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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?

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In the context of computer generated music, it can be more practical now than ever. –  Dave Dec 10 '12 at 22:21
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@WheatWilliams Speak for yourself, Wheat. I find the math very elucidating. I'm not suggesting the OP shouldn't listen to recordings or experiment as you suggest, but some of us find math to be a powerful tool for understanding these matters. –  Alex Basson Dec 11 '12 at 2:48
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@Dave exactly. I'm asking this in a distant relation to my computoser.com project (currently it uses MIDI, which is based on the 12-TET, but it's worth knowing the possibilities) –  Bozho Dec 11 '12 at 6:14
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Note that the definition of what is "A Major" (relative to whatever is your C note) itself changes under just intonation. The circle of fifths breaks and you actually have an infinite number of keys. –  Kaz Sep 1 '13 at 3:49
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Wendy Carlos had a computer-based synthesis system that allowed rapid updating of the tuning tables. She wrote and recorded many pieces in just intonation. With this system she could start in C major just intonation and suddenly modulate to F major just intonation by adjusting the pitches. She could go around the circle of fifths in just intonation! The results can be heard on her 1986 release "Beauty in the Beast". (Yes, it's really a spiral of fifths; the C you end up on is not quite the C you started on.) –  Mark Lutton Sep 2 '13 at 1:29

8 Answers 8

up vote 25 down vote accepted

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.

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thanks. It can't, if you start from one fundamental frequency. But if you start from multiple fundamental frequencies, and end up with hundreds of keys, it will potentially be in tune for e very key. Provided there's someone able to play it :) Right? –  Bozho Dec 10 '12 at 14:44
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@Bozho Well, in a sense, this is exactly what string players and vocalists do. Since a violin has no frets, the player can adjust her intonation as needed for each note, taking the context of the harmony into consideration. When a violinist plays an F# in the key of G, she'll play it slightly shaper than if the key is, say, A. But for fixed-pitched instruments like keyboards, this quickly becomes impractical. –  Alex Basson Dec 10 '12 at 14:47
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All wind instruments have the ability to fine-tune individual notes through a combination of techniques, in fact. Orchestral players are always on the lookout for 5th and 3rd chord members to raise or lower, respectively, to their justly-tuned equivalents, even if they're not explicitly marked "-14c". –  NReilingh Dec 10 '12 at 17:36
    
+1 Fantastic answer and discussion! –  lukecyca Dec 14 '12 at 4:22
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Actually, the real problem behind tempering is not how poorly just perfect fifths (via the third harmonic) line up with octaves (via the second harmonic), but how poorly just major thirds (via the fifth harmonic) lines up with octaves. To see this, note that the equal-tempered fifth is only a few cents short of just while the equal-tempered major third is about 14 cents greater than just. The essence of the reasoning is the same, though. –  oliTUTilo Jan 12 at 4:38

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.

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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.

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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.

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A comment re recorder - the pitch is somewhat dependent on breath pressure. More pressure sharpens the note; less flattens. Soft-wood recorders (e.g. pear, maple) are more prone to changing pitch than hardwood instruments (e.g. ebony). In our consort we use breath pressure extensively to tune chords to just tuning. –  kiwiron Jul 13 at 4:17

Using a keyed instrument with Just Intonation creates a bunch of puzzles that need to be solved. You are either faced with observing limits on navigating from place to place, or doing "comma pumps" (equating near by intervals, or bend/vibrato between them because they are close enough).

The problem isn't really Just Intonation though. It's caused by trying to play on an instrument that has a set of keys (and notating it as such), rather than being continuous. In other words, named keys may be a bad interface for Just Intonation.

On a fretless instrument, JI is not only practical, but the sensible way to navigate. Stopping a string to an existing note and playing a seventh harmonic there (ie: notePitchInHz * 7) is completely natural and can be described easily enough, but that note doesn't have an obvious 'name'.

Besides just labelling keys, Just Intonation might be the only viable way of doing relative pitch in a general way: Imagine that you had buttons on a monotonic instrument labeled like: /2, *2, /3, *3, /5, *5 ....

People already use pitch lattices which were derived in this way; like horizontal is *3, vertical is *2, etc.

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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.

enter image description here

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.

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Here's a huge list of 700 pitches within an octave, ordered by frequency difference, with their respective names, when they exist (Just Intonation people seem to don't care much about names): kylegann.com/Octave.html and here is the same table with the actual decimal representations, instead of just fractions: a3c8e3f1dc0bac4f596b4c29df042f945b58fc7e.googledrive.com/host/… –  fiatjaf Oct 5 at 19:01

I have come across an even more astonishing system for producing pure intervals on a guitar. A Turkish guitarist, Tolgahan Çoğulu, has patented a system for building a guitar that has channels under each string position that allows the quick installation or removal of any number of tiny partial frets, each one string-space wide, which can be adjusted up or down to any arbitrary microtonal position by hammering on them with a small "spudger" tool.

The performer would be able to recalibrate all the fret positions and intervals of the entire fingerboard any time the performer wishes to play in a different tuning system.

Apparently it was developed for the Turkish style of music called maqam, which uses quarter-tone intervals not found in Western music. But the luthier also demonstrates its use in Western music that uses equal temperament or meantone tuning systems, and mentions that it would be useful for playing Western Renaissance or Baroque pieces.

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In these two videos, he provides a technical description, narrated in English, and demonstrates the use of his instrument in playing excerpts from several different traditional compositions from different historical periods in Turkish and Western music.

His website indicates that he will build and sell many styles of guitars and other fretted instruments (not just classical guitar) by special order, but few details are provided.

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That's clever and completely awesome. On the viol we use tied-on frets, but they're not as effective as the big raised solid frets on a modern guitar, so this is a really neat solution to a problem most people don't even know exists. –  Matthew Walton Jul 15 at 12:33

A relatively new company in Sweden, True Temperament, retrofits electric, acoustic and classical guitars with new necks or fingerboards with heavily modified fret positions that are designed to improve intonation.

If I understand their intent, their "Thidell" design is for playing with something closer to pure intervals, but chiefly in the most common guitar keys of E, A, and D. The further you get away from those keys, the less accurate the intonation gets.

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They also have several other designs for producing other kinds of intonation more suited to other purposes. For instance, they make an entirely different fret layout for playing in the keys more commonly found in jazz.

This looks like a compromise that might work. I have not seen, heard or tried any of their necks or instruments but there are demo videos and audio on the web site.

The most extreme example is this special-order option, a fingerboard that they claim enables playing pure just intonation in only one key (again, if I understand the intent correctly -- all this is very complicated.)

Notice that there are 14 frets to the octave, because apparently (I have not worked through the music theory) certain chords require a sharper or flatter major or minor third than can be provided by only 12 fret positions. So based on the chord, you could choose a G# or an Ab which have distinctly different microtonal pitches, for example, depending on which pitch would produce the correct, in-tune interval in that particular chord.

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Steve Vai likes these - I posted up something on these last year. I plan on getting one of my guitars retrofitted with a Thidel - just for fun –  Dr Mayhem Aug 30 '13 at 18:50

I want to make an addition to all these excellent answers.

With just intonation, it's not possible to make all the chords just. Not even in a single key.

Let's look at the common just major scale based on I, IV and V just major triads:

C 1:1 D 9:8 E 5:4 F 4:3 G 3:2 A 5:3 B 15:8

In this scale, I, IV, V major triads (4:5:6) and iii and vi minor triads (10:12:15) are just.

But ii minor triad is out of tune: D-F interval is 32:27 instead of 5:6. This is ~294 cents vs 316, which is worse than the equal tempered 300.

Worse yet, D-A interval is 40:27 instead of 3:2; 680 cents vs 702, again way worse than the equal tempered 700.

One way to fix it is to flatten D down to 10:9 but this will break the V major triad. There's simply no way of making them all just without adding more notes. Not even in a single key.

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Thanks for adding this answer, as I was wondering about this very thing recently. –  Bradd Szonye Jul 13 at 21:11
    
In Pythagorean tuning, you can fix the "wolf" intervals by having separate keys (or frets or holes or whatever) for enharmonic notes. So, for example, the P5 is in 3:2 ratio, but the enharmonic d6 is 262144:177147. But this doesn't work in 5-limit JI because "the same" interval needs to have a different frequency ratio depending on context: If the M3 should be in 5:4 ratio, then M2 needs to be 9:8 half the time and 10:9 the other half. –  dan04 Jul 15 at 10:13

The foundations of existing music theory were build when scientific data about sound perception were absent and they were inclined to number mysticism which source was that consonant music intervals correspond division of string in ratios of small integers. Now the following facts are known: -the sound signal of basic existing music instruments may be considered as sum of fundamental frequency and harmonics which frequencies are multiples of fundamental frequency and which intensity decreases quickly in common case . -the ear may be considered as bank of strong overlapping band filters which diapasons roughly correspond one music tone and hence ratio 1.122 (or 1/1.122=0,891) -the sensation of dissonance arises when simultaneous existing frequencies are in the same diapason. By it strongest dissonance sensation arises if their distance is ca ½ of semitone that is ratio 1,029)

It is possible with help of these knowledges to come to following conclusions: - intervals with ratios of small integers are consonant as for them and their first (strongest) harmonics ratious don't belong to dissonant values. About their harmonics it is apparent that the less numbers in intervals' ratios the greater must be numbers of notes' harmonics for attainment ratio of their their frequencies as by tone interval or less. But the greater numbers of harmonic the less their intensities and weacker the sensation of corresponding dissonance. For example: for 5 and 7 harmonics if interval is 3/2- 3*5/(2*7)=15/14=1.07, for 3 and 5 harmonics if interval interval is7/4- 7*3/(4*5)=21/20=1.05 That is in second case more favorable for dissonances ratio is obtained for more strong harmonics (3and 5 instead 5 and 7). The question why just intonation is impractical is very convincing considered in the article „Renaisance „Just information“ Attainable Standard or Utopian Dream? ( http://www.medieval.org/emfaq/zarlino/article1.html )

Yuri Vilenkin

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