Before ACOUSTICS, a little fun fact: PARTIALS AND SCALES

kensuguro · Post by **kensuguro** » Sun May 05, 2002 10:12 am

If you don't mind my CAPITALIZING. Anyway, while the word is still out about simple sin waves and harmonics, I thought I'd just jot down a new one about how harmonic partials help us logically understand the equal temperament scale. This one will be a quick one.

So here it goes. It's kind of hard to differenciate between the words "harmonics" and "harmonic series", I'm kind of vague on this so if anyone knows what I'm talking about, please give it a jab. I say this because while a octaves are called harmonics, sounds of the harmonic series are also called harmonics as well.. and technically these 2 are somewhat different.

How different? Octaves are simply 2 times the preceedign note, so it's

220, 440, 880, 1760

Fair enough. But then, the harmonic series is derived by 2, 3, 4 times the root note, so it goes

220, 440, 660, 880

So the octave harmonics are actually included in the harmonic series, but they're not necessarily the same thing. It's usual for classic people to say "fifth harmonic", while computer musicians say "fifth partial". Just a minor language thing I guess. It seems though, that "partial" comes from an additive synthesis background, and so it points at a specific sine wave, not necessarily the harmonic.. who knows. I'm starting to sound dumb here.

Now, looking at the numbers more closely, you'll find something peculiar. The octaves go

220, 440, 880, 1760

But the partials, harmonic series, or whatever goes

220, 440, 660, 880, 1100

The point is that while the distance between the octaves increase as it goes higher, the notes in the harmonic series increase at a constant rate. To understand the true meaning of this, we need to understand how to build the chromatic scale. (I know, this whole thing is a bit TRIVIAL

)

Taking a look at the octave again:

220, 440, 880, 1760

We can see that the distance is increasing, which means each note of the chromatic scale has to cover much more than the preceeding octave. But there's one definite rule. The distance doubles every octave, and it takes 12 notes to complete that ratio. Thing is, tho, this isn't a problem of say, (440-220)/12=1 semitone. It's natural to think that way, but things are quite a bit more complicated.

Well, it happened to be so complex that I don't feel like going into the details. Here's the cheese. Square roots are the key to this calculation because:

(12th root of 2)to the Nth power

Every time N reaches 12, a 2 pops out. Ain't that smart? And then you give this equation a root note of some sort, say 220.

[(12th root of 2)to the Nth power]*220

And walla! That was quick, you just have to count how many notes you want to go from a low A and stick that into N.. like 12 would yield this:

[(12th root of 2)to the 12th power]*220
= 2*220
= 440

And there you are, an octave higher. Actually, I just skipped a large portion of the tasty stuff so IF anyone is interested, I can go into details.

Now that we've got the nuts and bolts to make the chromatic scale, let's compare that to the harmonic series, partial.. or we might as well call it the funkadelic juice for all I care.

So here goes the harmonic series again (for the billionth time!):

220, 440, 660, 880, 1100

Now this in semitones would be (so 220 is 0):

0, 12, 19, 24, 28

Let's check out the intervals:

12, 7, 5, 4

And these intervals, in more humanly understandable terms are:

Octave, Fifth, Forth, Third

So there, you get your fundamental tones of scale, in order of importance. And using these intervals, it is possible to map out the entire chromatic scale without going into square roots and other gibberish calculations.

------------------------
But there's one drawback.
------------------------

The harmonic series and the chromatic scale aren't totally the same. Like the Third that appears in the harmonic series is slightly off tune when compared to the scale derived from the 12th root equation. Here are the distance between the harmonic series in precise semitones calculated with the 12th root equation:

0,
12,
19.019550008653874200,
27.863137138648348200,

Obviously, the harmonic series harmonize extremely well, because the the tones are internally existant in most sounds. (the harmonics) But it's interesting to think that the chords we usually play, are slightly off from the "straight on" harmonic. Of course, that's why many people are interested in alternate tunings, whether it be just a source of authentic atmosphere, or a more pure harmony.

So this whole scale, tuning bit has a mathematical ring and an artistic ring to it.. Not sure if it rings in a equal temperament third or a harmonic third tho.

[ This Message was edited by: kensuguro on 2002-05-05 12:58 ]

kensuguro · Post by **kensuguro** » Sun May 05, 2002 10:35 am

Also, I should add that the higher you go in the harmonic series, the smaller the intervals, as the first few implies:

12, 7, 5, 4
The numbers are droppin, and quite fast. It's not too long before you bump into microtones, where the intervals are smaller than a semitone. Some people see a very big possibility in that route of thought. I haven't experimented with these, but it's still an intrigueing thought nonetheless.

An easy way to fiddle with the harmonic series would be to use the partials module in mod2. It's apparent that after the first few partials, you get out of tune tones.

But remember, to go through the harmonic scale, you can't just play it. It has to be pure. A waay too easy method to play with it would be.. hehe... y'all know this by heart! Drive the tee-bee tree-oh-tree through a ultra resonant lowpass filter, and then mega clip, overdrive, fuzz, chop, fry, add the chilli the signal, and then slooowly turn up the cutoff, and that sweep is the pure harmonic series. I think most of us here can even hum that sweep from memory!

But anyway, it's cool to know that that sweep we're quite familiar with, cannot be purely replicated by playing the notes. That's pretty cool, don't ya think?

What's even cooler is that Pythagoras, who found the octave and the fifth, and several other notes, ALSO knew that sweep. But in his case, he found them through string harmonics. Quite a techno dude I'd say. It must have been so fun back in those days.. I mean, it's like "WOW, that guy's music truely has a new sound to it.. I mean, it's like a new NOTE!" It must have been easy to sell records back then. hehe.

[ This Message was edited by: kensuguro on 2002-05-05 11:40 ]

kensuguro · Post by **kensuguro** » Sun May 05, 2002 11:00 am

And those who know will understand... that I'm trying EXTREMELY hard to steer away from logarythms.

Here's the idea folks, as you go up the harmonic scale, the intervals keep decreasing right? They'll keep decreasing, but will never reach zero. Mathematically, it will only reach zero when you're at the infinitieth harmonic. A totally useless, and un-fun, fact, but hey, to be called a Pulsar Geek, you gotta be a geek! haha!

wayne · Post by **wayne** » Sun May 05, 2002 11:16 am

Goodonya, Ken

dblbass · Post by **dblbass** » Tue May 07, 2002 12:19 am

Good stuff, Ken.

Here's another way to show that the tunings used in modern piano scales do not follow the perfect intervals of the harmonic series:

Accept on faith two simple assertions, and then I can show this easily without any nasty logarythms.

First, as Ken points out (and we all know) an octave is simply a perfect doubling of frequency. The ear can hear this interval remarkably accurately, and all modern musical instuments play octaves which are, well, octaves.

Second, the ear can also hear the perfect fifth (which should be seven perfect half-steps piled on top of one another) very accurately. The perfect fifth that the ear wants to percieve is a ratio of 3 to 2.

(Test this. Whip up a couple of tunable sine oscillators in a Mod Window, and tune them till your ear hears the sweetest fifth, and then check the frequencies. Should be 3:2)

Now in a modern instrument (say a piano) we do have all fifths tuned to sound the same ratio. Are these perfect 3:2 fifths? Easy to check using the cycle of fifths.

Start with any note, say a "C", and say that its frequency is some number, C. Go up a perfect fifth to C * 3/2. If our piano plays perfect fifths we should be at "G". Up another perfect fifth, then down a perfect octave gets us to C * 3/2 * 3/2 * 1/2. We should now be at "D" , two half steps above the original "C" on our piano.

Keep doing this, up a fifth, up another fifth, then down an octave (* 3/2, * 3/2 and *1/2). If in fact the fifths on our our piano are perfect, we'll cycle through all the notes : C, G, D, A, E, B, F#, C#, G#=Ab, Eb, Bb, F, and finally to C', an octave above where we started. One more * 1/2 should put us back to C.

Well, as most of you will know, we will get there if we play the keys on the piano, but we will not if we do the maths of perfect fifths.

Why not? First check the maths: We have ended up with C * (a big number / another big number). The numerator is twelve 3's multiplied together, or 531,441, and the denominator is nineteen 2's mutiplied together, or 524,228. Clearly C * 531,441 / 524,228 = C * 1.013643 does not equal C. Off by a little more than one percent.

(Alternatively, we could just go steadily up by fifths without adjusting down the octaves every other time, then the question would become whether we have reached a pitch which is seven octaves above C. Same difference.)

The only possible explanation for the discrepency is that our piano doesn't actually have perfect fifths. In fact, they're all just a tiny amount less than the 3:2 that the ear normally prefers. Or to tie this back to what Ken says, the octave series and the harmonic series (based on perfect, integral intervals, just like the cycle of perfect fifths) don't really coincide.

naturally, any book on scale theory and the equal tempered scale will explain this - I just like doing it without logs, just some plain old multiplication and division.

And of course the Greeks knew this millenia ago. The discrepency (scaled down to the half-step) is known as the "Comma of Pythagorus". Clearly, those Greeks were the original Geeks.

kensuguro · Post by **kensuguro** » Tue May 07, 2002 11:35 pm

heeye, dblbass, glad you're back with us. Was wondering what you were up to.. Can you cover the physcho acousctic section rushing a little into the cognitive science area? I believe you have a better background in them compared to what I tought myself.

dblbass · Post by **dblbass** » Wed May 08, 2002 7:46 pm

Hey Ken,

Took a new day-job. kinda busy, kinda fun, but took me away from my studio and the planet for a while.

As for psychoacoustics, its nothing I'd claim any particular expertise in, its just a topic I've read a few books on and thought about alot. Happy to provide anything I can, but I suspect we've got more expertise with others in our community.

at0m · Post by **at0m** » Wed Sep 18, 2002 8:31 am

More about tuning and scales:

http://www.xs4all.nl/~huygensf/scala/
http://www-math.cudenver.edu/~jstarret/microtone.html

at0m · Post by **at0m** » Sun Sep 22, 2002 8:29 pm

This curve has nothing to do with psycho acoustics. Psycho acoustics is how your emotions are linked to certain frequencies. As you will have noticed, some chords sound sad or lousy and some more happy or energetic.

You're the musicians, you name the chords

Some more on acoustics @ http://groups.yahoo.com/group/acoustics/links

[edit] I'm sorry, Ken. But as someone mentioned that, I tried to explain a little.

[ This Message was edited by: at0mic on 2002-09-22 23:24 ]

kensuguro · Post by **kensuguro** » Sun Sep 22, 2002 10:16 pm

? psycho acoustics?
I meant like a psycho acoustics section as a different article. not a part of THIS (partials) article.