INTRODUCTION

Good intonation is essential to good musical performances. A performance can not be successful without a reasonably good expression of intonation. When dealing with tonal music, good intonation lays the foundation of the musical meaning. Some musicians instinctively play with good intonation, while others need assistance. There are certain facts, besides our direct sense of intonation, that would help musicians to overcome problems of intonation. Among these the knowledge of temperament is most helpful. Also, to understand the role of the difference tone (a third tone produced when two tones sound simultaneously) in tonal music will facilitate making tuning decisions.

Even though there are at least half a dozen different tuning systems known to Western musicians, only three of them have real and persistent influences on tonal music: the Pythagorean, Just Intonation, and Equal Temperament. The first part of this study gives some information which instrumentlists could utilize as parameters to adjust the pitch in various situations. A study of the difference tone and the tonal music based on mathematical methods and our physical perception is presented in the second part.

PART ONE
THREE INFLUENTIAL TUNING SYSTEMS --
STRUCTURE AND ACOUSTICS

Overview

Most instrumental musicians tune by ear. Today, even though there have been many advances in technology, human ears are still the most reliable instruments to determine whether or not intervals seem to be in tune. Theory alone does not completely address this matter of tuning. While it fails to account totally for human hearing in setting up tuning systems, theory does explain some of the criteria of music from different periods as well as demonstrate different musical means for obtaining the desired effects. It has been proved that adjustments are necessary to the tuning technicians' ear e.g. with the phenomenon of the "octave stretch" in tuning keyboards.

This chapter presents a study concerning the development of three tuning systems: the Pythagorean, Just Temperament and Equal Temperament. Since each of the three systems has its own tuning problems, other temperaments, such as the mean-tone and irregular temperaments, were also developed to improve the acoustics in playing tonal music---especially for keyboard and fretted instruments. Instrumental music, however, does not follow such settings. Thus, it is beneficial for musicians to acquaint themselves with this knowledge. Equipped with these tools, they can more easily spot and solve tuning problems that tend to be tricky, if not mystical.

Pythagorean

The Pythagorean is the oldest tuning system in European music heritage. According to legend, "Pythagorus discovered concords in the simple ratios among the divisions of a sounding string." Because the ancient Greeks believed numbers were the answer to all things, including the physical and the spiritual, music was assigned with numbers. They even tried to explain universal harmony with it: ". . . so the system of musical sounds and rhythms, being ordered by numbers, exemplified the harmony of the cosmos and corresponded to it."

The ancient Greeks recognized three primary concords: fourth, fifth and octave. The corresponding ratios are:

If a vibrating string or air column sounds a C at its full length, it will sound an F with three quarters of its length, G with two thirds and C an octave higher with half of its length. Because intervals with these ratios do not cause beats, they are called pure intervals. Pythagorean scales are built upon pure fourths and fifths.

The ratios mentioned above are related to the string length, but modern acoustics prefers to use the ratios of frequency. To convert the length into frequency ratios, just invert them. Thus, in relation to the fundamental tone, the frequency ratios of the fourth, fifth and octave would be:

This method will be used throughout this document.

A basic way of constructing a Pythagorean scale is to stack up twelve fifths from a given note. Using F, the result produces all of the twelve notes for a complete chromatic scale:

F C G D A E B F# C# G# D# A#

If we set the value of C with the number one, the frequency ratios of this scale are as follows:

According to this specific Pythagorean scale, all the fifths are pure except one wolf fifth, between Bb (A#) and F. This impure fifth is a Pythagorean comma smaller than a pure fifth (about 23 cents) in order to end the scale at a C, which is exactly one octave above the starting C. In other words, each pure fifth is about two cents too large. Depending on the practical situation, i.e. where one starts the circle of fifths, the wolf fifth can occur anywhere in the scale.

Pythagorean tuning was developed at a time when melodic writing prevailed. Lindley says: "Among regular tuning systems Pythagorean intonation has the largest major 2nds and 3rds. Melodically the large major 2nds are handsome and the incisiveness of the small minor 2nds is of potential expressive value." Thus he suggests that the Pythagorean scale has its charm in expressing music linearly. Lloyd echoes this assumption: "It would appear that the scale used in plainsong and by string players in unaccompanied melody tends to approximate to this type of intonation." The Pythagorean scale tends to exaggerate all of the major and minor intervals, not just the seconds and thirds. The leading tone effect is also more satisfying with the narrow minor second: ". . . and tend[s] to 'anticipate' resolution of dissonance through semitone steps."

Just Temperament

The term 'Just Temperament' is not to be confused with 'Just Intonation.' The former refers to a scheme of tuning, and the latter means "playing or singing in tune." Even though many theorists had believed that hearing satisfaction could be had by the arithmetical relationships among intervals, scientific research revealed a different view. Our ears tend to fall on the side of justly tuned intervals, but it is only an approximation.

Many theorists believe that the harmonic series (Example 1) is the basis of tonality, but some dispute this for various reasons. The theory of harmonics, however, gives us a fundamental understanding of some of the tuning criteria. Also, the difference tone theory is best explained with the intervallic ratios derived from the harmonic series.

Example 1. Natural Harmonic Series (from the 1st through 16th)

From the first sixteen harmonics, one can compose the following tonal intervals with just ratios:

Two Just scales, major and minor, can be established with these data:

Two intervals, the major second and minor seventh, can be determined with different ratios. Most musicians will choose the one that achieves greater listening pleasure. For example, in a major key, using the larger second (9/8) keeps all three major triads as pure whereas using the smaller one (10/9) keeps all three minor triads pure.

The Just Temperament set up by the above procedure can produce a pleasing harmony because all the intervals are pure. When modulation is involved, some intervals will go out of tune. The more distant the keys, the more intervals will be out of tune. For example, if it modulates from C major to G major, the ratios of the intervals will be:

The only interval changed is the major second. If it modulates to D major, the ratios will look like this:

Now, there are two intervals changed, the major second and the perfect fifth. This shows why Just temperament is not a perfect tuning solution when the modulation occurs.

Equal Temperament

Compared to the other two tuning systems, Equal Temperament is quite young, but it is by no means less important. J. S. Bach's Well-Tempered Clavier (1722) was one of the first compositions to demonstrate the power of Equal Temperament and flexibility of modulation. It gave much confidence and encouragement to the followers of Equal Temperament, although "well-tempered" can be either "good or nearly equal temperament, as well as truly equal temperament."

The basic idea of building a scale with twelve equal semitones in an octave is to have a ratio that can reach the value of two after being applied twelve times. In 1581 Vicenzo Galilei suggested an already known ratio 18/17 to be used on the lute. If we take an Equal Temperament semitone as 100 cents, an octave should have 1200 cents. After mathematical conversion, the figure 18/17 equals 98.95 cents, practically indiscernible from 100 cents. Later, with advanced mathematics, this ratio was set to be (approximately 1.0595). The result of using this method is a group of mistuned intervals except the octave in exchange for freedom of modulation. Appendix B shows clearly the discrepancies that Equal Temperament has in tuning tonal intervals.

Instrumental musicians do not stay with a fixed temperament since all the tuning systems have certain intonation defects. They compensate the pitch differences with the help of their ear. Since it is a lot easier to have a reference scale setup as a starting point rather than having to skip around when modulating through different keys, Equal Temperament seems to be the best choice as its setting never varies in any key. Musicians need only to know the relative deviations from the pure tonal intervals to the corresponding Equal Temperament ones, then adjust according to their skill and practical needs.

Summary

For musicians, their common goal is to play "intervals of concords in tune with ideal accuracy." It is the Just Intonation, associated with pure intervals, that satisfy their ear. A rigid temperament just would not fulfill all the musical needs. Equal Temperament was designed for expediency, i.e., for modulating easily on a keyboard. It is the desired tuning because a key change does not effect its setting in any respect. Just Temperament gives much more pleasing chordal effects in tonal harmony (only in the keys it is set up for), but for melodic expression, the Pythagorean is superior.

Obtaining perfect intonation can be a complicated task. There are other factors affecting intonation such as pitch shifting, intensity of a tone, masking, and octave stretch. Learning variations in tuning equips musicians with the facility to handle different situations. This does not guarantee success, but allows them to be more successful.

 PART TWO
USING THE DIFFERENCE TONE AS THE PARAMETER OF INTONATION
IN
PLAYING TONAL MUSIC

Theory of The Difference Tone

The difference tone and the sum tone are called 'combination tones' or 'resultant tones' in musical acoustics. They are both produced when two tones sound simultaneously. The difference tone was discovered as early as 1714 by Tartini, but was first described in G. A. Sorge's book, Vorgemach der musicalischen Composition, in 1745-47. It was then recognized as Tartini's tone.

The combination tones are produced by the non-linearity of our ear. Acoustically they do not exist; the scientific instruments can not detect them. Mathematically, the frequency of the sum tone can be attained by adding the main frequencies of the two initial tones; the frequency of the difference tone is the difference of the two. For example, two sounds f1 and f2, with f2 being higher, will produce a sum tone at ( f1+ f2 ) and a difference tone at ( f2 - f1 ). As a matter of fact, the first overtone of the lower frequency (twice the frequency of the fundamental, 2f1) also produces a distinctive difference tone with the higher frequency f2, which is pitched at ( 2f1 - f2 ) and called the cubic difference tone. Because the sum tone is usually not as audible as the difference tone (some scholars even agree that sum tones do not exist), its effect in the tonal structure can be neglected.

Let us look at an example: if there are two sounds with frequencies of 400 Hz and 500 Hz respectively, the frequency of the sum tone would be 400+500=900 Hz and the simple difference tone would be 500-400=100 Hz. The frequency of the cubic difference tone would be 2x400-500=300 Hz.

To make the process of getting difference tones theoretically from tonal intervals more comprehensive, using the ratios mentioned in the first chapter would be the most advantageous as it simplifies the numbers and shows the relationship of the difference tones in the tonal structure. The frequency is useful for comparing the difference between temperaments. If one wants to know the corresponding frequencies, multiply the ratios with the selected tonic frequency to get the desired answers. Example 2 is a table of converting the ratios of notes into frequencies in the key of C according to Just Intonation and is based on c'=264.

Example 2. Table of Conversion from Ratios to Frequencies in the
Key of C (c'=264 Hz) using Just Temperament

Difference Tone's Tonal Implications

Chords Produced by Tonal Intervals

In tonal music structure there is a center in each key called tonic. The resulting acoustics of other individual notes along with the tonic form an interesting study. With some calculations, it is found that every tonal interval which involves the tonic produces a precise difference tone or cubic difference tone to match the tonal content. In other words, the frequencies of the difference tones give definitions to tonal intervals. Paul Hindemith in his The Craft of Musical Composition uses the difference tones to analyze the composition of all the tonal chords. Example 3 and Example 4 show how the difference tones fit in the tonal structure. The C is consistently used as the lower note because of its tonic function.

Example 3. Table of Simple and Cubic Difference Tones
Derived from Tonal Intervals.

Example 4. Example 3. Diagram of Simple and Cubic Difference Tones
Derived from Tonal Intervals.

The process of finding out the corresponding pitches of difference tones is purely arithmetical. If by multiplying two to a ratio we can match a note in the current octave range, then the difference tone has the same note name but would be an octave lower. If four can satisfy the same process, the real difference tone would be two octaves lower. Without much trouble, using the number eight would result in a distance of three octaves.

One cannot dispute that, from this revelation, the tonal structure and the harmonic series compliment each other's existence, because all the simple difference tones and cubic difference tones of tonal intervals fit perfectly into that framework. Even the flat Bb, which appears twice, and the flat Ab coincide with the ratios in C harmonic series.

Minor Triad and Its Tonal Origin

The harmonic series has been regarded as the origin of the tonality since François Rameau (1683-1764). The fourth, fifth and sixth harmonics derived from C harmonic series form a major triad, and the tenth, twelfth and fifteenth harmonics compose a minor triad. To some scholars, there are doubts about establishing the status of the minor triad in the tonality with the harmonic theory. The difference tone theory lends weight to that question.

From Example 5 we can observe that the major triad C-E-G in a C harmonic series generates many Cs in various registers and a G (C-E generates a C and a G; E-G generates two Cs; C--G generates two Cs). With such consistency, the stability and smoothness of this chord is predictable. The minor triad E-G-B derived from the same harmonic series also bears a root C but a more complex chord: E-G generates two Cs; G-B generates a G and a D; E-B generates two Es. It is a C major ninth chord in its first inversion (Example 6).

The result of the above discussion hints to us as to why music in a minor key will often end with a Picardy Third (a major triad), but it is rare to find music in a major key which ends on a minor triad.

Example 5. Chord Produced by C Major Triad.

Example 6. Chord Produced by E Minor Triad.

The Effect of The Difference Tones

The combination tones are generally inaudible, but flute and recorder players often experience this phenomenon in their duets. The coloratura and obbligato flute combination frequently has this trio effect. From theoretical calculations and chordal analysis we learned that the difference tone is not only a physiological phenomenon but also a musical one. The flute parts from the twenty fourth through thirty second measures in Beethoven's Leonore Overture No. 3 demonstrate the importance of good intonation which produces matching difference tones in the chordal structure.

Problems

Each tonal interval produces specific difference tones. That means the difference tones could create conflict with these chords of which they are not a part, although it is not common. John Krell mentions a good example in Kincadiana with the problems that are raised by the difference tones in the final chords of Strauss' Thus Spake Zarathustra.

Summary

The purpose of this paper does not lie in explaining the origin of harmonic tonality but serves as a reference for playing pure tonal harmony. There is an observation that can be easily made: The difference tone approach in playing tonal music corresponds to the principles laid out by Just Intonation. When pure intervals are being played, the difference tones, if audible, often strengthen tonality.

CONCLUSION

Most instrumentalists adjust the intonation by their instinct which has been acquired through experience. This study provides knowledge that should allow them to speed up the learning process. The discussions of various tuning systems show what is expected in order to achieve certain desired harmonic effects. In other words, particular interval distances can be determined without much confusion. Even though they cannot be heard very often in ensemble, the difference tones create a chordal basis for a harmonious, if not pure, acoustics.

There is no one perfect solution to playing tonal music in tune; each temperament has its own deficiencies just as each individual instrument has intonation defects. Scientific instruments, such as electronic tuners and oscilloscopes, cannot guarantee that musicians flawlessly match the intonation in ensemble situations. It is always up to the players' ear to decide a satisfying pitch at a split moment.

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