Why Is Just Intonation Impractical?

Question

I've read about the debate of "just intonation" vs 12-tone equal temperament. And nowhere it was clearly stated why just intonation is impractical. Here are my assumptions. Please let me know if I am correct.

Just intonation frequencies are based on the harmonic series. A fundamental tone is picked and then all of its harmonics are transposed within one octave (that is, in the range of the first two harmonics). The first N harmonics account for 12 different notes in that range.

However, if an instrument is tuned with the frequencies obtained in the above manner, the instrument only sounds good in one key. In other keys it sounds out of tune (because the frequency ratios for the intervals are not simple integer fractions like 3/2). For that reason the 12-TET tuning system was developed, so that the same strings can be reused in all keys without sounding out of tune (and without the need to re-tune the instrument when changing the key).

What is not clear is why this is the case. The Harmonic series should produce harmonic sounds. At first it looks they don't and therefore a "hack" is needed.

My guess (please refer me to a source explaining it) is that the harmonic-series-derived tones sound good in the key based on the tone that was chosen for the fundamental frequency for a given series. So if we choose C3 as the fundamental frequency, all intervals will be OK in C-major, but will be out-of-tune in A-major. For them to "work" in A-major, we need to pick A3 as the fundamental frequency and calculate and transpose the harmonics. Thus the 12 (or 24, or whatever) notes will have slightly different frequencies depending on the key. The compromise of 12-TET is made so that an instrument doesn't need hundreds of keys/strings in order to play in multiple keys.

Is that correct?

Answer 1

Yes, you're right. As for why the harmonic series doesn't produce notes that work in all keys, the simple answer is that the math just doesn't add up.

Let's work out the math for just intonation: Suppose you choose X Hz for the fundamental frequency and go from there. Then the octave above the fundamental should have frequency 2 X Hz. Meanwhile, the perfect fifth above X will have frequency (3/2) X Hz. The perfect fifth above that will have frequency (3/2)(3/2) X = (9/4) X Hz. Continuing on the cycle of fifths, you can easily see that every pitch generated this way will have frequency (3/2)^ n X Hz for some exponent n.

If there are twelve tones in the chromatic scale, then (3/2)^12 X should be some whole number of octaves above X, i.e. (3/2)^12 must equal a power of two. But this is impossible because no power of 2 can have 3 in its prime factorization, as all powers of 3/2 must have. Indeed, if you don't insist that the chromatic scale have twelve tones, you still can't make the math work: (3/2)^ n != 2^ m for any positive integer values of n and m.

Is it close, though? Not close enough. (3/2)^12 = 129.74, and the closest power of 2 is 2^7 = 128. In practical terms, this means that the A one octave above A440 is 440 * 129.74 / 64 = 892 Hz, which is definitely audibly distinct from the pure 880 Hz you'd expect. The math just doesn't work---just intonation cannot produce a set of pitches that work well in all keys.

Answer 2

Alex Basson has given you a great introduction to the mathematics. Let me approach the answer from a different perspective, that of the performing musician in a historical context.

Setting the mathematics aside, to put it simply, just intonation is what happens when you have a group of singers performing a capella, or a string quartet, or any other ensemble of monophonic instruments that can inflect or bend their pitch. But as soon as you insert a conventional piano or guitar (which are tuned to 12-tone equal temperament) into the ensemble, all the other instruments and performers will shift from just intonation into equal temperament so as not to clash with the guitar or piano. Singers and string players don't consciously think about it; it just happens.

There are also instruments in existence that play only in pure just intonation. These are instruments that can only play one scale in one key, and no extra notes outside of that. They include the natural trumpet or bugle (which have no keys, no valves, and no vent holes), or certain designs of the recorder, or the bagpipes.

Just intonation is extremely impractical for instruments that play chords (guitar or piano), or any instrument with fixed pitches which cannot bend, such as vibraphone or marimba.

How many keys do you want in an octave on your keyboard? In the Baroque period, 12-tone equal temperament had not yet been invented. Although the early harpsichords and organs had 12 notes to the octave, they used various tuning schemes that were based on just intonation. Each instrument could only be played successfully in a few keys with the tuning scheme in use.

To expand on that, innovative designers in the 1500s and 1600s built a few organs and harpsichords with between 14 and 36 different pitches/keys within one octave to be able to play in something closer to just intonation in many keys. (The previous link shows many different historical designs for keyboards that could play intervals that were closer to just intonation.)

To say that learning to play a keyboard with that many keys in an octave was an added difficulty to the keyboardist is an understatement. It also meant that harpsichords and organs had to have extra strings and extra pipes to play the extra pitches, adding significantly to the cost and the mechanical difficulties of building and maintaining the instrument.

This problem was largely resolved when the "Well-Tempered" tuning was invented and subsequently championed by J. S. Bach. Later on, true 12-tone equal temperament was developed. Around this time, most keyboard musicians lost interest in keyboards with extra keys/pitches for approximating just intervals in various keys.

In the modern era

there have been several designs for a just-intuned keyboard for electronic musical instruments, with many more than 12 keys/notes in an octave.

I know of one electric guitarist, Jon Catler, who plays guitars built with extra frets to make 31 equal-tempered notes in an octave. His purpose is to play conventional tonal music that enable a skilled performer to get close to just-intoned intervals in many keys; he's not composing and playing exotic non-Western scales or music. Lately he's been recording on a new guitar he designed with 64 notes in an octave that he says achieves just intonation in all keys.

Below are pictures of two guitar designs which he sells, and below that is a video demonstration, playing a guitar of yet a third design.

Here is a link to microtonal guitar fretboard designs.

Not many guitarists would want to learn to play one of those instruments. Take a close look at those frets on those fingerboards and you will see why just intonation on a guitar is impractical for anybody but a select few avant-garde musicians who want to go to the tremendous trouble to develop a very complicated playing technique in the name of creating more pure intervals.

Answer 3

Here is a table that I have adapted from one in Wikipedia that illustrates how just intonation differs from 12-tone equal temperament.

In modern instrument tuning, an octave is divided into 1200 cents. There are 100 cents in an equal-tempered half-step, and all half-steps are equal in their distance apart.

However, in just intonation, not all half-steps are the same size. This table explains the discrepancies, and shows you just how out-of-tune certain musical intervals are on the 12-tone equal tempered piano, organ, synthesizer or guitar.

As you can see, in 12-tone equal temperament, all intervals except the octave are slightly out-of-tune. The intervals that are the most noticeably out-of-tune are the tritone, the minor third, the major sixth, the major third, and the minor sixth.

Also note that just-intoned intervals cannot be expressed as integer values of cents in the first place. The cent is a convenient mathematical measuring unit based on 12-tone equal temperament. So the unit of the cent really has nothing to do with pure frequency ratios.