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# «Michael Kimber A bright and talented graduate student, about to complete her doctoral degree in violin performance and pedagogy and headed for her ...»

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Notice that both the C and G strings have a G harmonic. When the two strings are perfectly in tune with each other, their G harmonics match, and the sounds of the two strings blend smoothly when played together. When the two strings aren’t quite in tune with each other, their G harmonics clash, and we hear, or at least feel, a roughness, and if we listen intently, we hear a pulsing or beating. As we get the string closer and closer in tune, the beats slow down; when they stop, we’re in tune.

Notice in the C harmonic series that the interval from the 2nd harmonic partial (C) to the 3rd harmonic partial (G) is a perfect fifth (P5), the interval between two open strings. In a P5, the frequency of the upper note is 3/2, or 1.5 times that of the lower note. Every time we go up a P5, as from one string to the next, the frequency increases by 1.5 times.

Let’s look at the tuning of the open strings of viola and violin:

Multiplying the frequency by 1.5 each time we go up a string, we arrive at an E that is

5.0625 times the frequency of C. Now, wait a minute! Didn’t we just see that the E in the harmonic series of C is exactly 5 times C? That’s right! There are two ways to tune E. We saw this when we had to use “E” with open A but “E-” with open G in double stops. The harmonic E, which I call E-, is a bit lower in pitch than the E arrived at by tuning a series of P5s such as the open strings. The differenced is small, but you can hear it, and you can feel how you need to move your first finger to get from E in tune with open A (and with open E) and E- in tune with open G (and with open C).

This difference between E and E- (and other such pairs of notes) is called the syntonic comma. We will often have to deal with it when playing double stops and chords.

Knowing that it exists (and surprisingly, most musicians do not know!) will help us.

If we continue tuning by P5s in both directions from the open strings, we eventually arrive at a pair of enharmonic notes – notes of different names that most musicians consider to be two names for the same note – for example, G sharp and A flat – because on the piano both notes are played by the same key. However, if we tune by true or pure P5s of the ratio 3/2, G sharp comes out slightly higher than A flat!

–  –  –

Why don’t A flat and G sharp turn out to be the same, as they are on the piano? It’s because powers of 1.5 will never be quite equal to powers of 2. To say it in musical terms, no number of pure P5s will ever exactly equal any number of P8s (octaves).

Specifically, the twelfth power of 1.5 (twelve P5s) is 129.746337891, greater than 128, which is the seventh power of 2 (seven P8s). The so-called “circle” of fifths that should neatly close on the same pitch (albeit with a different name) is not a circle, but a spiral!

The pitch difference between a pair of enharmonic notes such as A flat and G sharp is called the Pythagorean comma, which is only a tiny bit larger than the syntonic comma.

We call the scale generated by pure P5s Pythagorean intonation. This is the scale that we use most of the time. We also call it melodic intonation. When we sometimes depart from it to play double stops and chords involving notes such as E-, we are using just intonation, which we also call harmonic intonation.

Seeing that enharmonic pairs of notes such as G# and Ab differ in pitch by almost exactly the same amount as pairs of melodically and harmonically tuned notes such as E and E-, we can visualize an enharmonic cluster of pitches as shown at the bottom of this page, melodic and harmonic intonation operating in tandem.

How does a piano tuner deal with these differences when tuning a piano? Isn’t he trying to get the piano exactly in tune? Actually, no. The piano tuner is setting up a tuning called equal temperament (ET), which is actually a carefully measured mistuning! Equal temperament has 1.4983071 P5s instead of 1.5 P5s. Each ET P5 is 1/12 of a Pythagorean comma narrower than a true or pure P5, so that the “circle” of fifths can end exactly where it began, and Ab and G# turn out to be the same. The resulting ET scale has twelve equal half steps or semitones, making it possible for the piano to be played in all keys, both sharp and flat, and the ET P5 is so close to the true or pure P5 that its mistuning is almost inaudible, so it might seem to be the ideal solution. Even though no ET interval is exactly in tune (except the P8), we accept ET on the piano because we’re used to it, and as it never gives us perfection, we don’t realize what we’re missing! Yes, it may be the best solution for pianos and other instruments of fixed pitch (the pitch cannot be changed by the player), but string players and other musicians who can adjust their pitch are always searching for what sounds best, and as we have seen, that requires making adjustments!

Here’s a diagram that could be considered a magnified map of a “single” note. We often talk of the twelve tones of a scale of half steps (a chromatic scale), but it would be more correct to speak of twelve zones, not tones! Each zone includes a pair of enharmonic notes in Pythagorean (melodic) intonation and a pair of enharmonic notes in just (harmonic) intonation. The sizes of commas and other intervals are given in cents.

There are 100 cents in each semitone of ET (that is, from one note on a piano to the next). (Do not confuse cents with frequencies. Every octave is 1200 cents, but the frequency from lower to upper note of an octave is always doubled. A 220 Hz to A 440 Hz is an octave;

so is A 440 Hz to a 880 Hz, or A 110 Hz to A 220 Hz, but all are 1200 cents in size.) You might find it interesting to compare the sizes of the intervals between notes in the Pythagorean, just, and ET scales in C major. Numbers are rounded to the nearest cent.

–  –  –

To play in tune, we obviously do not need to memorize numbers. I provide the numbers

only to illustrate the kinds of adjustments that we sometimes need to make. For example:

1) Pythagorean whole steps are slightly larger, half steps noticeably smaller, than in ET.

2) Just intonation has two sizes of whole step, and larger half steps than Pyth. or ET.

3) The just scale has E-, A-, and B-, each of which is quite noticeably lower than E, A, and B in either Pythagorean intonation or equal temperament.

4) A- is in tune harmonically with E-, C, and F, but not in tune with D. To be in tune with D, A- has to be adjusted to A.

In conclusion, I would like to say that I was pleased to see the doctoral student mentioned at the beginning of this essay refer to “the fluidity of tuning and intonation.” While she may not feel she has a “complete grasp,” she does recognize that good intonation is fluid.

I haven’t specifically answered her question as to “how string quartets tune different intervals within a chord based on the harmonic progressions.” Do they actually do this?

I was about to end this essay on a negative note when I remembered a classic harmonic progression that has sometimes been used as an argument against the practicality of playing in just intonation, but which actually demonstrates beautifully what she referred to as “the fluidity of tuning and intonation…based on the harmonic progressions.” The argument goes like this: In the key of C, the IV chord in JI would be F A- C. The ii chord would then “have to be” D- F A- (keeping the common tone A- and lowering D to D-), since D F A- would have a root that’s a comma out of tune with the other chord tones. Then the V chord would “have to be” G- B-- D-, and the I chord would be C- E-G-, a comma too low! The solution to this dilemma is ridiculously obvious: instead of lowering D to D- in the ii chord, why not raise A- to A? Then the ii chord will be D F+ A, much preferable to string players who love to be in tune with their open strings. The interesting point here is that this requires a member of the quartet (the violist, of course!) to change the pitch of a common tone from A- to A! The effect of the comma shift in this progression is actually quite lovely, and even more so when progressing from IV to V/V (from F A- C to D F#- A, the F rising to F#- and the A- rising to A) – impossible in ET!

I hope this introduction to the “wonders of intonation” (not, as Partch wrote, “the problems of intonation”) will be ear-opening, inspiring, and helpful as you learn to play

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