Microtonal music has been traditionally defined as music that employs intonation systems containing intervals smaller than the 12 equal temperment half-step. In more informal contexts, the term has been used to describe any music that uses intonation systems containing intervals of any size other than those that can be described by 12-tone equal temperment.
I have made a distinction between what I consider to be two types of intonation systems: synthetic and organic. I am not 100% satisfied with these designstions, but they help to label my concepts. A synthetic system would be one that, to a greater or lesser extent, uses some sort of artifice, usually some sort of complicated mathematics, to arrive at the relationships beween intervals within a system; an organic system would be arrived at because the uses of the system, whether for personal or cultural reasons, or some combination of both, thought that the relationships "sounded good".
Just intonation. This is not actually one system but a family of systems, normally based around a "pure interval" such as the 5th. In a just system, the pure interval would not be tempered, as in 12tet, but kept pure. In 12tet, a perfect fifth, being 7 half-steps, can be described by the formula: n^7 (n to the 7th power, or n*n*n*n*n*n*n), where n=12th root of 2. In a just intonation system, the formula is much simpler. The perfect fifth has a frequency ration of 3:2. This is much easier, but one quickly runs into a problem. It is usually considered desirable to have a system that also has octaves, a frequncy ratio of 2:1. So, if we have perfect 5ths, with a ratio of 3:2, and wish to continue stacking 5ths, our next tone has a ratio to our starting pitch of 9:4, the next in the series is 27:8. Now we can compress them into a single octave by multiplying the second term in each ratio by a power of 2. Thus, our first three tones are described by the following ratios:
3:2 9:8 27:16
If we had used "C" as our starting pitch, these pitches would be G, D, and A respectively.
Usually one continues this process until one has a complete chromatic scale. Here's where we get into problems. Let's continue the process around the circle of 5ths:
| Pitch | Ratio |
|---|---|
| G | 3:2 |
| D | 9:8 |
| A | 27:16 |
| E | 81:64 |
| B | 243:128 |
| F# | 729:512 |
| C# | 2,187:2,048 |
| G# | 6,561:4,096 |
| D#/Eb | 19,683:16,384 |
| Bb | 59,049:32,768 |
| F | 177,147:131,072 |
| C | 531,441:524,288 |
But wait! Those last numbers should be the same! But they're not. Now matter how far you go, no power of 3 is ever going to equal any power of 2 (okay except the 0 power, but that's our starting point). So some adjustment must be made. The difference is called a "comma". This interval was known back into antiquity, and is the basis for much of classical Persian and Indian music.
I also used the most hideous method of arriving at our end result, using only the cicle of 5ths. Normally one would use 4ths, and other recognised intervals as well. But the end result is the same: one cannot have pure intervals and an equal intonation system. The two are mutually exclusive.
Just so you know, if one were to describe a "pure" 5th in terms of a 12tet system, it would be approx. 702c.
Some have attempted to minimize the effect of the comma by continuing the subdivision of the octave past 12, thus spreading the comma out, so to speak. The most notorious example is the 43-tone just system used by Harry Partch.
14-tone equal temperment. This is my favorite. It works on the same priciples as 12tet, but the octave is divided into 14 equal parts instead of 12. The intervals are approx. 85.7c apart.
This will be limited to the systems with which I am personally well acquainted, ie., those of Central Java and of West Africa.
Slendro. This is a pentatonic (five note) scale used in Java and Bali, in the Republic of Indonesia. It is spoken of as being an "equi-distant" scale, that is, that the intervals are more or less the same size. If this were strictly true, the intervals would all be 240c. In reality, the intervals generally range from 220c to 260c. If they were all 240c, then the closest interval possible to the perfect 5th (700c) would be 720c. The general practice is to adjust some of the intervals so that at least one of the "5th"s is pretty close to pure. Which intervals are adjusted and by how much is left to the art and taste of the person doing the tuning, and as a result, every gamelan orchestra from these islands is tuned just a little bit differently.
Pelog. This is a 7-tone scale from the island of Java. The intervals in this scale are of a wide variety of sizes, normally ranging from 90c to 350c or more. Again, the exact sizes of the intervals is left to the taste of the tuner, and each orchestra is a little bit different.
See the technical documentation for the instruments of the orchestra that I play in on our web-site:
Balafon. The balafon is a xylophone indigenous to West Africa, specifically those areas that were once part of the Mandinka Empire. The balafon uses a 7-tone scale, and like slendro, it is usually spoken of as being "equi-distant". Hugh Tracey refers to this as "the minor whole-tone scale". In theory, then, the tones would all be ca. 171.4c apart. The reality is, of course, somewhat different. The instruments are tuned by ear, according to the skill and taste of the artisan. Below are some samples of balafon scales, measured in cents, taken in the field by Lynn Jessup, and documented in her book "The Mandinka Balafon: An Introduction with Notation for Teaching" (Xylo Publications, La Mesa, CA. 1983). The "ideal" scale is that where all the tones are 171.4c apart. It is included for comparison.
"Ideal" scale 0 0 0 0 171 166 166 174 343 309 320 362 514 499 465 529 685 687 618 713 857 837 844 879 1028 1026 1010 1024
Notice that all of the intervals are larger than a half-step (100c), but none so large as a whole-step (200c). Although there is considerable deviation between the samples, all of these insruments would be consisdered "in tune" by players within that musical culture.
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