Cameron Bobro

music and thoughts on this and that

How Our Vibrating Onion Of Sound Relates To Music

Now that we have a basic understanding of how sound acts in air, let’s talk about how these physics relate to music.

Earlier we mentioned in passing a couple of cases in which the relationships between the vibrations in the air, the relationships of our “innertones”, coincide, perfectly or nearly so, with modern Western 12-tone equal temperament.

Here’s a neat thing about 12-tone equal temperament. Play any tone. From there, you will always have other tones within 12-tone equal temperament (“12-tET”) which vibrate at twice the rate (2:1), three times the rate (3:1), and tones which coincide with the vibration relationships we find in our “innertones” of 3:2 4:3, and the octaves (half, twice, four times, one eighth, etc. the rates) of these such as 8:3, 2:3, etc.

Now, while the 2:1 (and 1:2, 4:1, etc.) relationships are exact in 12-tET, right in tune with the vibrations in the air, the 3:2 and 4:3 (the perfect fifth and perfect fourth) and their octaves (a fourth an octave higher, 8:3, for example) are just a wee hair off. This is called “tempering”, and it’s why we call it 12-tone equal temperament.

It’s a very clever bit of math, first documented by a Chinese prince half a century before appearing in Europe. It’s clever because if you try to make the vibrations of twelve tones all match the vibrations of 3:2 and 4:3, the pure fifths and fourths, happening in the air, you get eleven pure fifths (and corresponding pure fourths) and one real stinker, the famous “wolf”.
This wolf fifth is “lower” (slower, larger) than a fifth vibrating at a rate of 3:2. (The corresponding fourth is higher/faster/smaller).

So what, you might say. Well, think back to our vibrations within vibrations. When we play a single tone on our cello, or voice, or horn, the vibrations in the air all fit nicely within each other. Let’s say we have another instrument playing along, and the fundamental of the tone it is playing a perfect fifth “higher” than, that is, at a relative rate of 3:2 to, our tone of 100 Hz. The vibrations created by this sound will also fit nicely within those of our first sound. The fundamental of this sound is 150 Hz of course, it’s second harmonic at 300 hz, then 450 Hz, etc.

tone 1 tone 2 (in vibrations per second)

300… 300


600… 600

…and so on.

You can see how these vibrations, these, all match up or nestle inside each other harmoniously.

If we tuned that second instrument to that “wolf fifth”, however, it would vibrate at 147.9810552817716 times a second. You see that the harmonics of this tone will *not* fit in. If you imagine the vibrations in the air, you can “see” that the vibrations are moving at almost, but not quite the same rates. What this does is create new vibrations.

The second harmonic vibration of our wolf fifth tone is going to be conflicting with the third vibration of our original tone. It will be vibrating at 295.9621105635432 times a second, while the third harmonic of our original tone at 100 Hz. will be vibrating at 300 Hz.

These vibrations clash in the air and create what are called “beats”. In this case, the difference between the frequencies at that first point of clashing (second harmonic of “higher” tone with third harmonic of “lower” tone) is 4 vibrations per second, 4 Hz.

If you snap your fingers slowly and steadily, once a second, and sing evenly “january, february…”, that’s about exactly the rate of the beating. What it sounds like with musical instruments is a kind of throbbing, and with the other clashes within all the other “innertones” vibrating in the air (at the sixth harmonic of the first tone, for example), the effect taken all together is kind of a “howling” sound. And that is why it is called a wolf fifth.

But as I said earlier, the fifth of 12-tET is tempered and therefore a little “off”. Why doesn’t it howl as well? Well, if we played the 12-tET fifth along with our 100 Hz tone, we’d get our first and usually loudest clash between about 299.66 Hz and 300 Hz. The beat rate is only about 1/3 of a Hertz. That means it would take a full 3 seconds in this case (at these pitches)to complete a full beat, and the difference, the throbbing sound, is going to be almost completely unnoticed except in long smoothly sustained tones.

So here is where we find the acoustic connections between the 12-tET tuning system and the vibrations in the air: the vibrations that happen between all the second and third partials, and their simple multiples, are all very well in tune. Not perfect, but perfect enough in almost all cases, especially when you take into consideration that musicians with feeling and skill continually make little adjustments whenever possible- some vibrato, some more pressure on the guitar string playing the fifth (which raises the pitch to a pure 3:2) in a “power chord”, and so on.

For those of you into numerology and mysticism, you can see that the 12-tET system, with just a touch of compromise, fulfills the ancient Pythagorean ideal of everything, including music, being described by the numbers 1, 2 and 3.

But what about all the other tones, not just fourths, fifths and octaves, and all those resulting vibrations in the air?

Well, the price of ingeniously tweaking everything so that the vibrations between the first few (four, really) harmonic vibrations of musical tones
are all vibrating very well together is that the rest of the vibrations clash a lot.

If our two instruments were to play a 12-tET major third in harmony together, for example, the clash between the fifth harmonic component of the “lower” tone and the fourth harmonic component of the “higher” tone is very strong. This is the main reason you’ll hear some musicians complain that 12-tET is so “out of tune”. It is discordant if you value the “pure major third” which vibrates at the rate of 5:4. For several centuries in Western music, until about the time of Beethoven, the pure major third was the star of the show, so much so that people were willing to temper the fifth quite a bit and tolerate, not a wolf, but a rather wobbly fifth, in order to keep the 5:4 major thirds as pure as possible.

Surely there’s a way to keep your fourths, fifths and major thirds all pretty much perfectly in tune with the vibrations in the air, all at the same time, someone protests.

Indeed there is, and it’s been known for centuries, advocated by people from Isaac Newton to Schoenberg, and still in use to this day in the Turkish koma system. Instead of 12 perfect fifths, all you need is 53 pure fifths!

And here is where things get complicated, and paths diverge, and a myriad of approaches arise. We’ll talk about the main kinds of approaches in the next blog entry.

What “overtones”? Keeping Your Sound Understanding Sound

Now that you have read this:

let’s continue.

In order to continue, it is important that our basic understanding of what’s going on is solid. You will find that there are many half-baked ideas and plain old mistakes in music theory. Some digging will reveal that these ideas arise from, or depend on, failed understanding of sound vibrating in the air.

The words we use for things can be misleading. For example, in many languages, the harmonic vibrations we have been discussing are often called “overtones”. This seems to make sense in cultures that conceive of quicker rates of vibration as “higher” and slower rates of vibration “lower”. After all, since these vibrations are “higher” than the fundamental vibration, they are “over” the fundamental, right?

Well, no. As we discussed earlier, in the physical world these “overtones” are NOT happening “over”, but inside our big vibrating ball of sound.

Let’s go back to our cello playing a tone vibrating at 100 times per second. The speed of the sound itself is a steady 343.2 meters per second but within that sound as a whole we have vibrations, air pressure moving from high to low, happening 100 times a second. These means that if we were somehow able to see air pressure, and we took a photograph of the air at any given moment, we’d find alternating layers of high and low pressure within our “sound onion”, and a tone at 100 Hz would create high/low pairs of pressure areas 343.2 centimeters “thick”. We measure this on two dimensions and call it wavelength.

A complete cycle of high and low pressure is of course called a wave.

At any given moment, our 100 Hz tone is going to have high pressure zones 171.6 centimeters thick (which we call “long”) followed by low pressure zones 171.6 centimeters long. These pairs make a wavelength of 343.2 centimeters long.

Now, our cello tone is also producing vibrations and the rates 200, 300, 400, 500… times a second (called Hz as you remember).

That means that within our 343.2 centimeter wavelength, these moving “onion layers” of air pressure, we will find air being pressured and released at these “harmonic” rates, and we will have layers within the largest, fundamental layers, which have wavelengths proportionally smaller. Our 200 Hz harmonic, vibrating at a rate of 2:1, the “second harmonic”, has a wavelength half that of the fundamental, 171.6. That means of course that a high/low pair of pressure, a complete wavelength, matches, is “in tune with”, a single low or high pressure zone of the fundamental (“1:1”). So, two full wavelengths of the second harmonic fit into one wavelength of the first harmonic, the “fundamental”.

Three vibrations of the third harmonic, 3:1, fit into the vibration of 1:1, and four vibrations of the fourth harmonic, and so on.

As you can see, “overtone” is not a very good word at all. Innertones would be much better.

Once again, we see that the African concept of what we call “high” tones being “smaller” is really a better description.

Keep these things in mind when we continue!

A Sound Introduction to “Harmonics”

Let’s say you have a bowed string, or a sung voice.

Imagine, if you will, a sphere emanating from it, an invisible sphere in the air. The sphere is not a simple solid: it is made of many spheres, layered like an onion, and the layers are not stationary, but moving outwards. The layers are regions of lower and higher pressure in the air, all nestled within each other. The layers are in a continual state of change, now high in pressure, now low. The air is being compressed and released from pressure.

Now, here’s an odd thing about sound. The sound as a whole is moving outward from the instrument at a steady rate, about 334 meters a second in a comfortably warm room. This, the speed of sound, varies with the density of the medium through which the sound is traveling. The sound travels faster through more dense material. Through a piece of metal, it might travel even fifteen times as quickly. But it is consistent according to the environment. With a little variation depending on temperature and humidity (therefore the density) of the air, the sound as a whole travels with such consistent speed that if you have some kind of timer, you can project a sound and listen for when it returns to your ears, and reckon with deadly precision the distance of whatever it is that is causing your projected sound to bounce back.

Just ask any submarine, or bat.

Even we humans have great skill doing this, which is called “echo location” or “sonar”. If you don’t believe me, have someone you trust with your ankles and nose lead you blindfolded into more or less empty rooms of different sizes. Clap your hands. You will know immediately whether the rooms are large or small. You will certainly be able to rank the rooms according to size, even if you’re unsure of exactly how big they are. With practice, such as the experiences of a recording engineer, you will even be able to estimate the size, and something of the shape, of the rooms, blind.

In order to proof that it is echo location, and not magic or some unknown sense other than hearing that you are using, you need not only modus ponens (if this, then that) but modus tollens (if not this, then not that). And sure enough, if you go into an anechoic chamber or the “dead” room of a recording studio, or a room filled with all kinds of sound-reflecting surfaces, it is difficult or impossible to judge by blind echo location the size of the room.

But you have another skill, which submarines don’t have (I don’t know if bats do).  If you play or sing a sustained tone, then just by listening, you will have a very good idea as to the general nature of the materials the room is made of. You will know right away if the room is made of something like stone or of soft wood covered with heavy tapestries. Why? Because your sound is made of much more than a single instance of low and high pressure zones moving outward at a steady rate.

Within the musical sound, such as the one we had at the beginning, there is additional movement to the simple steady movement of the sound as a whole moving outward (and back to you when reflected by some object or surface).

These are changes in pressure within the sound as whole, high or low, which happen at different rates. The sound as a whole is moving steadily, but within it your “living onion layers” of changes in sound pressure are happening at a great number of different rates.

When you make a sound, you are actually making sounds within sounds within sounds… The sound as whole moves at a certain speed, which is consistent for every sound source in your environment, whether it’s a tuba or a piccolo. But your changes of pressure within the sound happen at an infinite number of rates. For a “high-pitched” sound, there are many changes in pressure, happening rapidly and, viewed like onion-layers, densely packed together. For a “low-pitched” sound, say, playing on the lowest string of a cello, the areas of low and high pressure within the sound are large and widely spaced, as long as your arm or longer. Within those “thick layers”,large areas, there are smaller, more rapidly changing, areas of pressure- the higher parts of the sound.

We can tell whether we’re in a cathedral or the back of a ’70s love van not just by echo location, but by the fact, learned from experience, that stone walls reflect  “higher”, quicker/smaller/more densely packed parts of a sound back to our ears, and deep shag wall-to-wall carpet absorbs them.

If you think about it, you will see why it’s so easy to kill the “bright” “high” parts sound in a room, and difficult to dampen down the “low” parts. Carpets and wall hangings will physically break up and diffuse the closely-spaced changes in air pressure, so they won’t reflect back, but the large, widely-spaced “low” sounds are going to be minimally altered by such treatments: they “go right through them”. The situation isn’t helped by the fact that, for several reasons we won’t go in to now, the “low” sounds usually have a lot more sheer energy to them.

You might be asking yourself why I keep putting “high” and “low” into quotes. Well, if you’re not familiar with the Sapir-Whorf hypothesis, look here:

“High” and “low” are cultural artifacts. They are technically accurate as far as higher (faster) and lower (slower) rates, but they are incomplete descriptions at best. In many African languages, they are “big” and “small”. In some ways this is a better description of the physics of sound. “Lower” sounds are in fact “bigger” than “higher” sounds. The ancient Greeks used “tense” and “slack”, obviously based on the tension of voice or string. There may be other cultural terms/conceptions.

Now that you have a mental picture of what’s going on, which is changes in pressure at many different rates within the steady movement of the sound as a whole, small within big, let’s go on to “harmonics”.

For reasons I can explain in another blog post if anyone’s interested, the rates of the sounds within sounds, the changes in air pressure within a sound, created by the human voice and stringed and wind instruments, are eerily consistent and “logical”. They are so amazingly consistent that they’ve even been treated with what can only be called religious awe over the thousands of years they’ve been observed and documented.

Let’s call the changes of air pressure due to sound “vibrations”. The measurable fact is that if you play on your cello a tone vibrating at 100 time a second (100 Hertz, aka Hz), you will also be creating in the air vibrations at 200 Hz, 300, 400, 500…

Of course everything in our directly observed universe happens in a stochastic, “+/-” manner, but these vibrations are so close to “perfect” in the rate of their movement that they defy measurement otherwise. Our ears also immediately detect, with spooky accuracy, when these vibrations-within-vibrations are not moving according to this pattern. We usually call the sounds not moving this way metallic, woody, noisy. The sounds working in this way we call “harmonic”, and the sounds not working this way we call “inharmonic”.

Please note that no one is making any kind of value judgment here. In fact if you were to pick a single archetypal, boiled down, essential, example of “human music” to send as a recording to an alien species, would you not send something like a singing voice (harmonic sound) accompanied by a drum (inharmonic sound)?

Now let’s take a look at how these physical phenomena relate to music. Let’s take the standard tuning of the Western world, twelve-tone equal temperament, and see how it relates to these “harmonics”, these vibrations.

Let’s say we play an A at 440 Hz on our stringed instrument. Vibrating within that sound we will also find a vibration at 880 Hz. This is twice the rate of the tone we were playing (which is called the fundamental), and is called the “second harmonic”. This vibration within a vibration is vibrating at a rate of 2:1 (880:440)to the fundamental. This relationship is called the octave and sure enough we find on, for example, our piano, precisely this same rate of vibration. For reasons we won’t go into now, this is usually considered as the “same note, only higher/lower”. (If you visualize the living layers of the sound onion, you might see why, though).

We also find within that sound vibrating 440 times a second (440 Hz) a vibration, a regular change in pressure low/high, at 1320 times a second (440 * 3). This is vibrating at 3 times the rate of the fundamental, at a rate of 3:1, and is called the third harmonic. Now when we go to the piano, we find this rate of vibration as a note (“an octave and fifth”) as well. But our piano is vibrating a tiny touch slower than 1320 Hz (we’ll get into why later). Nevertheless, it’s so close that it is obvious where that note came from: it comes from the third harmonic partial. It sounds “in tune” for the simple reason that it literally, physically is in tune, in fact the same as, the third harmonic partial of the tone below. If you listen to an acapella vocal group, or Indian music, you might very well hear that tone exactly, with amazing accuracy, at 1320 Hz, a perfect 3:1 relationship.

Aha, someone might say. I was watching Star Wars for the thirtieth time and I recognize that sound that you’re calling “3:1”. It was the “same” sound, but it was not so high compared to the lower tone. And here it is on my piano. Do…Sol! Yes- that’s the same relationship, except “an octave lower”, that is, vibrating at 1/2 the rate of that 3:1 harmonic. It is vibrating in relation to the lower tone at the rate of 3:2, which is the relationship between the third harmonic and the second harmonic. It “fits in perfectly” inside our “vibrating onion layers”.

So, says our someone reasonably, this is how my piano is tuned- I should be able to find all the tones corresponding to these relationships between the vibrations within vibrations! For example, when I sing “here comes the bride”, that’s the relationship between the third harmonic partial and the fourth (4:3)

Well, yes and no. The piano is tuned to a tempered system. The vibrations of the second partial, 2:1, 1:2, 4:1, etc, are exact. The relationships between the second and third partial and the second partial, 3:1, 3:2, 4:3, 8:3, are so close that they’re usually considered exact. As you look for more relations between the vibrations within vibrations and the tones you find on your piano, you’ll find that they’re further and further off of relationships between the harmonics, and eventually “inharmonic”. We’ll go into the why of that later.

There is other music, not traditional western music on instruments of mechanically fixed pitch, which does fit right in with the vibrations within vibrations. And of course there is music which, like western music, partially fits exactly in to the vibrations and partially doesn’t.

That’s enough for now. Hopefully from this you have a fundamentally sound image of “harmonics”, even if it’s a bit poetic, and are thinking about how how these vibrations within vibrations do or might relate to music.

A very important thing is to not forget Sapir-Whorf. Try to visualize so that you don’t get led astray by language.

Low-budget thickening/stereo mic technique

Recently I’ve been working on low-budget home-recording “empowerment” techniques.

Here is an effective recording technique I’ve stumbled on. It is related to some techniques studio engineers have used for decades to give a preternatural presence to electric guitars and drum kits. In this case we are going to look at getting a more out of the gear you already have
in your low-budget home/bedroom studio setup.

Let’s say you are recording an acoustic instrument. Your room might be good or great, but chances are in a bedroom studio this is not the case. You’ve probably figured out how to make to use of and treat your recording space so that it does not noticeably intrude on your sounds, or sounds okay. Even if your space sounds excellent, though, it is not likely that it sounds “big”!

Recording with an additional, more distant, microphone, called a room mic, or with a stereo pair of microphones, are techniques which are many times out of the question in home recording because the room is so often simply too small or too “boxy” for anything but close micing with a single

So, in order to get thicker sound and more stereo sounds out of your space, here’s something you can do.

Chances are you are recording to DAW software and have a little USB interface with two or four channels. Maybe you have a mixer, or even more outboard gear. For this technique all you need is two microphone channel inputs. And chances are you have but one “good” microphone, likely an
inexpensive but decent sounding Chinese-made LDC (large diaphragm condenser).

What you need here are two mics which are very different from each other. An LDC and a dynamic mic are ideal. I’ve been using an expensive Gefell LDC and an ultra-cheap Electrovoice knock-off of the classic Shure SM57 dynamic mic.

Your “one good” mic might also be a ribbon microphone or a nice dynamic like an SM7b. In that  case you might want to use a cheap LDC as your second mic. It doesn’t matter as long as the mics are very different in character. Two very different-sounding mics of the same general kind will work, too. NB: If one of your mics is a ribbon mic and the other a condenser, make sure you know what you are doing, because most inexpensive recording interfaces, and mixers for that matter, have
global phantom power. Your condenser mic needs it but a vintage ribbon mic can be damaged by it, and any ribbon mic new or old can be damaged by phantom power if you have rag-tag cables.

Let’s call your good mic, whatever kind it is, your money mic, and the second mic the support mic.

Set up your money mic in the sweetest spot you can find. This is surely what you already do.

Now set up the support mic so the diaphragm is very close to the diaphragm of the money mic. A gooseneck mic stand is great for this.

Arm two record tracks in your DAW, one with input from the money mic, the other from the support mic. Play your instrument and check your recording levels. You’ll want the two roughly in the same ballpark. If you’re using an LDC and a dynamic, the LDC is going to be putting out a much
hotter signal. You don’t need to get the dynamic up to the same level, just get it into a decent range above the noise floor. That is, you don’t want the dynamic whispering along, only to turn it up later and get a mass of noise along with it.

Make sure neither track is clipping no matter how loudly you play your instrument. If your LDC is ticking along at about -20dB RMS without hitting the red, and your dynamic is running about -24dB RMS, you’re perfectly fine. If you are going to be using this technique to multitrack and thicken
up some spindly or whispy instrument, don’t worry if your levels are moving as low as -30dB RMS. Why? Because if you pile up a half a dozen track of dental-floss lute recorded hot, you’re going to have to turn them down *anyway*, and with inexpensive gear one thing you do not want to be
doing, except as a lo-fi effect, is cranking the gain way up all the way up on a dinky interface. Try it and you’ll hear what I mean.

Now record some playing, listening to what you have. Make sure the two tracks are NOT panned: you’ll be listening in mono. Move the support mic, not the money mic, around until you get a sound you like. You’ll find spots that are very phase-y. It may be that putting the support mic at an angle makes for a better sound, it may be that getting it as close as possible and pointing in the same direction as the money mic gets the best sound.

Don’t worry if the support mic played solo doesn’t sound great. In fact, in the most likely scenario that you’re supporting a bright LDC with an inexpensive dynamic mic, you *want* the dynamic to be darker and more “compressed” sounding (which it is surely going to be).

At this point, if you have outboard analog options, either in the mics or on channels of your interface or mixer, to highpass/low-cut mud and room-boom, do it. This can save you some headaches later, and will make it easier to get the mics sounding good together. It will also help with common problems in bedroom studios.

Once you have the money mic sounding as good as possible when soloed, and the two mics playing together both panned to center sounding as good as possible, go ahead and do your recording.

Try panning the mics hard left and right. Put your DAW into mono with both mics panned. Notice that the sound doesn’t collapse or turn into a washy mess in mono. Why? Because you already took care of the mono sound at the very beginning. You’re just panning two distinct mono tracks which
you already know will work together in mono, because that’s how you set them up!

What you have might sound like you want just as it is. In that case, rock on!

Now download this magnificent free plugin:

Put the Sound Delay plugin as insert on the support mic track.

If you are using this technique for multitrack thickening, in other words layering up with multiple takes of the same thing, don’t put Sound Delay in as an insert. Put it on a submix buss, and route all your support mic tracks to that buss. This will save time and CPU cycles.

Now, *listening in mono*, tap your support mic out in time/distance with Sound Delay. Sound Delay lets you choose meters, feet, or milliseconds. One foot and one millesecond are very close to the same distance/time for sound (a foot is about 9/10 of a millisecond long). I use feet so I can
imagine the support mic moving away to become a room mic. You can move it out farther than the size of your room. As you listen, you’ll hear the color and feel of the sound change. From zero up to about 30-33 milliseconds, a little more in feet, or up to about 10 meters, you’ll hear one
sound, thickening or thinning and changing color according to phase relations and cancellations.

Some spots will sound bad, some good. You’ll know when you’ve found your spot.

Beyond about 33 milliseconds, the support mic will start sounding like an echo, or “delay”. If that’s what you want, go for it, tap it well the hell out if you want.

You’re doing this in mono. Why? No surprises when your tune is played back in mono. And unless you can force your listeners only, ever, to listen on headphones, every recording is eventually heard in mono or with some imbalance or collapsing of the stereo image. People wonder why their
mix sounds good at home and on headphones and then sounds poor elsewhere, and one reason is what is called “mono compatibility”.

This is the time to EQ for color, too, whether both tracks together routed to a single buss, or with insert effects. If you are layering up, just route the support mic buss to the same submix buss as the money mic. You can also use both inserts and a submix buss. For example, you might
want to do some surgical work on the tracks individually, with a more neutral EQ, say, tame the highs on the LDC a bit with a high shelf, then use a more colored or saturated EQ on both track together, on the submix buss.

EQ changes phase relationships, so we’re continuing to work in mono.

Once you have a sound you’re happy with, leave mono and pan things around to your pleasure. You’ll find that positions between hard left and right take some juggling. For example if you want your sound to be perceived about halfway to the left, you might have to pan your money mic three-quarters of the way to the left and your support mic all the way out to hard right.

One thickening method is track the same thing twice, then pan the mics of each take opposite. 1st take money mic left, support mic right, 2nd take the other way around. You can get a good thick stereo sound out of a single very thin instrument this way, even without using compression. Of course you must play with some precision to do this, but you may also be surprised by how much variation between the takes you can get away with.

As you have already figured out, there are countless variations. You can put some reverb on an auxillary bus and send more to the support mic than to the money mic. Just remember to continually check everything in mono.

And if you can’t get something satisfactory, you still have your mono money mic track as usual.

I hope this has been helpful to someone! Please leave comments below.

Here is a photo of the most makeshift variation of this technique I’ve used. Usually I use a Gefell M900 on-axis LDC and an EV sm57-type mic right next to each other on boomstands, in a nice big room (hall actually). This is the bedroom version, with the M900 and a dreadful cheap condenser that has a very different sound. You can see it duct-taped to the wooden chair.


…and a link to song recorded in that very session:









Practical Microtonal Music, continued

Last time we put our hands on ears on the basic framework of probably most of the world’s music: the octave, the pure fourth, the pure fifth.

Any guitarist or performer of folk musics from around the planet going through the steps described in the previous blog entry is going to immediately realize that they have already spelled out the skeleton of countless different “pentatonic” scales.

Yes indeed. Just fret somewhere within the large spaces you have left over after playing the open string, the pure fourth, the pure fifth, and the octave and there you have some kind of pentatonic scale. If you fret two notes within those spaces rather than one, you’ll have some kind of seven-note scale.

Since we’re using simple divisions of string length which happily gives us simple relationships between the harmonic partials, how about let’s fret halfway between the pure fourth and the nut? That gives us a wide “second” of 8:7. You might recognize it from middle-eastern music, especially if you play the interval from 8:7 up to the pure fourth of 4:3. That interval between is 7:6, which sounds like a dark minor third or wide augmented second. That’s a very recognizable sound, you will know it.

Now fret halfway between the pure fifth of 3:2 and the octave of 2:1. That’s 12:7, which sounds like a wide major sixth.

If you play the scale you now have, 1:1, 8:7, 4:3, 3:2, 12:7, 2:1, your ear will know it right off. It’s a broad pentatonic scale- it might sound Asian, or even “cowboy” to you, but it won’t be strange in the slightest.

Go ahead and jam on it. Put some little bits of masking tape on your fingerboard for reference but use your ears. You’ll catch the harmonic “unity” of the scale as the harmonics in the air blend smoothly from one to the other. They do this because all the notes are closely related within the harmonics which are vibrating in the air.

By now you’ve realized that there must be a practically infinite number of tunings you can make this way, just using your ears and simple divisions of string lengths.


’til next installation, enjoy and explore on your own! If you find something that sounds great with this simple approach, be sure to tell me!

PS. If you put your finger halfway between the pure fifth and the nut, you get the pure minor third of 6:5.

Okay, a practical introduction to alternative tunings and “microtonality”!

Take a fretless bass or play a guitar with a slide. Any stringed instrument without frets or with a high enough action to use a slide will do. A zither or dulcimer is also ideal.

Up until a couple of hundred years ago, a monochord was standard. You can make one with two nails, a board, a single instrument string and a narrow piece of wood to insert under the string as a movable bridge. You can also build or buy very nice monochords or canons. A canon is just like a monochord, but with more strings so that you can work on simultaneous harmonies as well. The great number of different kinds of zithers and dulcimers in the world are descendants of the canon and are essentially still canons. It is not by chance that the middle-eastern hammered zither is called in Turkish the kanun and in Arabic qanun.

There is a lovely plan in Cris Forster’s massive tome on tuning and instrument building:

Here at the KIBLA institution we got our copy from Amazon It’s a great book.

Okay, now, however you do it, finger, slide, or movable bridge, fret halfway between the nut and the bridge. Every guitar player knows this spot. It’s the octave, and it’s where you play the octave harmonics, usually the strongest sounding harmonics on a guitar.

You are now doing something that people have been doing for thousands of years. The ancients Greeks documented these activities with great precision. Half the string length gives you the octave. It so happens that the octave vibrates at twice the frequency as the open string. 1/2 string length, twice the frequency. If your string is vibrating at 220 cycle per second, 220 Hertz, the octave is vibrating at 440 Hertz.

So when you see the frequency ratio “2:1”, that’s the octave and it’s half a string length from the bridge to the nut. And it has a strong harmonic.

It’s important to get this right away, for several reasons. First of all, these things were first discovered by earThe “math” of it comes after, and when there is math in tuning, it is practical.

“Hey, how do you make that sweet high pinging sound?”

“Just put your finger lightly halfway along the string length…”

If you look at a modern western guitar, you’ll see that the twelfth fret, the “octave”, is halfway between the nut and the bridge. There might be a microscopic difference from exactly halfway, because unlike the a canon with its movable bridge, you have to push the string of a guitar down a bit, which effectively slightly deforms the string. So, a well-intoned guitar will have some minuscule compensation for that. If you’re playing slide, you’ll find that playing lightly your pure octave will be about dead-on halfway and will go sharp if you push down hard. But, minus these very small variations, the octave, 2:1, is 1/2 the string length.

If you have two identical strings at identical tension, and one of them is exactly half the length of the other, it will sound one octave higher.

Some of the ancient Greeks freaked out about these things and extrapolated them into physics and metaphysics. Although they came up with a good deal of mystical speculation, in some ways they also hit the scientific nail dead on the head. What we’re talking about here is called the harmonic series. It was first discovered by listening to music and is one of the basic elements of math and physics.

Now, play halfway between the octave and the nut. This is 3/4 of the way from the bridge to the nut. The tone you play stopping the string at this point vibrates at 4/3 times frequency of the open string and is called 4:3. You will recognize it immediately: here comes the bride! It is the pure fourth, or “Fa” in solfeggio.

If you play 2/3 of the way from the bridge to the nut, you get 3:2, the perfect fifth. Sing the Star Wars theme. du-du-du-DUM-DUM…That’s a perfect fifth on DUM-DUM…

Measuring 2/3 of a string length is not so instantly easy as measuring halfway, but no problem. Just guestimate and play a “harmonic”. There’s a loud harmonic right there at 2/3 the string length, every guitarist knows it.

We now have the basic skeleton of most of the world’s music. Starting from the open string, which is called 1:1, we have the fourth at 4;3, a little higher the fifth at 3:2, and the octave at 2:1. AKA the whole string length, 3/4 the length, 2/3 the length, and half the length.

Do, Fa, Sol, do.

That’s enough for now. Once you grok how incredibly simple this is, you’ll understand how it is that we have literally thousands of years of practical, hands on “tuning theory” (read: music practice) on record. When Didymus recorded his semitone at 16:15 two thousand years ago or so, it wasn’t just “elegant mathematics” as some historian ignorantly called it. It was simple instructions for musicians. Play a perfect fourth, then halfway to the nut, then halfway to the nut. Voila, that’s the 16:15 semitone. In fact the entire tuning system of Didymus can be done by ear using such bonehead-simple instructions. Anyone who really tries it for themselves will realize that is patently a documentation of actual musical practice. And sounds wonderful. Sweet and folksy, really. About the year 900 CE Al Farabi not only documented tunings and created his own, he also documented the fact that he was measuring real instruments.

There’s no need to resort to mumbo-jumbo about the pyramids to study ancient music. And starting at these basics, we can have a real understanding of tunings today.

This is not about “math”.

Always play, always listen, always sing these things.









New old Clarinet!

We interrupt the microtonal musings to express joy at picking up a vintage (mid-20th century) Clemens Wurlitzer Eb Clarinet at an antiques market! Perfectly maintained (only the corks in the joints are worn) granadillo (African blackwood, Dahlbergia melonoxylon, actually a kind of dense black true rosewood), Öhler system. Plays beautifully.

Oh, but this is no interruption, for the clarinet is one of the many, many instruments which can be played microtonally. With cross-fingerings and embouchure, quartertones are long a standard technique in the classical repertoire. Especially with a soft reed and a more open mouthpiece you can have pretty much any pitch you desire. The Albert key system is  the best for this, and is standard for Turkish music, klezmer, and so on. The Böhm system also works but takes some fairly heavy alternate fingerings (all the clarinet on my upcoming album is a vintage lower-budget Böhm Bb clarinet- all the “synthesizer” sounds are also played with this instrument).

It seems in the half hour that I’ve been playing the Wurlitzer that the Öhler system is going to be better than the Böhm system, we shall see. We shall hear, I should say!

Next time I’m in Istanbul I’ll try to pick up an inexpensive olivewood Turkish clarinet (Albert system, very open mouthpiece- I saw many narrow-bored clarinets in Istanbul as well, something I’ve never tried). For now, just loving the tone and playability of this, my first serious clarinet.