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Vibrating strings produce (more or less) harmonic overtones. If two strings are tuned in some simple frequency ratio such as 3/2, 4/3 or 5/3, then those harmonic overtones match up nicely and avoid roughness. But if the two strings are tuned in some haphazard fashion then the overtones of each string won’t match up, causing the overtones to clash with each other.
We can actually plot out a graph which shows the interval between two strings and the corresponding dissonance. This is called a dissonance curve, and for a normal string it looks something like this:
Well, imagine a weird kind of string that produces inharmonic overtones, such that the dissonance curve looks different to the one above. Because the dissonance curve is different, you couldn’t play Air on the G String and expect it to sound good. You could however write new music that would fit with the novel dissonance curve.
Today, such a string is more than just a mathematical curiosity. It exists in the physical world.
“Inharmonic Strings and the Hyperpiano” (by Kevin Hobby and William Sethares) is a paper published in Applied Acoustics. The strings in their hyperpiano have a stretched out dissonance curve where the double-octave sounds most consonant and the octave becomes dissonant. Okay so maybe it’s not going to be used on every new pop record, but this kind of freaky instrument can produce game-changing new tonalities.
Since the dissonance curve is stretched out to the double-octave or “hyperoctave”, Kevin Hobby suggests we might try tuning a hyperpiano instrument to 12 equal divisions of the hyperoctave. Wait, isn’t that just 6-EDO – a whole tone scale? Actually, it isn’t! They may be identical tunings, but the octave is considered a dissonant interval on the hyperpiano, analogous to the tritone on a normal piano. So it makes a lot more sense to describe this tuning as 12 equal divisions of the hyperoctave. Really.
The ringing of the strange hyperpiano sounds like a death bell for the unwavering cult-like belief in pure ratios and true frequencies. Tuning and timbre are deeply linked. If we’re willing to experiment with new timbres then we can uncover new musical vocabulary for the future to come.
The next step is to explore all this for yourself – download the sampled hyperpiano and give it a play.
What is the meaning of ET and EDO, and are they interchangeable?
ET: Equal Temperament
EDO: Equal Divisions of the Octave
In practice, yes they are interchangeable. For example, 12-ET and 12-EDO both refer to the exact same tuning which has 12 equal notes per octave. But there is a slight difference in their meaning.
12-ET suggests that the tuning is a temperament, i.e. it tempers some other interval, usually a just interval. 12-ET tempers 81/80, the syntonic comma, and other intervals.
12-EDO suggests that an octave has been divided into 12 equal parts, but otherwise doesn’t imply that tempering is of importance.
Some people will even say ET for 12-ET, 19-ET and 31-ET, while using EDO for 8-EDO, 13-EDO and others. Perhaps because 8-EDO and 13-EDO are not thought of as temperaments, whereas 12-ET, 19-ET and 31-ET are all useful meantone temperaments.
Personally, I always use EDO in my own thinking and private communication with other microtonalists, but will use TET or ET when I need to be understood by a larger, mixed audience.
To complicate things further, some folks use ED2 or ED2/1 synonymously with EDO, because the octave is equal to the ratio 2/1. The good thing about this format is that we can generalise it for other scales that divide some interval into equal parts (e.g. EDphi, ED3/2, ED4). I welcome the move to this kind of generalised terminology that helps us describe more tunings with less words.
The world of xenharmonic jargon is often difficult to navigate. Once you get your head around it, you can forget about the tuning theory politics and remember that the important part is to make inspiring and enjoyable music!