Three Systems of Intonation

June 11th, 2013 | Posted by BaylaK in Goal 01: Sounds | Goal 02: Intonation | Practicing

Introduction

This article is a brief summary of the three primary intonation systems used by violinists today, and a suggestion for an ideal compromise tuning for the violin which will facilitate tuning doublestops as well as tuning to other instruments.

Good intonation is both an acoustic reality, with notes “agreeing” (i.e., creating overtones or undertones) with other notes being sounded simultaneously, and a kind of societal compromise, with musicians and audiences growing accustomed to and agreeing upon a certain delineation of pitches. Professional musicians must have finely tuned ears and accurate fingers, so that they can adapt to the intonation system being used by the players with whom they are playing; but true artists must understand functional harmony as well, at times choosing to shade a note up or down to create a certain tension or relaxation.

In my job at Boston University, I therefore find it helpful to teach three of the most used and useful intonation systems: expressive, equal-tempered, and just. My students learn to hear the clear changes in interval sizes and emotional moods when these different systems are employed.

I begin with descriptions of the two older systems still relevant for violinists today.

 

I. Historical systems of tuning

There are two important sets of musical relationships, both based on the overtone series, which are particularly relevant for the modern violinist.  The first, discovered by the Greek mathematician Pythagoras in around 530 B.C.E., was that certain perfect harmonies were formed by mathematical ratios of frequencies:  the octave, whose ratio is 2:1, the fifth, which is 3:2, and the fourth, which is 4:3. These ratios, and the beautiful pure sonorities which are the audible proof of their perfection, are the basis for typical violin tuning, with “open” fifths, which ring so deliciously.  However, a scale built solely upon these fifths results in some intervals, particularly thirds and sixths, which sound less than pleasing. In 1482 the Spanish theorist Bartolomeo Ramos de Pareja introduced just intonation, a system in which the major third is calculated by the ratio of 5:4, and other intervals are arrived at by using the Pythagorean ratios of the octave, fifth and fourth.  Tuned in this system, 3rds and 6ths will have a “third tone,” an acoustic phenomenon which greatly enhances the pleasure of the listener and the player. Present-day intonation takes advantage of parts of both of these historical tuning systems.[1]

To use either of these systems, listen for the tones beyond or underneath the tones you are playing. Perfect consonances (8ves, 4ths, and 5ths) which are perfectly in tune have no beats; listen for the “wow-wow-wow-wow,” which will change as you approach perfection, and cease when you reach it. 3rds and 6ths which are tuned in just intonation will produce an audible “third tone;” a minor third, for example, will produce the fifth below the top note (C#-E will give an A); a major third will produce its lower note an octave lower (C-E will yield a lower C). Likewise, a major sixth will give a note a fifth below its lower note (E-C# will give an A), and a minor sixth will give a bass note an octave below its top note (E-C will give a lower C).  Learning to play in tune means accessing the overtone series.

Problems: In both of these early systems, intervals do not add up over distance or are not consistently sized. In the Pythagorean system, a high C will not be the same note as a low C, and playing pieces in different keys will be impossible.  Even tuning your violin with perfect open fifths, as most of us were taught to do, will result in a G which is quite flat to your open E.  Playing a C major chord using G-G-C-E will be some kind of disaster, as you will be forced to choose between playing Cs and Gs which do not ring and agree with your G string, OR living with a C which is intolerably too low to make a nice-sounding major third with your E string. As medieval musicians discovered, some compromise must be found.

 

II. The system most in use today: Equal Temperament – the Ultimate Compromise!

Definition: This system divides the octave into twelve equal half-steps, ideal for keyboards, modulation, and even atonal music.  In fact, the development of this system in the sixteenth century made Western classical music possible.

To use equal temperament, tune your violin using a tuner/drone which can sound all tones.  Tune your open strings directly to the corresponding notes on the tuner.

Problems: This system does not take advantage of the acoustical properties of our instrument. The violin will not ring. The sound will be bland; intervals will seem indistinguishable and lack personality. Sharps will seem low, flats will seem high, and in general intervals will not be expressive enough.  Half-steps will seem lazy, especially at fast tempos. Fifths and fourths will be slightly too small to ring. Major thirds and minor sixths will be slightly too wide to get a third tone, minor thirds and major sixths slightly too narrow. Only the octaves will be pure, and you will most definitely match the piano.

 

III. Expressive or solo intonation – the String Player’s Secret Weapon!

Definition: This system, which is used primarily by string soloists (especially violinists), takes advantage of the string instrument’s ability to play in the cracks between equal tempered notes. Sharps are raised and flats are lowered to give the maximum expression for each interval.  An expressive half-step will have much more tension than its equal-tempered counterpart. Leading tones will truly and unmistakably lead: in G major, an expressively high F# will be dying to reach its G; in Bb major, an expressively low Bb will be quite close to the open A, resulting in an A which is dying to get to its Bb. Use of this intonation will raise the emotional temperature of music, exaggerating yearning, tension, sadness, etc. It will also heighten the profile of the player, helping him or her to come out of the texture. (This can be why students and even some soloists often end up playing generally sharp, by the way—they are trying to raise sharps, which will help them stand out from the orchestra, but stray into raising everything.) A side benefit is that this particular intonation will more often result in perfect intonation with open strings (for example, a B on the A-string will tune UP to the E string, not down to the D string), giving a generally richer and fuller sonority to the instrument as the sympathetic vibrations of open strings are activated.

To use expressive intonation, slightly raise the third and seventh tones in major scales (often the sixth as well); lower the third tones in minor scales. Slightly exaggerate all accidentals– put sharps higher and flats lower. Tune all notes possible to open strings using pure octaves and fourths and when necessary, combinations of these intervals, i.e., B on A string should be tuned to open E, G to G string, C to G to G string, etc.  Half-steps will be noticeably smaller and whole steps fractionally wider than in equal temperament. This will be particularly helpful in fast passagework, where the ear perceives half-steps differently.

Problems: this system should be used only for solo lines, not when tuning within a vertical or chordal structure. Also, the player must get used to hearing and tuning 3rds and 6ths NOT using just intonation; major 3rds will sound wide and minor 3rds narrow, in comparison to those tuned with the just intonation system. Lastly, the player will not blend as easily in an orchestral section, or even with other players who have been trained using equal temperament.  Expressive intonation is a powerful tool, so use it deliberately and only with discretion!

 

IV. Bayla Keyes’s Compromise Violin Tuning – recommended for all situations

To tune the violin in a way which will allow you the best of many possible worlds of intonation, purchase a tuner/drone which has a variable A and which sounds all tones.  (I use a Sorinuri Digital Metronome/Tuner #MTR-33.) Tune your A and D strings with the tuner/drone note at A=441; match the A of the drone exactly, then turn the dial to D and match your open D string to the drone D exactly. Tune G with tuner/drone at A=440, but with the dial turned to sound the note G. Tune E with tuner/drone at A=442, but with the dial turned to sound the note E.  (Note: If you don’t have a movable tuner, tune A to 440, tune D to A in a VERY TIGHT fifth, tune G to D in a SLIGHTLY TIGHT fifth, tune E to A in a SLIGHTLY TIGHT fifth.) Your goal is to squeeze your beautiful open fifths a little, but not as much as an equal-tempered piano would.  Note that your open strings will create fifths which are smaller than perfectly open fifths, but in the case of G-D and A-E, wider than those of the equal-tempered piano.

 

Open Fifths (Pythagorean)     Compromise Tuning (Keyes)  Equal Temperament (Piano)

E: tuner at A = 443                    E: tuner at A = 442                           E: tuner at A = 441

A: tuner at A = 441                   A: tuner at A = 441                            A: tuner at A = 441

D: tuner at A = 440-439         D: tuner at A = 441                            D: tuner at A = 441

G: tuner at A = 439-438         G: tuner at A = 440                           G: tuner at A = 441

 

Tuned using Open Fifths, the violin’s acoustical properties will be fully apparent; the instrument will ring gloriously. However, the G string (and possibly the D) will be noticeably flat to the piano, and the E will be noticeably sharp; the violinist will be likely to play flat on the bottom of the instrument, and sharp on the top. Furthermore, in C Major, a chord using open G and open E (i.e. G-G-C-E) will sound nasty, because the distances will be too wide. With the cello C string the problem is exacerbated.

Tuned using Equal Temperament, the violin’s open strings will match the piano perfectly, and playing in tune should be considerably easier, all over the instrument. However, the violin’s acoustical properties will be rendered mute; the instrument will sound like a cigar box. It will actually feel harder to produce a sound, because there will be no sympathetic vibrations from your fifths.

With Compromise Tuning, the violin will be able to have some ring from its outer fifths, while preventing the wide and disconcerting spread between G and E; furthermore it will be close enough to the piano pitches corresponding to its open strings to prevent discord when playing with piano. The G string will be SLIGHTLY flat to the piano and the E SLIGHTLY sharp, but not enough to trouble the listener. When playing with cello, the violin E will not be so disconcertingly high to the cello C string.

 

V. Remembering pitches vs. relating to a pitch environment

A player with consistent intonation can remember pitches, so that an A is an A no matter where on the instrument it is played. Working with a tuner/drone can help you become more exact in your pitch memory by toning the ear as if it were a muscle; if you use the tuner and experiment with both equal temperament and expressive intonation you will find your ear developing greater ability to distinguish ever finer degrees of pitch. From the point of view of reliability, it does not matter which system of intonation you prefer, as long as you are consistent and intentional. Your goal when practicing is to establish steadfast relationships in your own intervals and with your own instrument.

But intonation must always be relative as well.  In rehearsal and performance, when you are playing with other instruments, you MUST listen and adjust. Using equal temperament is advisable in many situations, e.g. orchestral playing, matching the piano or in combination with winds.  However you can expect to be using Pythagorean intervals as well, in order to make perfect ringing fourths and fifths with other musicians. You may find that you have to play certain notes “OUT” of tune, in order to be IN tune with another player. Strive to establish consistent intervals with others, listening to intervals in order to access the extra sonorities available to you through the sympathetic vibrations of perfect octaves, fourths, and fifths, or even the third tones to thirds and sixths created between you and another player. In your own practice, relational tuning by matching perfect intervals with a tuner/drone, as well as whenever you have an applicable open string, will help develop your ear quickly; in rehearsal and performance, you will be matching others instead.  In all cases, your ear is your constant guide and your first responder!

 

© 2010 Bayla Keyes



[1]  I highly recommend Stuart Isacoff’s wonderful book, Temperament, for an in-depth history of the development of equal temperament.

 

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