The Impossibility of Exact Intonation I. Introduction. Some fundamental facts about the way pitches are generated and heard by our ears, although discovered centuries ago, deserve to be more widely known among the string-playing community. This article is an attempt to publicize these facts in a form that all can understand and appreciate, and to emphasize with perhaps undue force some of their immediate consequences for practical performance. Here is the bottom line: "playing in tune" is always a relative, subjective concept. This article, like Gaul, is divided into three parts. The first, the Introduction, is what you are now reading. Do not let yourself be put off by the title of the second part: Theoretical Basis of Intonation! I have worked to make it easy reading for anyone who has access to a piano keyboard and a hand- held calculator, although neither of these is necessary for its understanding. The third part, Practical Consequences, discusses the compromises and adjustments necessary for any string player. It gives suggestions for experiments that can be carried out with instrument under chin or between legs to confirm the conclusions of the second part. II. Theoretical Basis of Intonation The facts to be sketched below were known to Pythagoras, a semi-legendary Greek philosopher who lived as long before the birth of Christ as Bach lived before us. Here are the two fundamental facts: A. For two notes to be in octave relationship(e.g. C and C', or F' and F") their vibrations must be in the rela- tionship 1:2. Example: The A in the octave above the pitchfork A of 440 vibrations per second vibrates at a rate of 880 per second, while the A in the octave below, the A two notes below middle C, vibrates at a rate of 220 vibrations per second. B. For two notes to be in a fifth relationship(e.g. C and G, or E and B') their vibrations must be in the relationship 2:3. Example: The E a fifth above our 440 A vibrates at a rate of 3/2 times 440, or 660 vibrations per second, while the D a fifth below the 440 A vibrates at a rate of 2/3 times 440, or 293 1/3 vibrations per second. String players, you can confirm these facts on your instruments: play your open A, then play the octave harmonic. You do this by suppressing the vibration of the whole length of the string, and causing the string to vibrate essentially with half the string length. Since half as much string has to move, it goes twice as fast; result, octave! Same goes for the "fifth" -- actually "twelth" -- harmonic. Touch the string at one-third, or two-thirds, of its length. You will get an octave-and-a-fifth above the open string pitch, and you see that one-third of the string, vibrating at three times the bottom pitch, gives this octave-and-a-fifth note. As explained above, half of the vibrations - 3/2 times the vibrations of the open string - gives the pitch of a fifth above. These relations seem right and proper - 2 and 3 are small numbers, which seems good - but they lead us into an awful mess, as shall now be explained. Go to a piano, and find the lowest C. Now go up by fifths: to G, to the D, then to the A(almost two octaves above where you started), then to E, to B, to Fsharp = Gflat, Dflat, Aflat, Eflat, Bflat, F, and, finally, C, a high C, seven octaves above the C at which we started. Now, whatever the number of vibrations per second of the low C is, the G above it is vibrating at 3/2 times that, the D above that at 3/2 times 3/2 times that . . . and by the time we get up to Bflat, F, and the high C seven octaves above, we have achieved a vibration rate of "3/2 multiplied by itself 12 times", or 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 ! But - even simpler - since that high C is seven octaves above the low C we started with, that high C must vibrate at a rate of 2 x 2 x 2 x 2 x 2 x 2 x 2 times the low C we started with. Now turn from the piano to your calculator. Punch in 1.5(that is, 3/2) times 1.5, times 1.5 , times 1.5, . . . until you've multiplied 1.5 times itself 12 times. I've just done it, my calculator reads 129.746 approximately. What this says is: whatever rate your low C is vibrating at, your C seven octaves above is vibrating at about 129.746 times that rate - _if you _by fifths!_ Let's go by octaves: punch in 2 times 2, times 2, times 2, . . until you've multiplied 2 by itself 7 times. What did you get? That's right, 128. What this says is: whatever rate your low C is vibrating at, your C seven octaves above is vibrating at 128 times that rate. And the horrible fact of the matter is: 128 is not 129.746. Not the same. Close, but no cigar! Bach knew this. To resolve this difficulty, he proposed, "We really can't monkey with the octave; we've got to compress the fifth. We've got to find a ratio _smaller_ than 3/2(or 1.5), a tiny bit smaller, so that this ratio times itself 12 times gives us 128, not 129.746." I've just worked it out on my calculator, the new ratio has got to be, not 1.5, but 1.4983. Any fifth on a piano has got to be "squished", by a factor of about .0017, almost one part in 500. Only with this tuning can "fifths" (we string players should invent a new term, "quasi-fifths" or "tempered fifths") be reconciled with octaves, can one achieve an (almost) correct intonation that enables . . . well, enables Bach to create a sequence of pieces written in all possible keys. In "The Well-Tempered Klavier" - parts I and II - he even did it twice. Alas, most pianos have worse intonation problems than this slight but universal fifth correction. Several weeks of enthusiastic Chopin or Brahms will drive some intervals even further out of whack . . . and tuners tend to tune the three strings comprising most piano notes ever so slightly out of tune, in order to "liven up" the tone. But if you play a fifth on a freshly tuned piano and listen carefully you will hear "beats"; on your native instrument this would cause you to reach for an appropriate peg . . . ! We've just seen that fifths and octaves can't live together happily. The same goes for all the other intervals. If you have good strings and a gentle bow you can test the "fourth" harmonic and the "fifth" harmonic, touching the string at 1/4 and then 1/5 its length. Since 4 = 2 x 2, the fourth harmonic gives the octave over the octave, two octaves over the fundamental tone. The fifth harmonic gives two octaves and a third. The soprano's high B can be achieved not only by stopping the E string on the violin with your fourth finger in first position, but by touching the G string gently a fifth along its length, about where, if you put your finger down firmly, you'd get the B below middle C. _But these aren't the same notes!_ Why not? Well, if you want the stopped B on the E string, that's a fifth above a fifth above a fifth above a fifth above G. By what we've just seen, that is 3/2 x 3/2 x 3/2 x 3/2 = 81/16 times the fundamental G frequency. The harmonic B, the fuzzy note on the G string, should be 5 times the G frequency. Again, 81/16 is not 5. III. Practical Consequences Now, string player, take up your cello, viola, or violin, and tune it very carefully. Then place your first finger on the E of your D string(first position, this is all very simple). Play your stopped D and open A strings together very softly and purely, adjusting your first finger until you have a _perfect_ E-A fourth. Work on it till you have no beats, an absolutely pure fourth. Got it? Good! Now _don't move that finger!_ Don't give it even the slightest wiggle . . . but adjust your bow till you are now bowing the stopped D and the open G. How does it sound? a bit fishy, no? Now I give you permission to move your first finger; lower that E - you'll be moving your finger only a tiny amount toward the nut - until that major sixth, the E(stopped D) and open G, really rings pure. Got it? Keep it! Don't move your finger! and go back to the E-A fourth. Pretty bad, huh? Let's face facts, gentle reader and string player: you have (at least) _two_ E's on your D string(first position, no less!) each of which can claim to be correct. Indeed, each of which _is_ correct! The E which is the fifth above A is not the E that fits into a C major chord whose fifth is G. Another example, again using the D string of the violin: the Fsharp(second finger first position) that is a beautiful fourth with the B(on the A string) that itself is a beautiful fourth with the open E . . . got it? This Fsharp is not the same note that is a beautiful minor third with the A string itself. Isadore Saslav, the former concertmaster of the Baltimore Symphony, has worked out at least two consonant pitch levels for almost every note the violin plays, using similar comparisons with open strings or harmonics. The moral is clear: the Eflat, or Gsharp, or B, or whatever, must be thought of in its harmonic context. Sometimes one Csharp is correct, sometimes another. And it is not always clear which. And so you must be courteous to your chamber music colleague who is playing what seems to you to be a sharp B, or a flat Dflat. It may not be slovenliness; there may be good reasons behind her choice of pitch. Discuss it with her; perhaps she will succeed in persuading you that in the harmonic context of the piece you are playing _you_ must be the one to raise your Gsharp, or lower your F. An exploration of these issues and an integration of them into your playing will increase the sensitivity of your ear and the subtlety of your musicianship. For Further Reading 1. Redfield, John, Music: A Science and an Art, Knopf, New York, 1928. 2. Jeans, James, Science and Music, Macmillan, New York, 1937 3. Righter, Charles, The Musical Comma, University of Iowa Press, Iowa City, 1972.