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 ear. The “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.