Visualizing Harmonics

I suffer from a special form of dyslexia, one that makes it easy to understand things (specifically: related facts), but conversely makes it hard to explain things verbally. I think I understand how music works, so to prove I'm not mad I've made a minor career of the unholy challenge of explaining music to unwary passers-by.

Gradually it's getting much easier to explain, as I build a portfolio of simple examples. Learning about the jammer and the Wicki-Hayden key pattern) was a major breakthrough.Herein I will try to show you why the jammer and music go well together.

How to show clearly where the harmonics are and how they fit together
If one lays out the harmonics on a linear, piano-keyboard fashion, how they inter-relate is absolutely anything but clear:

This is roughly how the inner ear hears ("feels" is a more accurate term)  a single note. Sounds in the raw, as it were, broken down into individual harmonics touching the ear's nerves gently in a line and spacings determined by basic count-on-your-fingers arithmetic. That may be how the ear "feels" the touch, but like the feel of a pen in the hand, wherein touches across several fingers gets integrated into a single sensation; the brain sees a single "pen".

One can fold the graph above over midway at the octave change point:

This is much better; you can see how there's a repeating pattern. The pattern of harmonics is heard as a single note.
This is also, partly, the way the nerves fold over on their way to the brain.

Now this only shows you the good stuff; where the consonance points are. Around each harmonic is a spot about a semi-tone flat or sharp where if a another harmonic is present then it sounds dissonant.
(an audio demo is needed & planned here)

Dissonances are important too
Dissonances can be pretty intense (one person singing 1 semi-tone flat can be heard in the choir of 100 people!). Here's a rough diagram:

So each harmonic is surrounded by a bunch of  traps: put a harmonic there unintentionally and you'll sound like a musical klutz. The brain even seems to think in harmonic terms, so oddly enough, the dissonant places are, in some ways, farther way from the root note than the "harmonic places", thus the green and red notes  (the root and the fifth; Do and So) are close in a special harmonic way, and the black areas are distant. This isn't cultural: this is very basic brain wiring.

A better way to show how we hear sounds
If we fold it up into a 'jammer' (technical name Wicki/Hayden) layout as shown in Easier to Play, it becomes a lot clearer and closer yet to what the brain is "seeing":

So the jammer layout has two added bonuses:

  1. It's much closer to the way your brain hears sounds than a linear (straight-line) layout and
  2. It's much harder to sound like a fumble-fingered klutz in it.

From this, much is obvious. One can, believe it or not, deduce all the world's common musical scales - and perhaps, invent modestly exotic new ones .
Stay tuned and I will show how these scales come to be.

Note: you can get much of the above by carefully reading Wikipedia's articles on the harmonic series and a few books, but I hope I'm making it clearer.

Why an interval sounds like it does

If one play two notes there is always a pitch interval between them (even if it's the same nominal pitch). The two sets of harmonics overlay each other and interact in simple ways to produce a complex result.

A Major Third
To give one of the simplest and commonest examples, consider playing a note (the root, i.e. C) and places a second note exactly 1/3 higher in frequency (an Interval of a "perfect (major) third", i.e. an E), then some of the harmonics in the note that's a third higher will sit right on top of the roots' harmonics, the 5th, 10th and 20th to be exact.

This will sound nice and smooth (consonant) and the overlapping harmonics will sound especially interesting. Even a small deviation in overlap either way, but especially down in pitch (flat) won't sound as good (not really bad as in evil), just louder and obnoxious.

Now, in music, everything is relative: one can aim at lots of different harmonic series in the root. Below is shown the interval of a perfect minor third, some of its harmonics overlay the strong 6th, 12th, 24th and interestingly,  the 7th and 14th harmonics of the root (these last are not well tuned).
The odd thing is that it's kind of like friends: while you may get along fine with Susan, and just swimmingly with Ann, yet find Ann and Susan fight like cats that caught the same mouse; The minor third and major third just don't get along in the same chord. Yet a millisecond is all that's needed to separate them.

Making music interesting
Play a root note with a major third, then instantly switch to a minor third, and the overlaps change: we've switched to overlaps on the 6th, 12th, 24th, 7th and 14th. The ear doesn't care, but your brains goes 'hey - what happened to the root note?'. It takes about a second to figure it out, so as long as you keep changing about once a second, you're in business.
(dear reader: an audio demo is needed here - care to craft one for me?)

Making the simplest chord

Here you can kinda see the overylay pattern of a Major Chord (root position):

Things get interesting to the ear - and that means the start, the barest start, of harmony.

Naturally, there is more to it all. In particular, I've omitted the gory details dissonance, the critical effect of volume, and what happens when more than two harmonics interact.

Useful implications
  1. When one plays an instrument, there's a lot to listen for.
    Here's the point of all the above apparently abstract information on harmonics: I believe it will be very helpful to both novices and teachers (saving months, possibly years of work) to construct demos and directly show what to listen for, rather than having to work it out by many hours of trial and error. 
  2. Is it possible that most of the many people who say "I can't sing", yet can spot a single sour note in a performance are just people who never learned point 1?
  3. One should be able to design a better piano, that matches the above layout. This is the basis of the Jammer.