Equal Temperament vs Just vs Earth Tuning

Click on the link to listen to three versions of the same performance of Beethoven’s Moonlight Sonata on piano, tuned three different ways.

Three versions of the Beethoven Moonlight Sonata
  1. One in Equal Temperament. 
  2. The second in the “just” tuning I have developed based on natural resonances such as the ancient helek (3.33333 of today’s seconds) equivalent to the time it takes our Earth to rotate 1/72nd of a degree, as an E-flat. 
  3. The third recording is in “standard” equal temperament where the A note = 440 Hz

There’s nothing like a side by side comparison of the way we’re told music should be tuned, versus the way that harmonics propagate naturally, and based on the frequencies I’ve discovered to resonate with the fabric of the way in which our planet rotates and generates its magnetic field.

For me, the the first one is ok but doesn’t fully resonate. The second one interacts with the room when you listen – it sounds louder and fuller. The 3rd one is just horrific to me – a bunch of noise, especially after listening to the others. No wonder Beethoven seems to some to be haunted by ghosts and teetering into disonance – when all we hear of him is recorded in the shitty Equal Temperament 440 hz “standard” that was introduced just in time for World War 2, as it happens.

I hope you can enjoy how music is supposed to sound. Feel it in your gut and let me know in the comments!

Thanks to Edward Grant for this performance: www.classicalarchives.com/midi/composer/2156.html

Music based on the 3 and 5

The “cycle of fifths” is of course a circle, which presents a problem for anyone believing that nature is based on fundamental frequencies because at some point, you go past the “fundamental” and you’re still going around the cycle. So, where do you begin?!

What we do know, we think, is that there are at least two frequencies which specific resonant properties with the Earth, being an F at 5.4 and 10.8 vibrations per second (Hz), and a B-flat at 7.2 Hz. See the home-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 B-flat is the “foundational frequency”, or if there’s a frequency below that which generates B-flat, or a frequency below that which generates that, or what? 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 that 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 3rd harmonic below F, so we know nature is using those harmonics – but how far down do we go? Below Bb is Ab, then C#, then F#, then B, then E. Where do we stop? The problem being, when you get back to where you started, when you use the cycle of 3rd harmonics the frequency you start and end at are not octaves of each other, even though they both should be, say, Bb. 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 after the 4th harmonic (another octave) the next strongest harmonic is the 5th harmonic (AKA our musical major-third intervaldo-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:

  1. F# and C#
  2. Ab and Eb
  3. Bb and F
  4. C and G
  5. 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:

  1. Vibrations per second
  2. Vibrations per helek (1 ancient helek = 3.333 recurring moderns seconds)
  3. 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
  4. Number of seconds per vibration (for the really low vibrations)
  5. 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 DĂĽrer 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!

  1. Playing the white notes only – they all sound great together!
  2. Playing the black notes only – they all sound great together!
  3. 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:

  1. The frequencies I found on my tone generator (7.2 Hz for Bb, and 5.4 and 10.8 Hz for F)
  2. E-flat as 9.6 Hz or 0.3 Hz as a measure of the earth’s rotation
  3. C# as the time it takes the earth to rotate 1 degree
  4. 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)
  5. Plus we have A as 432 Hz – which everyone loves, and D as 288 Hz – both related to the dimensions of the planet.
  6. Our B is also the frequency of 72 degrees of rotation of the Earth;
  7. 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!

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!