Harmonic Exploration

CYMATICS

As you saw on the home page, vibration can make matter – such as salt, sand, or water – hold a standing pattern, revealing an intersection between something from another dimension and the vibrating medium.

Here is a beautiful video showing many of the ways in which sound affects water, matter and air:

When I first saw this in 2010, my mind was truly blown. Why is this not taught to every 5 year old child – to set them off on a lifetime of knowledge and understanding about how the universe actually works?!

As the chart below indicates, the video shows many points where interesting, geometrical shapes form, corresponding to frequencies in the harmonic series derived from our F and B-flat “still point” frequencies:

But the first two frequencies at least are almost exact matches to the harmonic frequencies based on the two frequencies I had discovered on my tone generator. Perhaps the universal resonance of B-flat detected by NASA (and its perfect 5th harmonic, the F note) does indeed imbue our daily lives with a resonance which inter-plays with every vibration and sound we feel and hear  – and this was also known to the ancients, somehow.

The Music of the Spheres

In Jacob Bronowski’s TV series,  The Ascent of Man – episode 4, “Music of the Spheres”, Bronowski demonstrates the physics of musical harmony advanced by Pythagoras and his disciples on the Greek island of Samos; showing how a stretched, vibrating string will yield different musical harmonics of the original vibration when touched at various whole-number divisions along its length.

When I watched this, aged 17, as a guitar player I already knew the interesting effect of “playing harmonics” – just touching the little finger to the string, right above the 12^th fret, without even pressing down – to yield a singing, pure tone, one octave above the note of the original string; going from a low D to a higher D, for example.

Some say Pythagoras learned this from his time in Egypt – that touching a vibrating string at whole integer divisions of its length creates still nodes – equi-distant along the length of the string – like “mini” “strings within the string” – resonating at harmonic overtones of that fundamental tone.  In other words, the building blocks of melody are all encompassed in each fundamental string.  You don’t just get the octave, you also get the musical 5th interval, the major 3rd, the 7th and the 9th harmonics, depending on where along the string’s length you touch with your finger.  Every string is a little symphony all by itself.

So, maybe there’s a “string” – like the “monochord” of yore –  from which all the harmonics of universal energy emanate.

The Harmonic Series

One Monday morning in early 1994 – being between jobs – I went to the trouble of playing harmonics at measured distances along the guitar string, and figuring out what note was generated where.

• Touching the string at the mid-point gives us an octave because the string is now vibrating in two parts, each at twice the original rate.
• Playing a harmonic, one third of the distance along the string creates two “still point” nodes – three equal divisions of the string – all vibrating at three times the rate of the original, to provide what is called the “fifth” in music (because it actually takes 5 notes up the scale to go from the first note to this note)
• Touching the string a quarter of the way along its length generates another octave
• At a 5th of the length of the string, the musical third will be generated
• etc.

Image Source: https://en.wikipedia.org/wiki/Harmonic

Here are all the harmonic notes  – generated by playing harmonics on a B-flat string – starting 1/2 way along the string, going to a 9th division of the string:

And, in sequence:

The harmonics described above are all “over-tones” – vibrating faster than the original note.  As players of stringed instrument know, the easiest harmonics to play are:

1. Octaves (at a point half-way, or a quarter way, down the length of the string)
2. Fifths (at a point a third of the way down the string)
3. Thirds (a fifth of the way along the string).
4. “Seventh” harmonics (played at one 7th the string length) are more difficult to play – and after that it becomes difficult to get the harmonic to ring at all. They’re just not so prevalent in the fundamental note.  (Unless you’re Eddie Van Halen with a Marshall stack playing micro-harmonics).

Modern “fast Fourier” spectral analysis of sound bears this out – that the more esoteric harmonics are fainter and less resonant.  Here’s a diagram of the relative amplitude of harmonics generated by a violin:

• Four Octaves of the fundamental note (G)
• Two Fifths (D)
• One Major 3rd (B) harmonic
• One 7^th (F)
• And some un-marked notes which appear to be another Major 3rd (B), another fifth (D), another Octave (G) and an 11th harmonic (C-sharp), plus some micro-tonics

… all resonating from within the one string being played – the G.  And the relative loudness of these harmonics is in fact what differentiates the sound of a saxophone, for example, from a trumpet or a violin:

[Image source: University of New South Wales Physics Department]

The Rolling Stones!

What’s this bunch of reprobates doing in our voyage to the center of music?

It turns out that the 5-string “open-G” tuning taught to Stones guitarist Keith Richards by Ry Cooder in 1968 exactly follows the harmonic series.  In bold are the harmonics to which open 5-string guitar is tuned:

1. Octave
2. Octave
3. Fifth
4. Octave
5. Major Third
6. Fifth
7. Seventh
8. Octave
9. Ninth

It’s as though the five strings are tuned intentionally to ring out the natural harmonics that are present within the first string.  5 strings resonating as one:

1. G (fundamental)
2. D (fifth)
3. G (octave)
4. B (major third)
5. D (octave of fifth)

With the use of a capo placed across any fret you like, the fundamental note and the resonance of the entire instrument can be changed easily – to suit the inspiration of the song, while retaining the harmonic relationship between the strings – and then fretting and playing specific notes allows the inherent harmony of the vibration to be explored as rhythm and melody.

Brown Sugar, Tumbling Dice, Start Me Up – many of the hits from 1968 to the present day were written and recorded in this tuning.

Prior to that, Keith had also used 6-string open-D, open-E and open-E-flat tunings – used on such late-60s songs as Street Fighting Man, Jumpin’ Jack Flash and You Can’t Always Get What You Want – and also used by Elmore James and some of the blues and slide-guitar greats.

Keith Richards himself remarked in notes at the Exhibitionism exhibit that he is fascinated with “how one string makes another vibrate” – called sympathetic vibration.

This very insight opened up a realm of possibilities for me – in terms of playing chords while playing notes that are harmonically aligned to the fundamental resonance of the open notes.  This is the fundamental nature of music.  And, if we could find the right chords,  perhaps, this could be the fundamental music of nature!   

Keith himself has remarked that open-tuning is like a sitar – with a sort of drone note ringing in the background.  The enduring popularity of the Rolling Stones’ music, when Keith (and Mick!) have constructed songs around this approach, shows that these open tunings and the way of playing them, really “strikes a chord” with many people.

The “modes”

You get a different emotional feeling in a piece of music depending on which note of the scale it starts on – which, by the way, is usually the one it ends on – that’s how you know you’re back to the song’s point of rest – its “point of view”.

And the reason that certain music sounds happy or sad has to do with which note of the harmonic series that starting note is.  Depending on whether you start your piece of music on the first harmonic, the second harmonic, the fifth harmonic, etc – you get a very different feeling in the music.

If you have a piano handy, try playing only the white notes:

• Starting at a C – you get a nice, jolly, major scale – found in many Christmas carols
• Now, play the same white notes starting at an A – gives you a maudlin minor scale

Same notes – different starting note – different feeling.

This is not just because certain notes carry an emotional weight (although I believe they do) – but because the intervals – the gaps between the notes as you climb the scale from your starting note – are spaced differently depending on the starting point.

If you start a melody on the second note of the harmonic series (the A-note, if you’re still playing white notes on the piano), it forces the third note in the scale to be just a semitone above the second note – and that’s where the sound we recognize as sad comes from.  It seems to be a common, psycho-acoustic reaction across all cultures.  A scale that starts with the second note of the harmonic series is known as a “Minor” scale (or Aeolian mode).

If you start with the fourth harmonic (the C in our white-note example, above), you get a more optimistic, “major” mode.  The interval between the second and third note is a whole tone, instead of a semi-tone.  And the interval between the 7^th and octave is a semi-tone – it too has a wider step.  It sounds happy, complete, robust, confident, healthy – if a little proud.  This is the “Major” (or Ionian) mode.

There is a mode name for each of the seven starting positions in the harmonic series.  For example:

• If you start with the first note of the harmonic series, it’s called Mixolydian mode.   It sounds happy (major third), though a little poignant (minor 7^th).  But, being as this mixolydian “mode” is actually the natural harmonic series itself, the music played in this mode matches the “personality” of the universe itself, in my view: “happy” yet “poignant”
• Aeolian mode, (the familiar western “Minor” key – just doesn’t have the energy for a full major third, it also has a minor seventh.  It has humility (minor 7^th) but generally lacks “get-up-and-go”.  After a while, it’s quite exhausting, like a friend who comes over and moans about their life for a few hours.  It’s a relief when it’s over
• Major (Ionian mode) sounds pompous and over-blown after a while.  One needs a little humility (a minor 7^th, perhaps) as the antidote.
• Phrygian mode starts at the 6^th note of the harmonic series – it is the basis for Flamenco – full of fire and passion, but ultimately, tragic.  It has a minor third, a minor sixth, a minor 7^th.  Everything is “minored out”.

But Western “classical” music, for whatever reason, only talks about Major (Ionian) and Minor (Aeolian).  You don’t see a piece by Beethoven in G-Phrygian – even though it may be – it will likely be called “G-minor”.  And you’re more likely to see a piece in Ab-Major than you are in Ab-Lydian – even though that may really be what’s going on.  Our culture tends to simplify and obfuscate.

Mixolydian mode is generally found in folk and country music.  Because it is the only mode that reflects the natural harmonic series by including a major-third and a minor, “dominant” 7^th – Mixolydian mode is the “natural” mode which describes the harmonics emanating from its fundamental note – so it is the mode we will be looking to to reflect the harmonics of our fundamental, universal tone.