure of time was the “helek” which equates to 3.333333 (recurring) seconds. Gives you more time to think, I suppose.

This measure of time was devised by dividing each of the 360 degrees by which the Earth turns every day into 72 parts to give a total of 25,920 helakim (plural of helek) per day.

First off, my frequency (and his) for C is 259.2. (By the way, each time you multiply a frequency by 10 you’re getting the Major Third of the original – so 25,920 also suggests a frequency for G# (C-D-E, E-F#-G#).

As the author points out, 25,920 also equates to the number of years in the Great Year – the time it takes for the world’s axial “wobble” to precess through 360 degrees, going through the twelve Ages – one for each of the Astrological signs. And it takes 72 years for the equinox to precess by one degree.

Two hours is 25,920/12 = 2160 helakim, and 2,160 years is the length of an Age. And this suggests a microcosm/macrocosm thing where we enjoy a tiny Age, or change of astrological sign, every two hours of the day.

And every day is, in effect, a mini Great Year as our position on the planet passes through all 12 astrological signs.

The Earth rotates 1 degree on its axis every 4 minutes (72 helakim x 3.333 seconds = 240 seconds = 4 minutes.) (3.333 recurring is a favourite number of the Free Masons but perhaps their big secret is simply the Helek. By the way, this video beautifully presents much of this.)

Every day, we turn 360 degrees, 4 minutes per degree = 1,440 minutes which is a number the author equates to F# as 1,440 Hz, which is an octave of 360 Hz – which is also the frequency I’ve found for F#.

Dividing the hour into seconds suggests B at 60 Hz.

He makes the root frequency for his scale 108 Hz because 360 degrees divided by 3.3333 seconds per helek = 108. 360 represents the full daily and annual rotation of the planet on its axis and around the zodiac. Now, 108 Hz is a sub-octave of 216 and 432 Hz – so that’s the frequency for A. Same as mine. So he sort of encompasses a whole year into A as the root note.

He then derives the harmonic series from this A using the “5-Limit” harmonic approach (which is simply deriving 5ths (multiply the frequency by 3) and the major third (multiply by 5).

The author also considers beats per minute as a starting point – so that the rhythm of the music and the music itself are aligned with the fabric of time.

He works into this a division of time (in seconds or helakim) by whole numbers: 2, 3, 5, 7, 9.

So he divides the day into seconds (60 seconds times 60 minutes times 12 hours) = 86,400 seconds, (a number which relates to A=432 Hz). He also starts with a notion of C at one cycle per second where its octaves would be 2 Hz, 4, 8, 16, 32, 64, 128, 256 Hz, etc.

And with one helek = 3.333-seconds, a minute is 18 helakim – which relates to D at 9, 18, 36, 72, 144, 288 Hz

As he says, “Using 5 Limit Tuning with the root set to A (at 216) rather than C, the frequencies of notes C4 (256), G4 (384), E4 (320), D4 (288), and B4 (240) are reducible to, respectively: 1, 3, 5, 9, and 15“. Meaning that 1, 3, 5, 9 are sub-octaves of the given frequencies, e.g. 9 x 2 x 2 x 2 x 2 x 2 = 288 Hz = D.

If, as I believe, the phenomenon I can demonstrate on my tone generator is indeed the phenomenon of sound interacting with the fabric of the universe, then what more powerful evidence than to find that these frequencies all tie back to a natural way of measuring time – at least on our Earth. Do check it out: https://ethnographicsblog.wordpress.com/2019/04/19/a-horological-and-mathematical-defense-of-philosophical-pitch/