One of the most important attributes of a well-trained musician is the ability to play perfectly in tune, something most advanced players routinely do. For many though, this is one of the most elusive skills; its rules and subtleties remain a mystery. Some have an intellectual understanding of what to do, but would like that knowledge to have a more integral and intuitive function in their performance. The following is intended for anyone who is interested in improving his or her knowledge of and proficiency for playing in tune. It will explain what intonation is and give information and exercises that will enable everyone to work to 1) hear exactly what is and what is not in tune, and 2) greatly improve their intonation while performing.
The first step is to define intonation. This will require an awareness of some of the rudimentary principles of the physics of sound.
BASIC ACOUSTICAL PRINCIPLES
A sound is made when a vibration in a medium causes an auditory sensation in the ear. The vibration causes waves, which may be described as quickly alternating amounts of compression and rarefaction of the medium, such as air or water (see Figure 1).
Figure 1 is a drawing of (a) a string at rest. When plucked (b-f), it vibrates, producing a reaction in the air medium. First (b-c), it pushes against the air and packs the molecules tighter than normal atmospheric pressure. Next (d-e) it recoils, creating a condition more like a vacuum, which is called rarefaction. The wave which results is transmitted to our ears as sound if the vibration which caused it is between about 16 and 20,000 complete cycles per second. Figure 2 is another illustration of sound production. A tuning fork attached to a piston in a tube creates compression and rarefaction as it vibrates.
A vibrating tuning fork, a screeching car, or a musical instrument being played all produce sound waves. Most musical instruments produce a sound called a pitch. For example, on a string instrument like a violin, the bow is drawn against a string, exciting it and making it act much like a pendulum. Once the string is set in motion it moves in one direction and comes to a complete rest before returning in the other direction and again coming to rest. This is one cycle, and depending on the number of times this process repeats itself per second (hertz or hz per second), it produces a given pitch. Many cycles per second will be heard as a high pitch; likewise, few cycles per second sound as a low pitch. Humans perceive vibrations between 16-20,000 hz as sound, so if the string vibrates between those numbers, we hear the movement as sound. If it vibrates slower than 16, we hear individual bits of sound. If the string vibrates faster than 16, we hear these individual cycles as one steady pitch. In the same way, our eyes cannot discern the individual frames of a related series of still photographs shown in quick succession to form the illusion of a moving picture. The number of vibrations of a sound nearly exactly determines its pitch. This is called frequency.
The other acoustical principle to be discussed is amplitude, which is related to the loudness of a sound. When a string of a fixed length is vibrated, it remains at essentially the same pitch, no matter how loud or soft it is played. If the string is making a soft sound, it was played with little force and is moving over a fairly small area rather slowly. If it is making a louder sound, it was played harder and is moving over a larger area at a faster rate. Therefore, the wave of the soft sound would look small when compared to the wave of the loud sound, and so the soft sound has a smaller amplitude than the loud sound. In a string of fixed length, the frequency will remain the same even if its amplitude is altered. Figure 3 illustrates two waves of the same frequency, but of different amplitudes. (Frequency = pitch; amplitude = loudness.)
Now we can better discuss intonation, which is the relationship of the vibrations of two or more pitches together.
INTONATION
Unison Let's begin by talking about two pitches played together in unison. The unison relationship is the best starting point because it is the simplest to understand and to hear. More complex intervals will be discussed later. Suppose two pitches of exactly the same frequency are played simultaneously, say, 176 Hz (which would sound as an f concert). Since both pitches are exactly the same number of vibrations, the notes are exactly in tune. Poor intonation would result if one of the notes were at a slightly different pitch and frequency. If one is at 176 Hz and the other is at 174 or 177, the notes are out of tune. But can people really hear if one note in a 'unison' is just one or two vibrations off? We can -- in fact, we can make even finer discriminations than that. How is this possible?
Resultants
The answer involves one of the most important tools of good intonation. When two notes are played together, they produce another pitch or resultant tone. In the case of a unison where the two notes are exactly in tune, they sound like one louder note because they reinforce each other. When one of the notes played is at the top of its wave, so is the other one. When it is at the bottom of its waveshape, so is the other one. This is called corresponding phase, and causes the sound to be amplified (although the pitch remains the same). Figure 4a illustrates this: two pitches of the same frequency but different amplitude are played together in corresponding phase. The amplitudes of both notes are added together to form one louder note. This is called constructive interference.
Figure 4b shows the reverse of the above situation: two unison pitches of unequal amplitude are played simultaneously exactly out of phase. Their sounds cancel each other out, but the louder pitch minus the softer note leaves some sound remaining. This relationship is known as destructive interference.
Finally, Figure 4c illustrates two pitches
of equal amplitude played exactly out of phase. The two amplitudes
cancel each other out completely, and so no sound is heard. Th
is
theoretical example is hard to achieve out of a laboratory; the
waveshapes must be exactly out of phase, and the listener must
be receiving the same amount of sound from both sources.
The concept of phase has an important application to intonation -- its practical aspects are familiar to any musician who has ever played in unison with another musician. If they were not exactly in unison -- say, one is playing at 176 Hz and the other at 174 Hz, they would have heard a rolling sound, often called 'beats', which is caused by the periodic constructive and destructive interference between the two frequencies. You hear, in addition to the two pitches, a woo-woo-woo-woo sound as the waves come into phase and back out. The resultant gets softer and louder as it goes out of and comes into phase. The amplitude produced by the two notes together changes and this is heard as beats. Figure 5 shows two notes of slightly different frequency (A and B dotted lines) and the resultant amplitude (heavy line) which varies between the sum of both amplitudes (louder) and the difference between the two amplitudes (softer). Also, Figure 6 is a pictorial representation of what is heard. It is a photograph of two window screens placed together. They go in and out of phase similar to the way that two pitches do when they are producing beats
PRACTICAL APPLICATIONS
Quite good hearing is essential to hear these beats, since they are fairly subtle, but if you listen carefully they will be apparent. If these beat tones are occurring in a unison, then somebody is out of tune. One of the two players must either raise or lower the pitch to get in tune. If both players are wildly varying their pitches it will be like a dog chasing its tail, and nothing will be accomplished. One person must remain steady and the other must move. Who should do what will be discussed later. Which way should a player move? Initially, it does not matter. If only one player moves, the beats will either get faster, which means that the notes are more out of tune, or the beats will get slower, which means that the pitches are getting closer. If by moving, the unison gets more out of tune, stop and go back the other way. If it is better, keep going. You must go through the place of calmness that signifies exact intonation to the place where the beats start to reappear. When this happens you have defined the point of exact intonation and now you can subtly go back to the middle point where you experienced the least beats. Now you are exactly in tune.
Almost invariably, people who have difficulty playing in tune either don't listen to the beats that arise, or if they do, they stop adjusting when the beats get fairly slow and never go all the way to get the notes precisely in tune. Some musicians who are truly adept at this may know exactly where the point of exactness is and not have to go through it, but it is much more likely that they will hit the bull's-eye, go through and return so quickly that no one hears.
Exercises
It is possible to practice to improve your intonation, using the information given above and some simple equipment. The pictures below show a metronome that not only gives tempos but also plays an A-440, and a Korg tuner. The tuner plays pitches and will also meter the pitches to show their intonation. It is useful but not necessary to have something such as this for the exercises that will be described next.
You will need two sound sources that can be played simultaneously. Eventually, one of these sources will be you playing your instrument, but in the beginning it is best to use two devices such as the oscillators pictured above. This is because when you are playing, some of your concentration goes into producing the sound; but for these exercises you will need all your concentration for listening and analyzing the beats. Also, it is difficult to keep a steady unchanging pitch when you play. One of the sound sources will remain on one pitch, but the other must be variable. Neither should play with any vibrato since this will mimic the beat pattern that you will be listening for and complicate things unnecessarily.
Set the first oscillator to a single steady pitch and adjust the other one to match it in unison. (The Korg tuner is especially good as a variable oscillator because it also has a meter that tells you where the pitch is.) Now, are there beats? If not, adjust the knob on the variable source to get some. Then adjust the knob to make the beats slower. Stop and listen. Are the beats slower or faster now? Follow the previous instructions to find the pitch and make the beats disappear. When the beats stop, the notes are exactly in tune.
It is often difficult to hear if a note is sharp or flat, so don't worry about which way to go initially. The thing to remember is that if you are the one who is to move and you are hearing beats, move. It will quickly become apparent that the beats are either faster or slower and then you will know what to do from there. After you have practiced this for a while it will become a natural part of your skills and you will intuitively know which way to move.
Now that you are successful at playing with two oscillators, practice with your instrument and an oscillator, or with another musician. Have the oscillator or other person keep a steady pitch and attempt to play an exact unison. If you can't make the beats stop, try another pitch; find a note you feel you will be able to correct. Practice just as you did with the two oscillators. If, however, you have trouble keeping a relatively steady pitch you should practice playing long tones with a tuner, getting the meter to stay completely still for long periods of time, to correct that.
A good way to get this process clear in your mind is to imagine that you are walking a tight rope. The tight rope with thin air on both sides is analogous to the 'in tune' area with beats to both sides. You might even try standing on one foot (as if on a tight rope) -- to keep your balance you will probably rock your arms back and forth in ever smaller arcs until you achieve relative stability. This is similar to searching for that spot where the beats are least apparent, and so balance and intonation are somewhat alike. It is interesting that the semicircular canals, which provide us with our sense of balance, are located in our ears, which, as everyone knows, give us our sense of hearing.
Now that you have practiced unisons until you feel comfortable with them, the next step is to play more complex intervals and, eventually, chords. But first some more physical concepts should be understood.
THE HARMONIC SERIES INTERVALS AND CHORDS
In pure tuning, as opposed to other tuning systems such as tempered or mean tone, the distances of the intervals are based on their relationships to each other in the natural harmonic series. This harmonic series is a set pattern of notes created by nearly all musical instruments as well as many other objects that make sound. You can buzz your lips into a length of hose and by changing lip tension create this pattern of notes. This process is the basic principle of all brass instruments. All woodwind, brass and string instruments can produce these harmonics, due to the physical properties of the instruments. It would be good to memorize this series because it is the basis for true intonation of intervals and many chords. You will also see interrelationships between this series of notes and the production of resultants.
Figure 8 shows the particular harmonic series for a tube about 11 and l/3 feet in length. This is the length of a French horn in F. When a note is a written C it would sound as a concert F. Any instrument, whether a piccolo or a tuba, would play this same series. The intervals remain the same, although the actual pitches will be different depending on the physical make-up of the instrument.
You will notice by studying Figure 8 that as the harmonics go higher, every note is closer to its following note than it was to its predecessor. Also, each note's vibration is derived by adding the number of vibrations of the first note, the root. In this series for the horn in F the bottom note is 44 Hz. The next note is 88, the next is 132, then 176 and so on, always adding 44 to get the frequency of the next highest note. This is true for any harmonic series. For an instrument whose root or fundamental is 440 the harmonics would go up 880, 1320, 1760 and so on. Following are some of the correspondences spoken of earlier and the reasons for having the harmonic series firmly in mind.
An octave is exactly a 2 to 1 ratio. You can see from the example above that every octave follows this ratio not only in vibration, but also in its placement in the series. The bottom written C is the first harmonic. It is one octave to the 2nd harmonic C. The next C is the 4th harmonic, followed by the 8th and then the 16th. Also, the numbers of vibrations for these octaves are 44 to 88 to 176 to 352 and so on. All of these octaves reduce to a 2 to 1 ratio. Likewise, since the first G appears at the 3rd harmonic, the next G will be at the 6th, and the following G at the 12th harmonic. These G octaves are also in a 2 to 1 ratio. Once you have found a ratio for any interval in the harmonic series, it will remain so throughout.
The most consonant intervals are those at the bottom of the harmonic series. Unison is the most consonant, followed by the octave, the fifth, the fourth, then the major third and minor third.
The ratio of the vibrations of all of these intervals is also given in this series. The octave is 2 to 1, and a fifth, as shown between the 3rd and 2nd harmonic, is a 3 to 2 ratio. A perfect fifth that is exactly in tune could by defined by both a musician and a physicist as one that is in exactly the ratio of 3 to 2, the 3 being the higher note, and the 2 the lower. If the higher note in a fifth is 150 Hz, the lower is 100. You will see that any perfect fifth is a 3 to 2 ratio. Take the 4th series C to the 6th series G. Again, a 3 to 2 ratio.
Figure 9 shows the ratios for a diatonic scale's intervals. All perfect 4ths form 4 to 3 ratios. Major 3rds are 5 to 4, and minor thirds are 6 to 5 and so on. For example, a major third above A 440 would be the C# at 550 Hz. A major chord is a 4:5:6 ratio. So a major chord with A 440 as the root would have C# 550 and E 660.
Now we can discuss the most interesting part of this, and the main reason for going into so much detail about these harmonics. This series of notes tells us what resultant note will be created when we play any interval.
Resultants
When a unison relationship is played, the resultant is another unison. When two notes in any other relationship are played together, a third note (different from either of the two original pitches) is created. There are two types of resultants: summation and difference tones. Here we are primarily concerned with difference tones, although summation tones are mentioned in connection with the wave form graph below (Figure 10). The rule for finding the difference tone of an interval is to first find the positions of the notes in the harmonic series, then subtract the lower number from the higher one. You could also subtract the number of the vibrations. The number found would be that of the resultant tone. For the unison that would be 1 minus 1, which equals zero; and so a unison is a special case, since it reproduces itself. Let's find the difference tones for some intervals.
For example, take the interval of a perfect fifth. By referring to Figure 8, we see that the fifth formed by the interval F 88 to C 132 can also be thought of as harmonic 2 to harmonic 3. By subtracting 2 from 3 we get one; and so the resultant tone for this interval would be the 1st harmonic, which is the octave below the lower note of the fifth. Likewise, for the fifth one octave higher, subtract: 6 - 4 = 2. The difference tone for a fifth is always the harmonic which is an octave below the bottom note of the fifth.
Some physicists say that our ears synthesize resultants, that the actual wave for that frequency is not present in the air. Although there is disagreement about how they are produced, the fact is they can be heard. Some people hear these resultants easily, others only rarely. It is true that there are varying abilities to hear these tones initially, but when people know what to listen for and strive to do so, a high degree of ability is acquired quite easily.
An analogy to help in understanding why these difference tones occur is to imagine two groups of people marching side by side. One group goes at 150 steps per minute, and the other at 100. The faster group takes three steps in exactly the same time as it takes the slower group to take two. If both groups start off in step together they will get back into step at intervals of 50 steps per minute (with periods of chaos in between). This is similar to what happens when a perfect fifth (3:2 ratio) is played.
Figure l0a below is a graph of a perfect fifth and its resultant. The heavy line is a C and the lighter unbroken line is a G a fifth above. The resultant is shown as a dotted line. Figure 10b shows how the dotted line from Figure 10a implies a waveform for a note with one half the frequency of the player C, that is, one octave below that C. If you wished to find the summation tones for this interval, you would add the positions of the notes in the harmonic series. In a fifth the 2nd and 3rd harmonics are played, therefore the additive resultant would be the fifth harmonic or E. These additive resultants are often impossible to hear, but when they can be utilized for tuning they are even more exact than the subtractive resultants.
When a chord is played more than one interval is involved, and so more than one resultant is produced. A major chord is a 4:5:6 ratio. Suppose the major chord is the one formed by the C, E and G which are the 4th, 5th and 6th harmonics shown in Figure 8. by subtracting to find the difference tones, we get 6 - 4 = 2, 6 - 5 = l, and 5 - 4 = 1. Therefore, two different resultants would be formed: the C one octave below the C in the chord, and also the C an octave below that. The strength of the lowest C resultant is doubled because it is formed by the difference of two intervals in the chord: G-E, and E-C. If you counted additive resultants, even more notes than this would be found.
When all of the notes of a chord are in tune and all the resultants are in relation with the notes actually played, resultants of resultants are produced. If you have ever-stood between two mirrors that are facing each other exactly and seen the reflections of reflections streaming off into infinity, you can imagine this aspect of a perfectly in tune chord.
Thus far we have been concerned with pure tone tuning only. However, another widely used tuning system is tempered tuning, which was developed around Bach's time to enable keyboards to play in all keys. On a piano, therefore, an E# and F natural are exactly the same. In pure tuning (also called just intonation) there is a small but important difference between the two. Tempered tuning is a valuable tool for all musicians and it is good for us all to be able to play tempered scales. Figure 11 shows scale degrees in cents and tells how pure and tempered tuning scales differ. Although piano tuning is more complex than this, basically a tuner divides a scale into octaves (these octaves are 2:1 ratios as in pure tone tuning) and then divides each octave into 12 equal half steps of 100 cents apiece (1200 total per octave). A tempered perfect fifth is 7 semi-tones, or 700 cents. It is 702 cents by just tuning, a minute difference. A tempered major third is 400 cents, while a pure major third is 386 cents, a rather hefty 14 cents lower or 14% of a semitone. Korg and StroboConn tuners are both built in tempered scales, but if you know how many cents to alter tones up or down you can play either tempered or pure scale degrees using those tuners.
PUTTING THE INFORMATION TO USE
In order to practice tuning using resultants it would be good to have a steady pitch source as mentioned earlier. If you can play a steady tone on your instrument and can play unisons exactly in tune, then you can progress to practicing octaves. Since 2 - 1 = 1, you get a reinforcement of the first harmonic from the resultant. Stop the beats completely; now the octave is exactly in tune. For some octaves you may not be able to hear beats even if the octave is out of tune; or you might not be able to stop the beats you do hear. Try not to get frustrated; sometimes room dimensions, humidity or temperature or a whole set of variables such as reeds will conspire against you. When you have more familiarity and sensitivity for tuning you will probably be able to hear all beats and adjust accordingly.
Once you feel comfortable with octaves, go on to fifths. Stay with these for a while because they offer many positive qualities for achieving good intonation. They are complex enough to be challenging to hear, but consonant enough that when you hit them correctly you will know it.
With someone or something that you know has a steady pitch, play scales at the interval of a perfect fifth. One person will play, say, a C and the other the G above it. Tune the interval and go on to the A to D interval. Tune and go on. Always try to tune each interval perfectly, unless you have decided after trying that you can't. Usually the person on top is the one to keep the pitch steady and the person with the lower note adjusts to match. This is because in a band or orchestra the first player must be free to find the prevailing pitch and then his or her section must match that. In chamber groups, for example a string quartet or brass or woodwind quintet, the pitch generally comes from the bottom, but the usual rule for orchestral and band musicians is, "Right or wrong, the first player is right." This does not mean that players in the section should not feel the right to disagree when needed. Sometimes even fine players have bad notes on their instruments and cannot get to the pitch of a colleague. All instruments are built somewhat out of tune, because in essence they are compromises. It takes sensitive players to subtly alter their instruments' pitches to play perfectly in tune.
The exercise that is given above will help you to hear and adapt your pitch so that you will have the facility for more difficult intervals. Ten minutes a day for a few weeks spent on this exercise will do more for your intonation than almost anything else. Be sure to practice this playing not only the top note, but also the bottom. They are both difficult in their own way.
Once you have mastered that exercise, try this one. You will need three players. This exercise takes very sensitive hearing and real cooperation. Play a major chord and try to get it in tune. Remember that the true tone major third is 14 cents lower than its tempered brother, so chances are that if the chord is out of tune but the root and fifth are in tune with each other, the third is high. Once the chord is in tune, the person playing the third should go down a half step to make a minor chord. Interestingly, true tuned major thirds are, as stated, lower than tempered but minor thirds are high. This is just the opposite of what logic would dictate. A minor third tempered is 300 cents, true tone is 316 or 16 cents higher. So the difference between a minor third and a major third is only 70 cents or .7 of a semitone.
After you have tuned the minor chord, the fifth should go down a half step, which produces a diminished chord. Tune. Now the root goes down one half step. Now you have a major chord one half step below the one you began with.
Continue with each person going down chromatically in turn, always staying at any chord until it is perfectly in tune with no beats. You will find that the minor and diminished chords are more difficult to tune than the major chords, but with practice you will be able to tune them also.
In doing this exercise you will realize that even though you have gotten your note and are holding it, this does not mean that you will be exactly in tune when the chord changes. All players must always be listening and adjusting. But you must adjust in moderation. If everyone is wandering around the note, then again it is like a dog chasing its own tail, and chaos is the result. You must be able to hear what it is that you want from the resultants and then subtly vary your pitch to achieve it. Usually at this degree of exactness you will all be tuning to the root of the chord, but all three players must be sensitive and adjust accordingly.
By knowing what to listen for and informed practicing, you will be able to play at a level of precise intonation that you could not have imagined only a short time before.
TWELVE TIPS FOR BETTER INTONATION
1. When taking a chord apart in order to tune it, you should always tune the most consonant intervals first and then go on to the more complex ones later. For instance, in a major seventh chord you should tune the roots first (this includes all octave doublings of the root pitch), then the fifths, then the thirds, and finally the sevenths. If the fifths are not in tune, then the thirds cannot possibly be.
2. People often jokingly say, "When in doubt use vibrato." This is actually good advice if used in moderation. When a group is having severe intonation problems it is a good idea to use a pitch vibrato. At least you will muddle up the bad sounds and will be in tune with someone at least part of the time.
3. Many players use pitch vibrato at all times. This is perfectly fine if it fits the musical style except when playing the tuning pitch. But the middle of the vibrato should be the exact middle of the pitch. Often players will make the bottom of their vibrato be the middle of the pitch. This means that some of the time they are in tune, but to a sensitive ear they are sharp.
4. For atonal music equal temperament is the preferred tuning method. Pure tuning is usually better in relatively diatonic, consonant music. In consonant music that is modulating in a chromatic manner, often the bass line will move in a relatively tempered manner and the treble lines will then tune to that using pure tuning. What mainly is heard is the relationships between the intervals, and if this is well in tune, the less precise tempered movement of the bass line will not be apparent. If the bass were to use only pure tone tuning in, say, mediant modulations, where they could be 14 cents low in relation to tempered notes, after only a few chord changes they would be hopelessly off pitch. Going back and forth between different tuning methods is an intuitive act which is part of good musicianship. The end result will be that which sounds the best.
5. Transcriptions of Bach Chorales have been made for large and small groups Playing them is an excellent way to improve overall intonation. If tuning is not good, fix the cadences by building the chords from the simple to complex intervals as mentioned in tip #1. The sensitivity required to play these chorales well will carry over to improve all aspects of performance.
6. Figure 11a below gives the deviation in cents from equal temperament.
Figure 11b is a list of the ratio of intervals and their number of cents. Note that the pure minor seventh can be played either 18 cents sharp (1018 cents above the root) or else 29 cents flat (971 cents). This latter placement is the same as that of the 7th harmonic in the harmonic series (see Figure 8, page 8), but it is so far flat that most instruments cannot get down to it, so 18 cents sharp, which is also acceptable, is used.
7. This is an illustration of the soundwaves and spectra of a tuning fork, clarinet and cornet. The reason the waves look different is that each instrument has a characteristic spectrum. The spectrum of a wave refers to the relative strengths of the overtones (fundamental and upper partials) that are present in that wave. That is why different instruments have characteristic sounds.
Waveforms
of tuning fork (top),
clarinet, and cornet, each at a frequency of
440, and at approximately the same intensity
8. It is good to know the pitch tendencies of certain instruments. For instance, clarinets tend to play intervals that are smaller at the extremes, i.e., flat at the top of its range, sharp at the bottom. Flutes, on the other hand, tend to expand their intervals at the extremes of range. Also some instruments tend to get sharp the louder they play, while others get flatter. A good player will compensate to minimize these tendencies. If problems still exist when playing with one of these instruments, however, you will at least have some idea how to work to fix the chords.
9. Pitch varies with temperature. Too hot or too cold will also cause intonation problems.
10. Since good hearing is important, it is a good idea to carefully clean your ears with a cotton swab or Q-tip. If you suspect that your hearing is not good, have it checked by a trained practitioner.
11. Scott Brubaker, who has played 2nd horn in the Metropolitan Opera orchestra since 1971 (certainly one of the best orchestras in the world and one that plays with exceptional intonation) made an excellent observation. He recommends spending a great deal of time practicing with a Korg or Boss Tuner, the kind that shows where your pitch is with lights and/or a meter. He recommends that you try to pin the meter exactly on the middle (0) on every note (precisely in-tune in equal temperament). When faced with the statement that some parts of the scale are various cents up or down from 0, he said something to the effect that, "if you are exactly in tune on the tempered scale, nobody is going to bother you about your intonation". In other words, that is a great place to start, and then you can shade for everything depending on the subtleties of what you hear later.
12. You might find yourself playing in a group that cannot play in tune. One of the curses of developing a better ear is that you will be more sensitive when things are out of tune. Try to keep a positive outlook. If the interference patterns are a quagmire of dissonance when they should be consonant, try not to be too disturbed. Pick someone and concentrate on playing in tune with that person; or play in equal temperament to minimize the instability.
However, after learning and applying these principles, you will certainly find yourself performing in groups that play with better intonation, and when there are difficulties you will be able to make positive suggestions to correct problems. After improving their intonation, musicians often find that pieces that have never gone well are suddenly presentable. Another bonus is that the sensitivity involved with good intonation carries over to all aspects of playing. One who is sensitive to pitch will be likely to be sensitive to colleagues' dynamics and phrasing. Listening creatively and knowing what to do with that information makes for much better playing at all levels.
This paper largely grew out of discussions with Mark Eubanks, 1st bassoonist of the Oregon Symphony. He had information to verbalize concepts that for me had been mostly intuitive. A number of his illustrations have been used. He also went over a rough draft of this paper and provided useful comments. Lewis Hayler, musician and physicist, also sent material used in this paper and provided many useful ideas. He drew Figures 10a and 10b which show the relationship between the notes of a perfect fifth. Dr. Lee C. Hein gave valuable physiological information. Lois Landin drew Figure 3. Alicyn Warren provided much needed editing, typing and technical information also.
Some of the best material that has been written about the physics of resultants is a series of two articles by Christopher Leuba, in the Brass Bulletin (#32, 1980, and #33, 1981). These articles go deeply into the actual computations for a thorough understanding of matters only alluded to in this paper. Lewis Hayler sent me excerpts from three physics books that discuss acoustics. They are:
John Rigden, Physics and the Sound of Music
Wilmer Bartholomew, Acoustics of Music (© 1945)
Llewelyn Lloyd, Music and Sound (© 1938)
Figure 8 is taken from The Horn and its Inner Acoustics by Dr. Willi Albi, published by Schilke Company. Alicyn Warren loaned me the book from which many of these illustrations and ideas are taken. It is:
Charles A. Culver, Musical Acoustics (© 1956), McGraw-Hill Book Co. Itzhak Bentov's Stalking the Wild Pendulum (© 1979) has interesting material about interference patterns and beat frequencies.
Richard Hein was for 25 years a free lance French hornist living in New York City. He played in various orchestras and chamber groups, made a number of recordings and toured with various musical groups throughout the United States, Canada and Europe. He still plays with the Linden Woodwind Quintet. He also taught privately and at various schools. He is a graduate of the Manhattan School of Music, where he had a full scholarship to study with Richard C. Moore. He now makes his living as a computer consultant in the New York City area.