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The following sentence is from an article titled Microtone from Grove Music.

The harmonic question is differently settled, of course, when microtones are conceived not as additions to the equal-tempered chromatic system but as basic intervals in other tunings – tunings which have customarily been developed not in order to make available intervals smaller than a semitone but to find better approximations to just intonation than 12-note equal temperament can deliver.

Although I read it several times, then looked up 'just intonation' on Britannica, I still didn't understand the author's suggestion.

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    It's not clear from the quotation whether the author is considering the anachronistic application of "microtonal" to historical tuning systems of the early 19th century and earlier or reserving it for systems developed in modern times. Can you describe how the "harmonic question" is "settled" in the earlier part of the article?
    – phoog
    May 23 at 8:33

6 Answers 6

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Music that uses tuning systems other than 12-tone equal temperament (12-TET) is often referred to as "microtonal". 12-TET arose as an approximation of Just Intonation so that it was possible to modulate to any key. However, it's not a very close approximation. Other tuning systems, often called "microtonal" systems, provide better approximations.

A very good explanation of the mathematical difference between 12-TET and Just Intonation can be found in Difference between equal temperament and just intonation. A Just Major 3rd and a 12-Tet Major 3rd are close mathematically, but still audibly different. A "microtonal Major 3rd" might be equal to, or at least closer to, a Just Major 3rd.

Many instances of this type of microtonality can be found on this site by searching the tag. One in particular that might be of use:

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To expand on what @Aaron already said:

music (tones) is based on partial harmonics. Take the first two partial harmonics (1:2 and 2:3) of any given tone and you get the octave (1:2, double the frequency) and the fifth (2:3, multiply the frequency by 1.5).

Now, go to the nearest piano and stack 7 octaves of any tone. i.e. C-c-c'-c''... Because each one has a frequency double that of the previous one, the first and the last tone should have a frequency relation of 1:128 (2 to the power of 7).

Next, try the same with fifths, i.e. C-G-D-A-E-B-F#-C#-G#-D#-A#-E#(F)-C. Because we have arrived at the same key we should have the same frequency relation between the first and the last tone as before. But when we calculate (1.5 to the power of 12) we find that the result is 129.7 and change. This difference is called the "Pythagorean Comma".

Using only the first and second partials one could construct all 12 tones of our tone system (use fifths to construct the tones, then octaves to have these tone all in the same register). Unfortunately only one scale would sound "in tune", the farther you get away from the original key it the circle of fifths the less properly tuned the scales sound. This is called "just tuning".

Now, you don't want to retune your instrument every time you modulate to a new key and hence ways to get around this were sought - and found. The basic idea is to distribute the difference as evenly as possible around all tones so that no scale sounds perfect anymore but no one sounds completely out of tune either. These forms of tunings are called "temperament".

The last (major) development was the "well-tempered temperament", which makes all the 12 major and minor keys available (a fact which excited J.S.Bach so much, he wrote a piece for a keyboard tuned that way, the "well-tempered keyboard". The well-tempered tuning was discovered in the 16th century by a Chinese mathematician, later re-discovered in the 17th century in Europe (Werckmeister, Kirnberger, ...).

Today we use a tuning quite similar (but not equal) to the well-tempered tuning, the "equal temperament", where the relations between semitones are uniformly 1:12th root of two.

On the upside we are able to modulate to absolutely any scale, as showcased i.e. in "Giant Steps" by John Coltrane. On the downside no interval is "pure" any more and when you play a major triad you will hear a "beat", which you wouldn't hear in a pure tuning.

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  • There seems to be confusion between meantone and Pythagorean tuning. In Pythagorean tuning, it's not true that "the farther you get away from the original key it the circle of fifths the less properly tuned the scales sound"; rather, all keys will sound similar except for the relative location of the single wolf fifth. The real issue is with the major thirds, even sharper in Pythagorean tuning than in equal temperament, which is what led to the development of meantone temperaments long before modulating to distant keys was an issue. These temperaments do favor keys with simpler key signatures.
    – phoog
    May 23 at 11:01
  • I don't see you this answers the OP
    – Aaron
    May 23 at 12:49
  • Try bringing up quarter-comma meantone tuning, too. There were also (apparently) a load of types of well-temperament, too.
    – Dekkadeci
    May 23 at 19:06
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Before 12tet came about, some instruments - such as harpsichords - whch are tuned specifically, would be tuned so that their notes sounded best in certain keys. Just Intonation and other temperaments were used , where the intervals sounded more harmonious, musical. This meant that playing pieces in other keys did not sound as harmonious, musical. The pitches of those notes are where this question comes from, I think - differing slightly from the pitches in common use today - with use of 12tet.

Thus the compromise of 12tet came about, where each note was tuned so that it could sound pretty much o.k. in any key. But it was a compromise, where each octave is split exactly equally into 12 parts, and we've basically got used to it.

Instruments such as violin, trombone, and voice will tend to gravitate back to just intonation tuning while playing, so for example, a C note in one key will be slightly at a different pitch from a C note in a different key.

Those two (or more) different C pitches are microtones, not the same as each other. It's rare that they would be used together in one piece - the C in a piece in key C will not be the same pitch as the C (M3) in a piece in key A♭, (or even C (P5) in key F), but they wouldn't occur in the same piece in one key only. That's to say, one pitch would be used that is appropriate for one harmony, a subtly different one for a different harmony, if that makes sense.

Trouble is, on an instrument tuned to J.I. it has to be one or the other, so 12tet gets used. There have been pieces written for microtonal music specially, but they are a different breed.

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  • "Just Intonation was used": not for tuning harpsichords, it wasn't. Rather, there were dozens of different unequal temperaments used in different times and places. Also, it's not correct to say "It's rare that they would be used together in one piece." The tuning of any given note can vary even in one key when played by strings or trombones or sung. To take the example of C, suppose you're in E-flat major. The third of the A flat major triad will be lower than the fifth of an F chord (whether minor, major, or dominant).
    – phoog
    May 23 at 8:24
  • @phoog - what I meant was that JI would be used as one tuning method for instruments that needed to be tuned (like pianos today). Of course other temperaments were used, none of them 12tet at the time. I take your point about what harmony is happening in a particular bar. See if the edit helps. Thanks for the explanation of the dv!
    – Tim
    May 23 at 8:46
  • But if you tune a keyboard to just intonation, it isn't useful even in one key because of the wolf fifth between the second and sixth scale degrees. Nobody ever tuned keyboards in pure just intonation (unless you count pythagorean tuning in the middle ages, where there were only 8 pitches per octave and the major third was a dissonance, so it's not at all the same thing as the 5-limit just intonation most people have in mind when they say "just intonation"). Another way of looking at this is to consider the need to move certain pitches even when staying in one key, which a keyboard can't do.
    – phoog
    May 23 at 8:53
  • @phoog - would my answer be best deleted? I think if you included info such as in these comments, it would make a good answer. I'm learning a lot ! OP seems to be homing in on JI.
    – Tim
    May 23 at 9:58
  • No, I wouldn't delete this answer. I would just rewrite the beginning of the second sentence so it doesn't say (or appear to say) that harpsichords were tuned in just intonation. (In fact, the family of temperaments that dominated from the late Renaissance to the middle baroque, "meantone" temperaments, had prominent major thirds that were pure "just" major thirds, or at least closer to pure than those of 12-tone ET, but in order to achieve this they had prominent fifths that were largely rather farther from pure than 12-tone ET, so hardly "just.")
    – phoog
    May 23 at 10:24
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Just in case Grove has a global perspective on things: Arabic music is often described as having "microtones" or "quarter tones". The actual case here is precisely what Grove describes: it does not concern additional intervals to the scale, but rather a different construction of the scale in a different basic tuning. For instance, you could have a scale C-D-E,-F-G-A-B-c where "E," denotes a half flat. In that case the normal E & Eb simply don't exist: the third scale degree is consistently the half flat.

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Whenever you play two notes at the same time, the difference in pitches is called an interval. Because of how pitch perception works, generally we care about the ratio of frequencies, rather than the difference.

Some intervals sound better than others. That's subjective, of course, but it's generally held that the best-sounding frequency ratios are the ones that match up with simple integer ratios - such as 2:1 (i.e., an octave), or 3:2 (the so-called "perfect fifth"), or 4:3 (the so-called "perfect fourth" - which you may notice is also the same as going up an octave and then down a perfect fifth, since 2/1 * 2/3 = 4/3). More complex combinations are usually held to sound okay, even if the numbers get big. For example, going up two perfect fourths and then down a perfect fifth gives you a ratio of 32:27, which is sort of close to the simpler 6:5.

But none of these ideal ratios is actually produced by instruments tuned in the normal modern way (12TET). The modern tuning is designed so that the spacings are the same no matter what key you're playing in; the ratios between adjacent notes are identical. It turns out that no matter how many notes you divide up the octave into, you can't get just-intonation results if you divide it up equally. But with 12 notes (a convention with centuries if not millennia of tradition behind it), you can get very close for some notes (2^(7/12) is about 1.4983), and acceptably close for most others, and it produces a sound that everyone (at least in contemporary Western cultures) is very familiar with.

"Microtones", generally, just means "notes chosen so that the intervals between notes are smaller than in 12TET". That is to say, more than 12 notes are used per octave. (Of course, as the number of notes increases, it becomes harder to design instruments or to write or perform music.)

There are any number of reasons you might want to do this, and depending on your reasons, you might choose to space the notes out in different ways. Equal spacing with a greater number of notes produces interesting results - for example, with 19 equally spaced notes, you get an extremely accurate approximation for the 6:5 interval. Some avant-garde composers, on the other hand, choose dissonance instead; with 14 or 15 notes, the closest approximations to the perfect fifth are quite far off.

However, if we have fixed the key that we want to play in (easy enough in the case of electronic music), then instead of equally spacing the notes, we could simply add notes that are intended to make it easier to get good approximations of the just-intonation intervals in certain keys.

One example of this is in the classical Indian system, each 12TET note, aside from the root and the perfect fifth, is essentially split into two slightly different variations.

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  • This is explicitly not what the question is about. OP understands that microtones can be used to modify, expand, or decorate 12-TET. OP wants to understand the quotation's other usage of the term "microtone" which is entirely independent of any reference to 12-TET.
    – Aaron
    May 24 at 1:22
  • It's not at all clear to me what OP understands, and "the question is about" what the author meant by a single, very long sentence. I tried to cover all the bases. That said, find better approximations to just intonation than 12-note equal temperament can deliver. sounds much more to me like "decorate 12TET so that JI intervals are also available" than "independently develop a microtonal scale with good JI approximations" - especially since it's being contrasted with make available intervals smaller than a semitone (i.e., "use nTET where n>12"). May 24 at 1:29
  • I think your answer would be greatly improved if it focused more on the "independently develop a microtonal scale" rather than explaining the differences between 12-TET and JI.
    – Aaron
    May 24 at 2:33
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Just intonation favours expressing intervals across the entire keyboard as ratios of small numbers:

  • (Perfect) octave - 2:1
  • Perfect fifth - 3:2
  • Perfect fourth - 4:3
  • Major third - 5:4
  • Minor third - 6:5

As alluded to in other answers, though, these small-number ratios conflict with each other when repeated across the entire keyboard and near-matches keep being found. One of the most important such examples is that 7 octaves tuned as 2:1 each almost equal 12 perfect fifths tuned as 3:2 each - the ratio difference between the 12 perfect fifths and the 7 octaves is (1.5)12⁄ 27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents. This is known as a Pythagorean comma.

12-tone equal temperament expresses all intervals as ratios of the 12th root of 2, so only octaves get nice, simple, and clean ratios that resemble just intonation precisely. For example, 12TET's ratio for a major third is equal to the cube root of 2 ≈ 1.259921, which is off from just intonation's 5:4 = 1.25 ratio by around 13.69 cents (a big enough difference to force a concert band instrument player to re-tune their instrument - I speak from experience playing in concert bands here).

Just intonation still holds a certain appeal for listeners because it produces particularly harmonious results that 12TET's more complex ratios and therefore increased tendency of subtle beats will not produce. Perhaps one of the most famous examples of that appeal is barbershop quartet music's penchant for wondrously ringing major-minor 7th (a.k.a. "dominant 7th") chords - chords that require just intonation in order to ring. Barbershop quartets use a 7:4 ratio for minor 7ths in order to get that ringing just right.

Tuning systems therefore will often use commas and portions thereof, all microtonal-sized, to approximate at least some of the best parts of just intonation while mitigating at least some of the worst parts (e.g. habit towards wolf fifths if you want to preserve all (perfect) octaves and are willing to compromise on "perfect" fifths). For example, quarter-comma meantone tuning prioritizes major thirds that sound justly tuned at the cost of leaving in a wolf fifth. The comma that quarter-comma meantone tuning cares about is the syntonic comma = 81:80 ≈ 21.51 cents (among others, this is the just intonation ratio difference between 4-perfect-fifths and 2-octaves-and-1-major-third - e.g. C4 to E6 - and between 1-octave-and-1-minor-third and 3-perfect-fourths - e.g. C4 to E♭5), and it flattens nearly all its "perfect" fifths by a quarter of that comma. Quarter-comma meantone tuning was notably more prominent in the 16th-17th centuries than today.

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