Thurmond Knight
Maker of Violins, Violas and 'Cellos

Tuning: Bent vs. Linear

A question was recently posted on  regarding tuning the cello’s lower strings slightly sharp in order to play in tune with the higher strings of the violin.

The following is the answer which Guilherme Carvalho presented.

"You've touched a particularly delicate point: tuning our instruments.

I have had (and heard) this same experience. The lower strings of the lower instruments always have to be tuned higher. This is also the case when you tune a piano. And the higher strings have to be tuned a little bit lower, so everything sounds really fine. Why should that be? Simple: we don't hear (mechanically) in a linear way, but we interpret what we hear linearly. This is somewhat similar to some illusions caused by the refraction of light. Ever noticed how a pole looks bent when you put it half-way in the water? That's because light doesn't follow a straight line as it leaves the water (different speeds in different media), but we interpret what our eyes catch as if it did. We're seeing (mechanically) a bent light beam, but we interpret it as a straight one. Likewise with sound: we hear "bent" intervals (up or down), but we interpret them as "straight" ones.

Of course, when we talk about sound perception, we're talking about something that varies almost continuously, not by sudden jumps as light passing from water to air. But the idea is similar. And this is also only perceptible beyond some distance in pitch (like the distance between violin E and cello C, or double bass E). But, to a lesser extent, this can be perceived in a single instrument. How easy is it to play really tuned when you tune your cello to *really perfect* fifths? I find it hard. Try this: play open A and E on second string (perfect fourth). Tune it really well. Then without moving your first finger at all, play the major sixth E and open G below that (3rd string). Is it as tuned as that last perfect fourth? I have found it is not.

Naturally, you may say that this is because a sixth is not a perfect interval, or because that sixth has a different function in the tonality "implied" by the previous fourth than it would in other tonalities. But I sustain the discrepancy goes beyond that (even if not to the point of considering "perfect" thirds and sixths, as did a violin teacher I knew). I've found that tuning the D string a little bit higher than a perfect fifth below open A, then open G just a bit higher than a perfect fifth below that open D, and so forth, solves this problem. By "just a bit" I mean something like 3 to 5 oscillations per second when you play both strings together. Physicists call these oscillations "Beats."

I also think that this tuning brings forth more and richer overtones to the sound, and makes playing in correct pitch easier, even when playing solo pieces or studies, even when double stops are not involved. I believe this happens because the whole instrument responds better, and you can perceive pitch also by the quality and richness of the sound. This actually opens the door to expressive tuning (like making minor thirds a bit smaller and "darker", and major thirds bigger and brighter)."

The following reply to the same question of tuning was posted by Jennifer Carter (from Tracy California):

"Mr. Carvalho has hit the nail on the head. The following is a mathematical analysis with the situation. What I'm going to say will be rather long-winded, very tedious, but hopefully helpful.

Like Mr. Carvalho has said, this interesting situation stems from a quirk in our musical system. A similar (well, not really but the analogy is there) dilemma faced the ancients when they were trying to devise a workable calendar. They could make a solar year, about 365 days long, but the lunar cycles were 28 days long and didn't fit in evenly with the solar cycle. The ancient Hebrews resolved this by adding a thirteenth month 7 times in 19 years. The Romans resolved it a different way: by creating months of fixed length that were independent of the moon, the very months that we have today. But to fix that, we have to add a leap day every four year.

But I digress. The problem in music is that there are two basic ways to divide the octave, and they don't go together exactly. Mr. Carvalho has described these two ways as linear and bent, and that's a good way to put it.

For another way to demonstrate the difference between the two tuning systems, play a major tenth on a piano. It sounds a bit off. Notice how a minor tenth sounds better. But if you have a fine string quartet tune and then play a major tenth, it will likely be perfectly in tune. This illustrates the difference between the so-called "Just Intonation" that string musicians have learned and the equal-tempered (linear) tuning of the piano (remember Bach's "The Well-Tempered Klavier"?)

The difference between the systems is what they are based on. In both, the octave is equal, with a 2:1 ratio, meaning that if A is 440 Hz, the A above it is 880 Hz, and the A below it is 220 Hz. But with equal temperment, the octave is divided into 12 EQUAL semi-tones, in a linear fashion:

A = 440.000 Hz
A# = 466.164
B = 493.883
C = 523.251
C# = 554.365
D = 587.330
D# = 622.254
E = 659.255
F = 698.456
F# = 739.990
G = 783.991
G# = 830.609
A = 880.000

Those are the frequencies of the pitches, equally tempered, like you'd find on a piano, harp, guitar, etc.

"Just Intonation" -- a vague term but oh well -- is based not on equal semi-tones but on bigger intervals with pure ratios. We already noted that the octave interval has a ratio of 2:1. The major third is 5:4, the perfect fourth is 4:3, and the perfect fifth is 3:2, to name a few.

If we apply these integer ratios to A=440, these are the frequencies we get:

A = 440 Hz
C# (M3) = 550
D (P4) = 586.667
E (P5) = 660
A (8va) = 880

Now compare those frequencies to the frequencies of the equally-tempered scale. Even though the octaves are the same, the pitches in between are compromised.

So back to your original question: If everyone in the quartet starts with A=440 and tunes using the equally-tempered tuning of a piano or an electric tuner, so that each string is 7 evenly-spaced semi-tones above or below the next string, these are the frequencies each string will have (let's hope my calculations were correct):

Violin E = 659.255 Hz
Violin/Viola A = 440
Violin/Viola D = 293.665
Cello A = 220
Violin/Viola G = 195.998
Cello D = 146.832
Viola C = 130.813
Cello G = 97.999
Cello C = 65.406

Now let's compare the cello C to the violin E. In order for the major third to sound in-tune to our ears, there should be that integer ratio of 5:4. If we assume that the E is perfectly "in tune" (a rather subjective term, we see) then the C should vibrate at a frequency of 65.926 Hz to sound the major third. That's a hair sharper than the C string is tuned to, but the difference is noticeable, especially at such a slow frequency -- it's nearly a quarter-tone.

In closing (finally!) this tuning difference is resolved by all musicians tuning all their strings so that all the intervals -- not just perfect fifths -- are "in tune," with the fifths a bit close, as Mr. Carvalho described. That's a tall order, of course, and even more challenging when playing with an equally-tempered instrument like a piano. But you learn with time. And besides, open-string tuning doesn't make you play out of tune; putting your fingers in the wrong places does."

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