Intro to Xenharmonics
Welcome to the Introduction to Xenharmonics page! This page will introduce a new comer to the world of alternate tunings and the various terms that pop up. You'll learn the basics of tuning theory with Just Intonation, temperaments, and Subgroups. Let's get started!
Xenharmonics is essentially the theoretical study of harmony as a whole, outside the scope of western traditional harmony. Xenharmony uses mathematical figures to look at music in a slightly more objective sense. But most of all, Xenharmonics is about working with tuning systems involving ratios completely outside the scope of 12 EDO. This is where the fun is, discovering brand new sounds and making music with them.
What is Just Intonation? Without overwhelming you with insanely complicated sciency crap, I'll try to explain. Just Intonation, often abbreviated "JI" is a system in which ratios are used to name intervals. An example would be say, "3/2" or "7/5" etc... These ratios refer to the portion of a sound making device such as a string or tube that is resonating with the left over portion being whatever is left in the bottom number to equal the top number. For example, if we play a guitar string open, we could say this is the "fundamental" or 1/1 meaning one of the whole is ringing. Duh, right? Easy! So if I say "2/1" then what part is ringing? If you're catching on, you should be able to figure out that 2/1 indicates that half the string i ringing. Why? Well, it seems a bit odd but ratios are read backwards so 2/1 is saying 1/2 of the string is dead and one half is vibrating. This means that you shortened the guitar string enough for the sound to be 2/1 apart from the fundamental. If we stop the string at two third's it's length than we get the interval 3/2 meaning two thirds is vibrating with one third cut off from ringing out.
Just Intonation sounds GOOD! It is the most natural occurring intervals along a string or tube. The scale that all fixed pitch western instruments are tuned to is not Just intonation. It is quite off from Just intonation but uses what are called "approximations." You see, just intonation is funny, it's very in tune sounding BUT if you tried modulating within the same set of notes that you used to produce some JI scale, then it would change or be more out of tune. Just intonation does not allow for modulation which is why we use temperaments most of the time instead.
Harmonics and Limits
If you play guitar, you probably know what "natural harmonics" are. Basically you rest your finger on the string lightly and let go once you strike the note which produces a chime like sound on the guitar string, I'm sure you've heard it or have heard of it before. Well harmonics are not only Just intonation but they are extremely concordant sounds that occur on an sound making device simply be striking it while touching it in a certain place. In addition, overtones occur on every instrument except may'be a pure sound wave. Overtones produce not only the texture on a particular instrument but they allow us to differentiate between different sounds in daily life.
In xenharmonic, there is a term called "harmonic limit". Limit is how high in the harmonic series a scale can represent accurately WITHOUT SKIPPING ANY RATIO CATEGORIES. In this case, accurate means close to the actual pitch value of Just intonation, inaccurate means well, further away. Take 12 Equal Temperament, this scale is 5-limit because it cannot represent ratios between 5 and 17 with any accuracy. So what does that mean?? I'll explain:
The unfortunate thing is, at the beginning, you will simply have to learn what things sound like. You cannot predict what something will sound like by reading a number. Developing a good ear for just intonation is crucial in understanding xenharmonic theory. But for now, just take my word for it. You'll also need to become somewhat familiar with the harmonic series which I'll write below omitting any repeated intervals:
1/1 2/1 3/2 5/4 7/4 11/8 13/8 15/8 17/16
Now 12 EDO has a 5-limit meaning that it only approximates in entirety, up to 5/4. However, 12 EDO also approximates 15/8 and 17/16 which are the 15th and 17th harmonics. The weird part is in the section between, 7, 11 and 13 are all new sounds. 7/4 is barely, and very horribly approximated in 12 EDO but 11/8 and 13/8 cannot even be badly rendered in 12 edo, the are too distant from the pitches available.