The following article was published in the Summer 1997 issue of the ITA Journal, Vol. 25, No. 3, pp 25-27.  It was edited for this page in April 2001.

THOUGHTS ON INTONATION

by

Michael Keener

Intonation is that mysterious, sometimes elusive concept that we as performers and teachers must face daily. Now, I realize that a discussion about intonation can fill volumes, and the can of worms that I am opening will go on forever. However, I will try to convey some of my ideas on the subject in the hope that we can talk more about a most important topic, which seems not to get much discussion except the often-heard phrases such as “play in tune” or “you’re out of tune.”  (I think that those kinds of comments sometimes come from those who can hear poor intonation but know neither what to do about it nor how to assist the members of an ensemble in improving their intonation.)

For as long as I can remember, I have known that the rules of intonation seem to be constantly changing, depending on the type of ensemble one is a part of or whether a fixed-pitch instrument (piano, organ, etc.) is involved. The main problem was that I did not know why the rules changed, or how to anticipate the changes. Since ignorance is, at times, bliss, I was able to fit into the situation, sometimes struggling and unsure, but usually able to deal with the situation.

One key to survival is the makeup of the ensemble. When performing with like-minded colleagues, the problems seem to work themselves out. When with a group who are unable to discern the finer points of intonation, the search is on for someone with whom to play in tune, with questionable results due to the simple odds against success.

Perhaps a definition would be a good starting point. Webster’s New World Dictionary (1990) defines intonation as, (2) “the manner of producing tones with regard to a given standard of pitch.” My Harvard Dictionary of Music (seventh printing, 1966) indicates, (1) “in ensemble performance, intonation denotes the singing or playing in tune, either as good or bad.” These seem to be some fair beginning points, but as we shall see, they also seem a bit simplistic, as well as vague. As usual, there is much more to the subject than initially meets the eye, or ear, in this case.    

 INTONATION AND THE PIANO

First let’s talk about fixed-pitch instruments. For the past eleven years my “money gig” has been tuning and servicing pianos. When learning to tune pianos, my experience on the trombone and knowledge of the uses of the harmonic series put me at a definite advantage over others in my class. At the same time, many aspects of trombone playing became clearer as I learned more about the piano and the production and components of piano tone and voicing. In a larger sense, I recognized what I refer to as a “universal law.” Also being an amateur radio operator, I had studied radio frequency electromagnetic waves. My “universal law” occurred to me when, as I learned to tune, I realized that waves of all types (sound, radio, water) act in similar ways. Tuning a piano string is very similar to tuning a radio-transmitting antenna, or an interval in a trombone duet, for that matter.  

An important component in  the description of piano tone production is the concept that the piano is not a harmonic instrument. As a matter of fact, a term frequently used among piano technicians is “inharmonicity.” Inharmonicity, in pianos, refers to any deviation from the theoretically perfect vibration of a string. Reasons for such deviation include the ratio of the length of a wire in proportion to its diameter, and the lack of true termination points of the speaking length of the wire that results in a constantly changing length of the wire.¹ In fact, inharmonicity in pianos deviates from the theoretical to a degree that we do not refer to harmonics, but partials, and the partial series. The partials make up the whole sound of a given note as the wire vibrates along its entire length while simultaneously dividing into segments of 1/2, 1/3, 1/4, 1/5, and so forth, of its full length.

One of the techniques that a tuner must use to make the piano sound in tune with itself, and compensate for inharmonicity, is to stretch the octaves both above and below the center, or temperament octave. This results in octaves that are slightly wider than a pure or “just-tuned" octaves. The impact that inharmonicity has on performers who must play our instruments with the piano is that we must hear the difference and adjust our intonation to fit that of the piano. However, some confusion appears from the difference in the intonation of the melodic line when playing alone in the practice room from the intonation demanded by the piano.

Perhaps a point of equal impact when playing with a fixed pitch instrument is the actual temperament used to tune the piano, or the manner in which the octave is divided among the twelve chromatic tones within the octave. I believe that our ears have, for about one hundred years, been polluted by hearing the equal temperament from pianos, organs, harpsichords, and the like. This constant exposure to a system of tuning in which no interval, save the unison, is pure has conflicted with our natural tendency of producing, and wanting to hear, perfect 5ths, 4ths and octaves, as well as just-tuned (pure) thirds, sixths, seconds and sevenths. What we hear in equal temperament is contracted (narrow) 5ths, expanded (wide) 4ths, expanded major 3rds and major 6ths and contracted minor 3rds and minor 6ths.  Is it any wonder that, as we get farther away from the beginning of the 20th century, we get more confused about just where to place the notes of the scale?

When we play alone or in an ensemble with no fixed pitch instrument involved, for example a Melodious Etude, we generally employ scales based on the melodic aspects of the composition (in a major key, the major diatonic scale), and just intonation (beatless intervals). With these parameters the intervals will vary as to the distance between consecutive notes of the scale. We can also make modulations smoothly and slip into the new key using the same principles. If, however, we are asked to play the same Melodious Etude with a piano that has been tuned to the equal temperament, each semitone will be equidistant, or 100 cents apart.² The obvious problem here for the wind or string player is note placement. Our ear wants to hear a perfect fifth, but the piano is producing a contracted fifth, for example.

Table 1 illustrates the differences in frequencies of the equal tempered scale and the diatonic major scale.³ Note that the percent deviation can be converted into “cents.”  At A-440 (note number 49 on the piano) 1 Hz equals approximately 4 cents. To get an idea of the difference of 4 cents play an A against a pitch standard (a tuning fork or electronic-generated sound) and try to produce a second A that beats once per second against the standard.

 Table 1:

     INTERVAL                RATIO                 FREQUENCIES             % OF SEMITONE DMS 

                 (Step of scale)                                      ET                 DMS             VARIES FROM ET

     I               C40               1:1                261.626        261.626

     II              D42               9:8                293.665        294.328             4% higher

     III             E44               5:4                329.628        327.032             14% lower

     IV             F45               4:3                349.228        348.834              2% lower

     V              G47               3:2               391.995        392.439             2% higher

     VI             A49               5:3                440.000        436.043             16% lower

     VII            B51             15:8                493.883        490.548             12% lower

     VIII          C52               2:1                523.252        523.252                              

                        (Note: ET= equal temperament; DMS= diatonic major scale)

Please don’t take the forgoing as my plea for the total disregard for, or elimination of  the equal temperament. Equal temperament is, at this point in time, probably still the best compromise for tuning a fixed pitch instrument on which will be performed music of many historical periods. My point here is to try to explain some of the problems that we face when playing the trombone, or other wind or stringed instrument, with a piano, organ, and the like. Besides, even without considering the equal temperament, adjustments would be necessary due to inharmonicity, although the deviations seem less in magnitude when one employs the historical temperaments (i.e., meantone or well temperament) which are inherently more harmonic in nature and do include more perfect intervals than the equal temperament.

Just as an aside, allow me to state my opinion that, contrary to what many of us were taught, J.S. Bach did not invent the equal temperament. His work for the “Well Tempered Clavier” was just that -- for a well-tempered instrument. In addition, or probably more central to the work, Bach took advantage of, and exploited the characters of the keys or key colors that are present in a well temperament, and other early temperaments.  Characters of the keys are absent in a truly equal- tempered instrument. The basis of the equal temperament is the equality of the size of the semitones, which results in all keys having the same, generic, character. The equal temperament was probably not practiced before the 20th century,⁴ which begs the question of how a piano should be tuned for performance of compositions written before 1900. (Another topic for another time.)

As you can see, there are problems to be encountered when playing with a fixed pitch instrument. As usual, in order to remedy these problems, we must remain flexible and open to change our regular playing patterns. Or, better yet, flexibility should be an integral part of our regular playing patterns. The piano is not going to adjust to our note placement, so we must be the ones to adjust, or risk a poor review or not being hired again due to poor intonation.

WIND AND STRING ENSEMBLE

Even when we remove the fixed pitch instrument and add other voices the situation remains very complex. We not only need to continue to concentrate on the melodic aspects of intonation, but also need to consider intonation from a harmonic point of view. Our main task, intonation-wise, is to line up the harmonics of the two, or more, voices so that there is no conflict among harmonics (overtones) as well as fundamentals. (See figure 2) When the harmonics do not correspond, we hear the beats produced by the out-of-phase pitches. To pull the interval into tune we must make the beats go away and restore calm to the situation by adjusting the placement of the notes.

Figure 2: The first measure represents the chord played by the section. Measure 2 shows the notes of the chord as whole notes, and the harmonics (overtones) as quarter notes (with no stems). The ties connect the common harmonics of the three notes.

      Once again the bombardment of equal temperament and its non-pure intervals comes into play. The precedents set by years of exposure give many the tendency to play major thirds wider than a just- tuned third (the third too high), and introduces beats that have come to seem correct to many ears.⁵ It seems as though some players think that, if the major third sounds so good, a higher major third will sound even better, especially if the note is the leading tone to the tonic note. Unfortunately, that track is taking us toward a derailment. Even though that tendency seems to encourage the raising of the third, we need to keep the note in the position of purity of the interval and beatless. The note should correspond to the harmonics of the other members of the chord structure that will allow all of the notes to ring as a unit. In so doing, the listener is not asked to endure the noise of the beats within the interval, and the performer can relax and enjoy the more beautiful sound of the interval and easier tone production, through which their individual sound characteristics can reach a fuller resonance.

Part of the problem that we encounter when we take this more harmonic approach is that the placement of the notes changes when the harmonies change. One analogy can be taken from the guitar. A fine-tuning of the guitar can be centered on the key in which the composition is written or begins. When the key changes, the guitar begins to sound out of tune because the notes are have shifted slightly to accommodate the new key, but the guitar, and its frets, have remained unchanged. A slight adjustment of the tuning pegs to clean up the intonation is sufficient until another key change occurs. Obviously, the ear of the performer and the quality of the instrument will determine how tolerable the intonation is at any given moment, however, the need to re-tune does arise. Fortunately, we as trombonists can compensate for these changes by constantly adjusting our pitch, but we need to remain mindful of the requirements and learn to anticipate changes.

BOTTOM LINE AND SUGGESTIONS

As usual, the bottom line for intonation is to believe your ears and adjust accordingly. Sometimes the adjustments might seem radical, but try them anyway. My guess is that many will be astounded by the difference in the actual placement of the slide for that 14 cent lowering of the major third. You might also be astounded by the quietness of that interval, but once you get it into your ear I am confident that you will not have it any other way.

A problem in pursuing this type of study is finding compatriots with whom to take the journey. If you are fortunate enough to have a regular duet partner or be a part of a section, trio or quartet, all involved will benefit. If not, network your way into a situation where you can try some of the concepts, even for the short term.

Another option is to record one line of simple duets and play duets with yourself, slowing the tempo in order to allow time for adjustment of the second line on playback. As you fill in the second line, you will discover that all of the adjustment must take place in the “live” line. If you are like me, you will complain that the “other player” doesn’t meet you half way, but you can learn a lot about your playing. Besides, we all occasionally have to sit next to players who won’t budge off of their notes, don’t we?

As far as suggestions for literature are concerned, I would recommend any simple duets of the type that would be used with beginning level students. Duets with plenty of sustained notes in both parts can give both players time to adjust, or playing any duet at a slower tempo can also work. Remember that, when you go from duets to trios, the addition of one line triples the number of intervals present in the harmony. By playing easy duets, trios and quartets when working on better intonation, the innocent whole note passages can present real challenges, and real rewards.

I would be interested to hear any feedback or ideas from others who have been working on intonation with this approach.

 ENDNOTES:

        1. For further information about inharmonicity see Piano Servicing  and Rebuilding byArthur Reblitz (The Vestal Press, Vestal, NY, 1976.)

        2.    One cent is 1/100 of a semitone, or one percent  of the distance between semitones in equal temperament.

        3.     Leuba, Christopher, A Study of Musical Intonation (Prospect Publications, 1962, revised 1977) p.11.

        4.      Jorgensen, Owen, Tuning (Michigan State University Press, East Lansing, MI, 1991) p. 4-7.

        5.   Keener, David L., conversations and interviews.

The Intonation Tutor