Tolerance (1980)

This is the age of precision and applied mathematics, so the kind of tolerance we are discussing here is in the engineer's sense of the term. Specifications or blueprints for some manufactured article will say + or - 5% or 5% tolerance, meaning that a certain distance or a certain part may be above or below its labelled value by that percentage. Or an engineering drawing will say + or - .003" meaning a certain part can be three-thousandths of an inch smaller or larger than the target value.

How does this apply to music? Well, before this era of the Do-It-Yourselfer, it didn't. Tolerances in musical instruments were the sacred untouchable property of the manufacturer, and pitch-tolerances of the professional tuner. Now more people than ever are making their own instruments, or changing them around -- why, even the guitar stores may stock customizing-parts, much as the auto supplies have been doing for quite a while.

Just lately, many non-12 tuning-systems which were impractical theories in the past, have become feasible and realizable on today's instruments. New precision tuning-devices are on the market, while some electronic organs have their own tuner built right in. With the violin family, tuning is mental habit instead of a hardware gadget. Up to now, "tuning" meant a more or less accurate approximation to 12 equally-spaced tones per octave -- the tolerances of mistuning have been "by guess and by gosh."

Piano tuners are taught how rather than why, and it takes about six months to master a routine which is something like a religious ritual, and which cannot be transferred to a non-twelve system. The way it is usually taught is such as to close the mind to even imagining what a 17- or 19- or 22-tone tuning-routine would be like. Theoretically the tolerance in piano-tuning is very tight; in practice ... well let's be charitable and drop the subject!

Guitars are pre-tuned by placing the fret according to freting-tables. These tables can be calculated to better than one part in a million on computers, but nobody in their right mind would expect that kind of accuracy on real guitars. Your friendly tone-bender in the rock band wouldn't even want it if he could have it. People buy all kinds of magic effects-boxes to keep the tones from being too steady, too dead. Synthesizers and electronic organs have all kinds of special circuits invented for them to liven up the tone, to loosen the tolerance. What a paradox! Just when modem technology attains precision in tuning, nobody really wants it; the manufacturer tells the chief engineer to stir in just the precise amount of out-of-tuneness that will sell the organs to the general public.

So a specific percentage tolerance for even the 12-tone equal temperament tuning is pretty much an exercise in futility. One book on electronic organs said 0.1%, which is one part in a thousand. Well maybe for the organbuilder, that is ideal; but when the organist puts on the tremolo it might be a quartertone wide, which is 3% or 30 times that amount. Who's kidding whom? One part in a thousand on a guitar fingerboard would be 1/2 mm, say 1/50 inch. A very careful maker might be lucky enough to position the frets that well most of the time, but that would not guarantee such precision in the pitches actually heard from that guitar in a real performance.

Pitch tolerances are thus not quite the same thing for the instrument-maker as they are for the performer--even when those two are the same person, as now happens. The gulf between theory and practice is even wider! More about that in a moment. Musical notation makes this even more complicated. Now let us consider "new" tuning systems other than the standard 12-tone-equal, and it really gets hairy.

It would be helpful if one could make a sharp division between fixed-pitch instruments (those with frets or keyboards, for example) and flexible-pitch instruments (violin family, voice, thereminvox, slide trombone, steel guitar, etc.). This would simplify matters, and would correspond to the digital-vs.-analog idea now familiar from computer technology. (Staff-notation and guitar tablature are digital as the piano and guitar are; voice and violin are analog.) Unfortunately the real world isn't all that neat. Pianos go out of tune and voice teachers try to install psychological frets and keyboards in their students' vocal cords. Although it has a keyboard, the clavichord can bend notes upward or execute vibrato. Although violins can play any pitch in between, violinists spend hours practicing scales so they can pretend to be digital.

Once we exercise our freedom of choice and start entertaining the idea of exploring all the possible tuning-systems, the theorists send us a chilling blast of Mathematics. Books on music theory are silent as the tomb, while computers are almost always deaf and mute. Hence the gulf between theory and practice. It is next to impossible to predict in advance how a tuning is going to sound, regardless of how carefully the mathematics behind it has been done. Thus most non-12 systems have been literally condemned without any HEARING.

So long as the mathematics is not applied in the construction or tuning of musical instruments, or programming a computer to perform musical sounds, the tolerances applied to the numbers expressing pitches or ratios will almost always remain unrealistically tight. The theorist is tempted to idealize--to think of music in terms of perfect instruments forever playing in perfect tune. As every composer and musician knows, this is so far from what really happens as to be utterly ridiculous. The existence of such a crude approximation to just ratios as the 12-tone equal temperament, and then the wide erratic variations in the sounds of this temperament on actual instruments, should be evidence enough of that.

This in turn reacts backward: practical musicians pooh-pooh theorists and inventors when they point out that new or different tunings might be better. The numbers and the fractions and decimals turn them off. So the theorists' attempLs to explain and prove and provide necessary information are cruelly rebuffed by the musical establishment. For instance, the theorist might state that a real true correct major third contains, not 4 keyboard-sernitones, but only 3.863137 of them. And maybe that there are an infinite number of figures after that they go on forever. Or if the theorist is so inclined he might put this another way: the ratio of a true major third is 4:5 (one tone makes four vibrations in exactly the time the other tone makes five), whereas the 12-tone tempered imitation major third has a ratio of 1 to the cube root of 2, approximately 1.259 and an infinite string of decimals after that. That is, you can make the keyboard major third look simpler by calling it 4 semitones exactly, or the just major third look simpler by saying its ratio (either of frequency or string-length) is 4 to 5, exactly. See what psychological attitude does?

With the current vogue of guitar, extending to other (12-tone of course) fretted instruments, the counting of semitones in an interval becomes much more obvious and easy than it is on a keyboard arranged as 7 white and 5 black keys. So the matter of attitude becomes still more important than it ever has been. The degrees of the staff stand for unequal intervals of the diatonic scale, whereas the frets on a guitar and the little squares or numbers in the guitar tablature stand for equal twelfths of an octave. This is bound to change our attitudes, whether we want to or not. And with that, our tolerance.

The theoretical numbers or the positions of those frets are uncompromising, creating an impression of exact point values. So with Partch's ratio-fractions for just (untempered) intervals, creating an impression of perfect precision, exactly-fixed pitches. But the real world of musical sound is not like that! Performers and listeners have tolerances, so instead of exact pitches, we have what the engineer would call "bands of frequencies" -- if Middle A is nominally 440 hertz, the actual pitches heard in a performance may vary up and down from that as far as 438 and 442 or even further; moreover, with singers in a chorus or violins playing in unison, or a many-rank pipe-organ, you will hear a "smear" or "spread" of all the possible pitches between say 437 and 443 Hz at once. This is true even when vibrato and tremolo are not under consideration. But the theoretical table will say "440.000" implying that A is exactly on the button, not deviating by one vibration in a thousand seconds, which is 16-2/3 minutes. In case you have wondered about those zeros after a decimal point, they imply that kind of accuracy. If a machine delivered that kind of pitches with narrow tolerance, it would be rejected as inhumanly dead. And that's what happens in practice.

It is either humiliating, or a declaration of one's living humanity, to sing into an electronic tuning device and find that one cannot keep the pitch steady enough to please the machine. Again a matter of attitude.

So don't be frightened or threatened by all the digital readouts and computer printouts and statements of precision in today's environment-theory and mathematical realization are necessary IN THEIR PROPER PLACE, which has to do with designing and building instruments and making them stay in tune, but a different aspect of mathematics will fit actual musical composition and performance better, and fortunately there is such a mathematics -- it deals with bands of frequencies and thus ranges of pitch instead of absolute points, and with allowable deviations from those ideal points (which is tolerance), and for that matter with deviations from exact timing, which is rhythmic tolerance. I am afraid that this term "microtones" used for new non-12 tuning-systems, is most unfortunate, because it scares musicians off -- in the metric system, "micro-" means one-millionth, 0.000001. Since that meaning often appears in magazines and newspapers, it implies that going to microtonal music will necessitate ruinously expensive and intolerably fussy exactitude which would wreck any musical performance and make it impossible to learn any new instruments. This is why I use "xenharmonics," "beyond twelve" and "non-12" so often. Microtones implies a precision that is impossible to attain and that we could not use if we had it!

Now let's get down to brass tacks: the tolerable deviations from ideal pitches. We cannot set any fixed values. It all depends on how fast the music goes--the tempo. This is exactly why accordions are played so terribly fast, so that the harshness of the 12-tone temperament as heard on accordion reeds will be less painful. Indeed, the beats of the distorted sharp major thirds and sixths are often masked by deliberately putting in another set of reeds tuned slightly sharp to beat with the main set, and perhaps a third set of reeds tuned slightly flat. A violin-and-piano piece, or a piano concerto with orchestra, sets up the same kind of pitched battle, and it has to sound out-of-tune no matter how hard everyone tries. The conflict between organ and orchestra, or organ and band, is so great that these possibilities are often discarded as hopeless. Indeed, piano and organ do not do well together either, since the octaves on a piano are tuned with a "stretch" whereas any stretch on an organ is very small. So what's that about twelve absolutely EQUALLY-spaced pitches? Malarkey! Baloney. Nonsense.

And don't you dare bring up Bach--he didn't have a set of tuning forks or a lab full of oscilloscopes and tuning-monitors--so his version of 12 equally-spaced pitches could have been anything at all. I really doubt that he intended the sterile rigid electric-typewriter-style harpsichord performances of his works today. Any more than he would have liked the blurred fuzzy piano performances in the late 19th century.

After 40 years of tuning the 12-tone temperament on organs and pianos and harpsichords, I am tired of it and want to escape the Schoenberg Squirrelcage, and help you to do so too. The easiest way is to have a guitar refretted to the 19-tone scale, and take it from there. Once this is done, other instruments can be tuned to such a guitar, and one can learn to sing in the 19-tone system, and very fortunately, 90% of ordinary sheet music can be played in that scale without alteration. Through 17 years of actual experiment, I have ascertained that the tolerance for deviation or mistuning of the 19-tone scale (and therefore such scales as 14 or 17 or 22) is about as wide as that for the familiar 12. This is not speculation on paper; this is what I and others have found out with real instruments. So why wait? Forget about that million-dollar "dream" laboratory and all the scientist's expensive gadgets, and start enjoying all the new resources and new moods and expressive powers of these non-12 scales. Concrete example: I fretted a guitar to 18-tone (Busoni's proposed third-tones) and can use this guitar as a 17 or a 19 without the theoretical errors from moving the bridge spoiling any performances. So you can have three systems for the price of one.

Not only that, but some people are experimenting with UNequal temperaments -- 17 and 19 in particular can be warped and twisted and stretched and squuz to get novel effects.

Beyond quartertones, 24 or more tones per octave, naturally enough the tolerances are going to get tighter. That's exactly why I suggest you start with 19. If your personality clashes with 19, then try 22, before going on to other systems. The 31-tone system is very important as a next stage in your explorations. It has a calm, restful mood. It is more suited to slower music where you can savor the novel harmonic progressions and subtle nuances. It represents a practical upper limit for fluent playing of fretted instruments, although 41-tone guitars exist.

With suitable tone-qualities and proper listening conditions, you can go to higher numbers of tones and smaller intervals with the tighter tolerances these will demand: 41 43 46 50 53 65 72 77... till one needs a computer to take care of the problems involved. Some of you will wonder why I haven't mentioned just intonation.

I had to deal with the easiest way for the newcomer to get into non-12 and do something about it, instead of being afraid to begin. So I had to start with systems affording comfortably wide tolerances. Doesn't that make sense? "Afford" goes into another meaning also: systems like 17 19 22 are within your means. If I insist that you start experimenting with 171 tones per octave, I am just wasting paper, regardless of the merits of 171, 559, and 612.

Now theoretically just intonation or untempered tuning or pure or exact tuning means an INFINITE number of notes-i.e., it should require instruments larger than the known universe! Which would obviously take all eternity to tune. No wonder the scoffers rush in with sneers and contempt. The purpose of temperaments is to reduce this impossible number of pitches to something practical. Does that mean I am asking you to wait for Heaven or a million more reincarnations before you can enjoy just intonation? Not at all: there are other ways of applying the Tolerance Principle and of using a just system with a practicable number of tones per octave.

Theoretical just intonation means no tolerance at all, so this obviously is unattainable perfection. Even if it could be attained here on earth, our ears would not be able to appreciate it. Therefore must we content ourselves with 12? Hardly! That kind of argument is cruel and unfair. In the real world, our ears can distinguish many fine shades of pitch in the middle register near the top of the treble staff, but at the upper and lower extremes, our discrimination is coarse indeed. If tones are sustained and rich in harmonics, we can distinguish slight differences of intervals; but if they are snappy and short and dry and lacking in overtones, such as on a xylophone, just intonation's refinements would escape us and be wasted money. That indicates that a reedy organ tone, or a sort of violin tone-quality on an electronic organ or computer, would be optimum for the extended versions of just intonation. Composers should aim for delicacy and subtlety which will come through in just tuning.

Just intonation ordinarily means a limited selection from the above-mentioned infinity of pitches. 12 per octave are not enough to do more than play trivial exercises and compare a major scale against a tempered major scale. 24 will do for a good many pieces of music; something like 40 or even 60 pitches would do better. Which 24 or which 40? That depends on the individual composition. With modern electronics we can do what we want. If you are cramped for facilities, start with a steel guitar tuned to just chords.

If the series of perfect fifths is carried out a short distance and then a few notes from the series of perfect major thirds are added, we run into an interval called the comma (81:80) which can be a nuisance in some pieces of music and an advantage in others. Then we run into a very tiny interval which Ellis called the Skhisma (about a 600th of an octave) on the very borderline of tuners to tune and listeners to hear. Frankly, I cannot think of any practical musical performance where it would be worth the trouble to provide two pitches a skhisma apart. As a composer I wouldn't even dream of calling for it. If I give you its ratio here: 32805:32768, I risk turning you and everybody off. Those numbers have no musical meaning, even though they are necessary in one 5 studies and calculations to understand the just system. If to the major thirds and fifths we add just harmonic sevenths to the network of tones, other tiny intervals occur as near-coincidences. Somewhat larger intervals also occur, and these suggest imitating them in temperaments such as the 31-tone. So the infinity of tones mentioned above can be contracted to something quite manageable, and still have some smooth, beatless, serene just intervals when the music needs them.

Back to the real world of now and what is available: the violin family can play any gradation of pitch no matter how tiny, but of course with limitations of human hands and ears. However, the violin family is very valuable for an extra ability: ELASTIC tuning. Suppose that in a string quartet a series of harmonies occurs which causes the comma to appear, and that shift of a comma at that point would be unpleasant to hear. (Kind of seasick.) Why, the players, almost on a subconscious level, will temper on the run, so to speak. They will bend their pitches in such a way as to get rid of that comma if it isn't wanted in that place. I can imagine, as composer, that sometimes I will want the comma heard plainly -- whether I can ever get a string quartet to do that, however, is doubtful. By "elastic tuning" I mean that the performers can alter the tuning of a chord WHILE IT IS BEING SUSTAINED. So this is dynamic tolerance.

It so happens that I built a special electronic organ back in 1962 which can retune itself and is just as elastic as the quartet. While a chord is sounding it will try to become "juster" than it started out to be. With modern computer developments it should be possible to have even greater tuning-elasticity. And the degree of etasticity will be variable.

I hope this discussion has helped you see that non-twelve tunings are feasible and affordable NOW, since the practical tolerances are reasonable enough.