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Temperament and the Saxophone Quartet

by SCPO Timothy E. Roberts

  • Introduction
  • Physics of Tuning
  • Tuning Systems
  • Conclusion
  • Glossary of Terms



The great variance in tone color and pitch among the four voices of the saxophone quartet can cause some musical directors to look unfavorably upon our unique genre of chamber music. Many band directors, unwilling or perhaps unprepared to meet the challenges of saxophone quartet tuning, may have decided to solve the problem by eliminating the quartet - or some of its voices - from their ensemble. Few bands regularly use the soprano saxophone, perhaps as a result of frustration or a lack of experience in improving the pitch of the saxophone quartet. Rather than avoiding programming the saxophone quartet in chamber concerts or steering clear of band arrangements that highlight the saxophone ensemble, we should try to understand and solve their unique pitch discrepancies.


Let's begin with a brief history of pitch standards and how they apply to the saxophone quartet. Musicologists can only trace pitch standards back to the late seventeenth and early eighteenth centuries. At that time, three standards existed: choir pitch, cornett pitch, and chamber pitch. Chamber pitch, or Kammerton, was the pitch standard most frequently employed by the instrumental wind and small chamber ensembles of the Baroque and early Classical periods. Kammerton was usually a half step lower than today's standard pitch, but it could vary widely in different places and at different times. The chamber pitch of a Mozart wind quintet performed in Salzburg could vary as much as a whole-tone from the pitch standard of a Bach Brandenburg Concerto performed a year later in Leipzig.


The standard represented by chamber pitch rose steadily in the early nineteenth century. In 1858 the Paris Academy recommended that the pitch of the A above middle C be fixed at 435 vibrations per second. In 1890, a musicology conference in Vienna gave international recognition to this standard. Over the last 100 years the standard slowly and indiscriminately rose, and now the A above middle C for all practical purposes is fixed at 440 vibrations per second. (Today there are some orchestras in Europe and the United States that tune to A=442, 443, or 444.) With today's technology, we can set a pitch standard of A= 440 for our quartets with the use of a simple and inexpensive electronic tuner.

Physics of Tuning

Now that we understand and have established a pitch standard for our ensemble, we can calculate the correct intervals for all the notes of all four quartet instruments. Various methods have been used over the centuries to determine these relative pitches. Just intonation is one system of tuning in which all the intervals are derived from the natural (pure) fifth and third. While most discussions of pitch and tuning use a vibrating string as an example, we'll use the saxophone family of instruments.


Pitch is produced by the frequency of the vibrations of the saxophone's air column. The key to understanding this concept is, the vibration's frequency is in inverse proportion to the vibrating length. Therefore, if the whole air column produces a sound vibrating at the frequency of 400, then halving the air column will produce a sound vibrating at twice the frequency, 800. An air stream that is two-thirds the length of the original column of air produces the frequency 400 x 3/2 or 600.


Musically, what are the relationships of the vibrations to the sounds produced? To illustrate, an air stream vibrating within the full length of the instrument produces the baritone saxophone's low Bb. Fingering a third line Bb halves the length of the air stream, thereby doubling the frequency of the vibration and producing an octave. Because the alto saxophone sounds one octave higher than the baritone, its low Bb also vibrates at twice the frequency of the baritone's low Bb. Fingering a bottom space F on the baritone (or playing a low Bb on the tenor) shortens the air stream by a third, resulting in 2/3 of the air stream vibrating. Using the rule above, the new note will vibrate 3/2 quicker than the full-length baritone air stream, producing a fifth.


Now, finger a low Eb to reduce the baritone's air stream by 1/4, leaving 3/4 of the air column vibrating 4/3 faster than the original to produce a fourth. Reduce the air stream by a fifth, leaving 4/5 of the air stream vibrating 5/4 faster to sound the low D, a major third above the original Bb. Knowing that the relationship of the saxophone voices above the baritone is a fifth (tenor), octave (alto), and an octave plus a fifth (soprano), intervals on each instrument and between instruments can be calculated by combining the ratios of each basic interval on the following chart:


Baritone/Alto Tuning Ratios


Baritone/Alto Tuning Ratios


Tuning Systems


Aside from being very technical and rather complicated, this early system was difficult to apply in actual practice. Just intonation results in intervals that sound "purer" than those in our present tuning system. (They haven't been "tempered.") Complications arise because interval ratios don't always remain constant. The vibration rate (ratio) of the major third (Bb to D) mentioned above is 5/4. The ratio of the major second, Bb to C, is 9/8. But, still using the low Bb as our point of reference, the ratio of a different major second, C to D, is NOT 9/8, but 10/9 (5/4 divided by 9/8)! Additionally, not all fifths are pure; calculating from the same Bb baritone note, the fifth of C to G has a ratio of 40/27 instead of the 3/2 frequency ratio. These types of discrepancies made modulation extremely difficult in the system of just intonation.


Pythagorean tuning, invented by the Greek philosopher Pythagoras of the sixth century BC, involves creating the various tones of a scale by placing successive spirals of fifths on top of each other (or, creating a chain of 3/2 ratios. Remember that 3/2 is the ratio of the fifth.) Although this method works reasonably well for diatonic melodies, its flaws are displayed when chromatic notes are introduced. Rather than a circle of fifths as we know it today, Pythagoras' spiral of fifths made each successive fifth a tiny bit sharper in pitch than the last. Enharmonic notes (B# and C, F# and Gb) are identical in pitch in our present system of tuning. However, they are almost a quarter tone apart in the Pythagorean method. Notice in the chart that the major third in the just intonation scale is represented by a ratio of 5/4, while the Pythagorean major third is 81/64. The difference between these two major thirds is represented by a ratio of 81/80, enough to make anyone with sensitive ears uncomfortable.


When we use the term temperament today, we refer to compromising or bringing together the tuning of the acoustically correct, natural systems discussed above. The best system of temperament was introduced in 1690 by Andreas Werkmeister and put into practical use by Bach's "Well-Tempered Clavier." It was not universally accepted in continental Europe until the mid-nineteenth century (interestingly concurrent with Adolphe Sax's invention of the saxophone!). Commonly referred to as equal temperament, it divides the octave into twelve half steps of equal size. Enharmonics such as B# and C are now expressed as an equal 1/1 ratio. Thus, no one key is favored at the expense of another, and modulations and accidentals may be employed without conflict. Notice that except for the octave, none of the intervals in equal temperament are identical with their counterparts in just intonation or Pythagorean tuning. Our contemporary ears have grown accustomed to these compromises of equal temperament.





Understanding these principals is one of the first and most important steps in helping today's saxophone quartets overcome the problems of properly tuning four unique-sounding instruments. Still, some superficial, problematic factors affect the saxophone's pitch. Leaky pads, improperly adjusted key openings, poor mouthpieces with unsuitable bores or warped facings, reeds that are too soft or too hard (requiring unnecessary embouchure adjustments), lack of breath support, tightening of the throat or embouchure in the high register, and loosening of the throat or embouchure in the low register are all recipes for ensemble pitch disasters.


Unless each member of the quartet has perfected the ability to discriminate between good and bad pitch in rehearsal and performance - and there are some who have - we can help enhance the overall musical outcome by simply listening to each other. When working on a section of music, avoid using only one note as a pitch reference. Slow triads, scales, simple melodic lines, and short Bach chorales will help your quartet develop an acute sense of pitch discrimination. With proper preparation and practice, you will find that a quartet that tunes together stays together.



Glossary of Terms


a sign of chromatic alteration used for single notes or measures, as opposed to those given in the key signature. The English names of accidentals are sharp, flat, double sharp, double flat, and natural.
music of the period c. 1600-1750. The Baroque era began with the rise of monody, opera, oratorio, cantata, and recitative. It closed with the deaths of Bach and Handel.
chamber pitch
(Kammerton)- the absolute pitch of a specific note, standardized in order to obtain identical pitches on all instruments. Today's standard pitch is usually a'=440.
choir pitch
(Chorton)- a pitch standard used during the Baroque period for church organs and sacred choral music.
the use of tones outside of the diatonic scale, for example, in the key of C: C, D, D#, F# rather than the diatonic C, D, E, F.
classical period
generally refers to the period from 1770-1830 during which the composers Haydn, Mozart, Beethoven, Schubert were active. Its complex development of style and form began around 1740 with composers in Mannheim, Germany, Vienna, Italy, and Bohemia.
cornet pitch
(Cornett-ton)- the pitch standard practiced before 1889 which town musicians used for their brass instruments.
refers to the natural scale consisting of five whole tones and two semitones as found within a key. For example, the C major scale of C, D, E, F, G, A, B, C is diatonic.
tones which are the same degree of the chromatic scale, but which are "spelled" differently, e.g. G# and Ab; and C# and Db.
equal temperament
the system of temperament in which the octave is divided into 12 equal semitones.
the difference in pitch between two notes.
denotes the singing or playing in tune as either good or bad; the matching (or lack thereof) of pitches with another player or singer.
just intonation
a method of tuning and intonation in which all the intervals are derived from the natural (pure) fifth and natural (pure) third.
see chamber pitch.
the change of key within a composition.
Pythagorean tuning
tuning - the tuning system based on the Pythagorean scale which derived all the tones from the interval of the pure fifth (3/2).
refers to a system of tuning in which the intervals deviate from the "pure," such as the acoustically correct intervals used in the Pythagorean scale and in just intonation.