Difference between equal temperament and just intonation

Question

I am learning Indian classical music and learned about just intonation and equal temperament. I want to know the difference between them. I know that Indian classical music uses just intonation and Western music uses equal temperament.

Can you give me some more info in layman language? You can also add some science.

Answer 1

Every note has a pitch, determined by the fundamental frequency of the sound wave that produces it. When you have two different notes, you have two different pitches, caused by two different frequencies. The distance between those pitches is called an interval, and corresponds to the ratio of the note's frequencies. For example, if one note is an octave above a second note, their frequencies will have the ratio 2:1.

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In Just Intonation, which was historically common in Western music at least into the Baroque period, the intervals are determined by forcing the notes to have frequencies that are small whole number ratios to each other. For example, a perfect fifth is 3:2, a perfect forth is 4:3, a major third is 5:4, and a minor third is 6:5. In historical western music, they usually stopped there (the interval 7:6 is not used), and all other intervals are then combinations of these intervals. The benefit of this system is that, because the frequencies are numerically related to one another, they sound very consonant, or pure to the ear. The downside is that those pure intervals only exist in one key; outside of that, the intervals start to break down.

For example, if you start at A♭, and go up a major third you get to C (with a ratio of 5:4 = 1.25). If you go up another major third, you reach E (5:4 x 5:4 = 25:16 = 1.5625). If you then go up one more major third, you reach G♯ (5:4 x 5:4 x 5:4 = 125:64 = 1.953125). As you can see, this doesn't quite reach the next highest A♭ (which must have a ratio of 2:1, by definition). The basic mathematical problem is that no matter how many times you repeat any of these ratios (other than the octave), you will never get back to a power of two, which means you will never get back to the starting pitch (in any octave). Which further means that you either have to have an infinite number of notes in your scale, or you have to add in an impure interval somewhere to get back to where you started.

If you want to play in lots of different keys and maintain these pure intervals, one possible solution is to use instruments that can freely vary their pitch, like violins or voices, so that you can always adjust to the correct just interval. This is often done subconsciously by singers, even in western music, just by finding what sounds good.

However, many instruments cannot do this; especially keyboards and fretted instruments, both of which have to have a predefined scale. One possibility for them is to start adding a bunch of extra notes: Make A♭ and G♯ be two different notes, for example. However, there could be an infinite number of such notes, so things can get pretty insane pretty quickly, like this keyboard that has 24 notes per octave:

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This approach hasn't really been pursued much in western music.

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Instead, the solution that Western music has devised is called 12-Tone Equal Temperament (12-TET). Rather than forcing frequency ratios to pure-sounding whole number ratios, it makes a compromise to divide each octave into twelve equally-sized intervals, and adds them together to get larger intervals. The frequency ratio required for this small interval (which happens to be a half step) is the 12th root of 2 (2^1/12 = 1.059463...), since multiplying it by itself 12 times gives you back 2 (an octave). In fact, the term "ratio" is a bit of a misnomer, since this is actually an irrational number.

If you combine 7 of these small intervals (2^7/12 = 1.4983...) you get a close approximation to a pure perfect fifth (3:2 = 1.50). If you combine four of them (2^4/12 = 1.2599...) you get a less-close approximation to a pure major third (5:4 = 1.250) which is still somewhat serviceable. But the key point is that by making all of the intervals equally out of tune, no key is any worse than the other. You do have to sacrifice the pureness of the intervals, but it is a relatively subtle effect to the untrained ear.

Here is a video that demonstrates both types of intervals:

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TLDR: Just intonation refers to using mathematically simple ratios between a pair pitches; justly tuned intervals are objectively more consonant than non-justly tuned intervals. However, it is mathematically impossible to construct a scale of fixed pitches in which every possible interval is justly tuned, and some may even sound quite bad. 12 TET is one way of compromising when tuning a scale; it equally detunes all intervals, so that no single interval is either perfectly just, nor terribly far away from being just.

Answer 2

A simple answer: One is not better than the other. They serve different purposes according to different musical traditions.

Just intonation is pure tuning according to the pure mathematical overtones produced by musical instruments. When an instrument is tuned to just intonation, it is more or less only able to play in one key, and to use limited chords and harmonies. This is perfectly suited to Indian music (which relies on elaborate melodies, drones and open intervals, but has no place for Western chord progressions and harmonies) and music of many other cultures.

12-tone equal temperament is a compromise developed in Western music that did not become widespread until about 150 years ago. In 12-tone equal temperament, the intervals between the notes are changed slightly to make them less pure. This enables an instrument so tuned to play in all Western keys and all modes, but with all intervals and chords being slightly out-of-tune to varying degrees. However, these compromises are deemed acceptable to Western musicians and listeners. This makes it possible for one instrument to play Western music that relies on complex Western chords and harmonies and which can modulate to different Western keys and modes at any time.

Classical Indian music would sound wrong and inauthentic if you played it on instruments tuned to 12-tone equal temperament (such as the Western guitar or piano). Some musicians and composers have worked out compromises for combining traditional Indian instruments with Western instruments (tuned to 12-tone equal temperament), but I suspect that these compositions lose some of the character that makes the music sound traditionally Indian.

Answer 3

Etymology, Math, and Acoustics

These two terms can mean different things depending on who you talk to, especially since they are not discussed nearly as thoroughly in music theory teachings. Naturally, this makes the terminology more vague and less agreed upon. You probably have a good sense of them already, though. On a basic level, just intonation is when a particular harmonic interval is ideal to the human ear.

For some people, this would be the members of an orchestra adjusting their intonation so that there is no beating in the chord. That could be called just. More commonly, it is the relation of intervals by whole number ratios. For more information on this, you need to understand some basic acoustics.

A sine wave is the most basic sort of sound that you can have. Flutes are quite close to a pure sine wave and you've most likely heard them produced electronically. When two sine waves interact, they can do so constructively or destructively. In the following image, there are three waves. The first is a sine wave representing our root and the second is a sine wave representing our fifth. The third is their interference:

If the third wave was played, you would hear both tones and they would be perfectly just. Now pay attention to the cycles. The first wave has two cycles in the same period that the second wave has three. At that point, they are perfectly in phase with each other and are both at the beginning of a new cycle. So these waves are said to have a 2:3 ratio. This is a just perfect fifth. So long as they remain just, their cycles will line up at a constant interval. This is a generally pleasing effect. Unfortunately, having all of our music set up like this is much more difficult than one might expect.

History

Pythagoras is commonly talked about in communities concerned with just intonation and microtonality and he is often credited with discoveries like diving a vibrating string in half to make an octave. Regardless of whether or not this is true, Pythagoras and his followers were heavily interested in music and believe its relationship to numbers to be divine. He attempted to stack fifths on top of one another to create scales which reflected these beliefs, but ran into a problem. The frequency ratio for an octave is 1:2, so a doubled frequency is an octave. So any octave of our root must be the root multiplied by a power of 2. Fifths are powers of 3. He considered this geometrically with pentagrams, but still soon came to an unfortunate conclusion. There is no way for a power of 2 to equal a power of 3. A circle of fifths will never close on an octave. There's a lot more information on Pythagorean tunings, but for now, let's move forward.

Musical tuning evolved and many tried to reconcile this circle of fifths problem. I'd like to point out that a circle of fifths is certainly not the only way to achieve just intonation. Regardless, people were very concerned. If you left that gap between the last fifth and the next octave, you weren't met with many problems in your root key. In fact, this is one way to derive a major pentatonic scale. However, shifting to other keys could get ugly quickly. The tuning that has won out in modern western music traditions is equal temperament.

More Math

If you flatten the fifth you're using in your circle of fifths, the circle will close. The fifth that we use as a generator is only a little flat of a just perfect fifth, gives us quite a few consonances to work with, and closes after 12 fifths. This is called tempering the fifth, ergo the name. With our fifth, it creates an equal temperament. Now, we hear frequencies logarithmically, meaning the difference in Hz between two intervals will be progressively larger as you move up in octaves. The difference between A4 and A5 is 440 Hz while the difference between A5 and A6 is twice that at 880 Hz. So to divide the octaves into equal sounding intervals, we use powers of fractions. This is essential the same as taking a root of a square, but I like this notation better.

For an octave divided into m equal parts, an interval of n steps will be 2^n/m. A minor third in 12 tet is 3 steps (semitones). So that interval would be 2^3/12. The nice thing about this is that you can transpose a melody to any key and and its intervals will have the same qualities. It also makes for easier construction of fretted instruments. And really, most people can't tell the difference because 12 tet is pretty close to just intonation.

Answer 4

If you want really clear as a bell, non-muddled tones, you want just intonation. We hear music in intervals. Just intonation is Pythagorean intervals, expressed as ratios. What we call the perfect fifth interval, 7 semitones, is a 3:2 frequency ratio. If you play it along with your root note, it will coincide in perfect clear harmony. However Pythagorean ratios get more complex, expressed as X^m / Y^n where you might end up with 128/27 or even (80*7) / (3*5) as a tone ratio in some scale.

The problem with such theoretically clean ratios on a piano divided into 12 semi-tones per octave is that stepping up 7 semi-tones from C to G may give you an exact 3:2 ratio, but stepping up 7 semitones from D to A will not. - Essentially your compositions are limited to playing in the key of C on a piano in the key of C if you want to play with your set of clean scale frequency intervals.

Bach, with his treatise 'The Well Tempered Clavier' solved this by uniformly fudging the interval between all 12 semitones. Any jump of 7 semitones starting from any key will produce the identical frequency interval. Unfortunately this arrangement is a compromise, and that interval is no longer a sweet clean exact 3:2 any more.

The advantage of having done this is that any song can instantly be transposed from one key to another without changing it's interval flavor, and that pianos no longer have to be built with a primary key in mind. Songs in the key of F# or Db all have the same muddy intervals between them now.

Recent synthesizers have brought back the possibility of just intonation, where the whole keyboard, or just individual chord notes, remap to the key or root of the moment to sound clear.

Specifically equal temperament provides a ratio of 2^(1/12):1 for each semitone. After 12 semitone multiplications you end up with a ratio of 2:1, an octave, the only pure ratio in equal temperament.

There's more beyond that, with stretch tunings and such. Just intonation is possible on a fretless string instrument. With frets you can get just intervals between strings, but you are stuck with equal-temperament between frets. Just intonation requires tuning your instrument for a specific key. With standard equal temperament, you can play further down the neck in another key and not end up with different sound quality intervals between notes.

Sitar music is largely about the shimmering color produced with all the drones more so than the jumps between notes, so just intonation is more critical to have there. Also there's nothing out of place spending an hour to tune up for a raga. That wouldn't fly playing rockabilly at a pub.

Each has it's place. You could conceivably play some beautiful shoegazing rock with just intonation, but only if you find a band willing to put in a large investment in time and gear to do so.

Pythagorean ratios are a world of study unto themselves, and probably a huge distraction towards acquiring useful generic composition/playing skills in the Western world.

Answer 5

A couple of corrections. Pythagoras did write about 3:2 fifths and did build scales from these. He did not invent the idea, just as he did not invent the stuff about triangles. He studied in Egypt, where the same theories are written on papyrus documents from 2000 years before Pythagoras, and most probably predate the documents we have.

Bach did not invent Equal Temperament, and did not write the 48 Preludes and Fugues for Equal Temperament. He wrote them for Well-Temperament.

Well Temperaments are those where a) the cycle of fifths meets b) flats and sharps coincide such that there are only 12 pitches per octave c) all keys are playable d) some keys are purer than others e) each key has its own set of intervals and character

There are many ways of distributing the comma such that these conditions are met, the most famous of which are those by Andreas Werckmeister, and the most recent one, causing some excitment, is by Lehman, which he based on doodles written on Bach manuscripts.

Temperaments are vital when considering polyphonic music and harmony. Music which is monophonic can much better be suited using pure tuning of intervals. Indian music should never use ET or any other temperament.

Answer 6

I will try to simplify this down as much as possible. The answers here either are not answering the fundamental questions, makes quantum leaps in assumptions, extremely long winded, make gross subjective judgements or are just flat out wrong.

So a little physics primer. All music instruments (including your voice) is some kind of a standing wave (e.g a wave on a string for guitars or air for flutes etc). All standing waves oscillate at certain frequencies called overtones (this is not a physics stack exchange so you can dive into what overtones are in https://en.wikipedia.org/wiki/Overtone). Whats important to know here however is that there exists a fundamental frequency f for a given standing wave and overtones are whole number multiples of this fundamental frequency).

Just intonation is an intonation scheme in which notes are arranged in fractions of these overtones. For instance the octave is the ratio between the first over tone and the fundamental (2f/f = 2). Similarly the perfect fifth is the ratio between the second overtone and the first overtone (i.e. 3f/2f = 1.5). The ratios are then multiplied by the fundamental and sorted in ascending order.

The reason this is done is to fit in as points of "consonance" in a given fixed amount of time. For instance if the fundamental is 10 hz, the perfect fifth is 15hz and in every 3 seconds, the waveforms are constructively interfering. Contrast to that for the octave of 20hz, its constructively interfering every 2 seconds. Thats why we find the octave more "consonant" than the perfect fifth. (Note this is a subjective intepretation, but anecdotally I find people correlate consonance with the frequency of constructive interference of waveforms in standing waves).

Now while this is the most optimal arrangement "consonant" arrangement of notes for a given fundamental, there is a big problem. This effectively fixes the fundamental. For instance if the fundamental is the note A4 (440Hz) then you can use B as the tonic (if you do, then the ratios between notes no longer follow the whole number ratios). You can plot just a just intontated scale with the frequencies and try changing the tonic from the fundamental to realize that this is indeed the case.

As western music became more complicated, there was a need to

1. Play music across multiple keys without changing instruments e.g. the sonata allegro form in Piano.

2. Play chords outside of the basic harmonization of a scale.

A just intonated system simply did not work in either of the scenarios. So instrument builders tried to adhoc fix intonations by tweaking the frequency values for certain notes.

The most elegant of the adhoc transformations resulted in this formula

f(n) / f(n - 1) = 2^(1/12)

where f(k) is the frequency of the kth note. This is equal tempered intonation.

This compromise made the ratios as close as possible to fractions of just intonation and kept the scale tonic invariant at the same time (for instance the perfect 5th of equal temparament has the ratio to tonic 2 ^ 7/12 = 1.4983... which is almost the same as 1.5).

It immediately solved the problems of western music at the time and was extremely close to a just intonation scale to have the same degree of consonance.

P.S

1. There is no peer reviewed literature in existence that I know of that A/B tests equal and just intonated identical pieces of music on large scale population sizes. Hence there is no existing proof its "Inauthentic" in any tangible sense.

2. There is a huge proliferation of equal tempered instruments in ICM. A notable example is the harmonium. Its popularity suggests that intonation is not the most central requirement of a musical system.