Constructing the Just Major Sixth

Since the goal is to construct just intervals, open string tuning to just fifths is equally acceptable. Passage (1), above shows how to construct just versions of the perfect fourth and major sixth. This is one of many passages and tests that the string player should also practice either a fifth higher or lower (e.g. by testing on A♮ the viola or B♮ on the violin). To enhance confidence in the just major sixth, the following passage should be played:

Observe that the highest note is a bit flat, and listen for a beat pattern when it is played against the highest open string. The tempo will depend on whether the open strings are just or equal tempered fifths. With just-tuned open strings, the tempo will be 163 beats per minute for the cello, with the viola and violin beating twice and thrice as fast, respectively. This passage was played by an amateur violist on all three instruments. Clear and pleasant beats at approximately the expected tempo were heard on the cello, but the two higher instruments made annoying dissonant sounds that seemed to be beating at or near the correct tempo. Six college-age non-musicians could recognize the presence of beats, but were unable to assign any tempo to it. A biology professor with a good sense of rhythm, and who often conducts for local amateur and high school musicals was able to hear a beat pattern of all three instruments that was roughly consistent with the theoretical tempo.

The beat pattern for each instrument was measured electronically using readily available technology: The output of a Radio Shack tie clip microphone was fed into the sound recorder application found on a Windows PC at a sample rate 44100 Hz. When a passage by the violin was replayed at exactly half the sample rate (i.e. down one octave), it sounded like a cello and made clearly identifiable beats. When the viola’s signal was played at half speed, and the beats were marginally discernible. The recorded signals were also analyzed by Matlab to produce the spectrograms shown in Figure 4. The four red horizontal patterns show the four harmonics for each instrument, with the fundamental appearing at the bottom. Note that the second harmonic beats at twice the fundamental’s beat frequency, with the third harmonic beating at thrice the fundamental’s frequency, exactly as one would expect. . Also note the absence of beats in part of the viola’s signal corresponds to a weak fundamental harmonic in the spectrograph. The spectrograph allowed us to count the beats and establish the tempo on each instrument with the following results:

Instrument	Tempo (beats/min)	Tempo Error	Fingered Pitch Error (%)
“ideal”	163	0	-21.5%
violin	109	-54	-14.4%
viola	132	-31	-17.4%
cello	169	+6	-22.3%

In this table, the “ideal” pitch would be a final fingered note is flat by 21.5%, and the three instruments did produce flattened notes. (Here the pitch deviation from equal temperament was calculated from the beat frequency.) The largest error in the beat tempo occurred for the violin, which beat triplets at a tempo of 109 beats per minute. An absence of beats is seen in the first two seconds of the viola’s spectrogram, and the presence of beats at the second harmonic suggests that the viola has a tendency to not sound at the fundamental harmonic at this pitch. This was confirmed by the viola’s weak strength in the fundamental as seen in a power spectrum that was also calculated by Matlab.

Back in 1965, when music seemed to play a more prominent role in public education, students in a middle school string orchestra played a game involving bow control. The goal was to hold a note on one bow as long as possible, which challenged the students’ ability to control the bow. An analogous (and more advanced game) might be to see who can make the best beat pattern in passage (3).