# What is 22 equal temperament? [closed]

I gave this song a listen here. This artist used 22 tone micro tonal in the time signature of 5/4. Here are some questions that came into my mind:

• How did this artist use 22 tone in his song?
• What makes it 22 tone and how does it work?
• Are there any key signatures?
• What does it mean by frequencies?
• How is it different from 12 tone? Any similarities?

I'm guessing it's a set of notes or chords that are used to make it 22 tone... If there are any song or piece examples to share as well, that would be great. (optional question: If you would to create a 22 tone piece, how would you do it?)

• You may find some helpful insights here: music.stackexchange.com/questions/72077/… – Aaron Nov 23 '20 at 0:03
• Seeing as you're on Sevish's Youtube channel already he has several videos and a detailed blogpost discussing his working process and the software he uses. In fact, he did a long interview on microtonal music on another channel just the other day but I haven't watched it yet. – Bob says reinstate Monica Nov 23 '20 at 10:43

In the common 12-edo tuning, the octave is divided in 12 equally-spaced steps (hence equal divisions of the octave). What's meant my “equal” is that the frequency ratio between subsequent notes is always the same, or equivalently the logarithm of the frequencies of neighbouring notes always have the same difference. The other way around, it means that the frequency of the i-th note can be computed as an exponential, specifically

fi = f0 · 2i12 = f0 · (12√2)i.

Tabulated, these frequencies are, starting from A440,

``````440.0 Hz, 466.2 Hz, 493.9 Hz, 523.3 Hz, 554.4 Hz, 587.3 Hz, 622.3 Hz, 659.3 Hz, 740.0 Hz, 784.0 Hz, 830.6 Hz, 880 Hz
``````

The main reason 12-edo is so common is that it is, without too many possible notes, a reasonably good approximation to 5-limit just intonation. For example, seven steps come out as

f7f0 = 2712 ≈ 1.4983

which is very close to 32 = 1.5. Therefore, a 12-edo perfect fifth sounds almost exactly like a JI fifth. It is slightly narrower, but almost imperceptibly.

Not quite as good is the approximation of thirds:

f4f0 = 2412 ≈ 1.2599

That one is audibly wider than the JI major third 54 = 1.25, but it still good enough to pass off as an approximation in many contexts.

12-edo is by no means unique in offering approximations to those JI intervals. With more steps, you can in particularly get better thirds, though the fifths typically get a little worse. Specifically, the ratios for fifth and major third in the best 5-limit tunings are

Because Western tonal music is essentially 5-limit, all of these tuning systems can be used for rendering most music, though there are various quirks to be aware of. 19-edo and 31-edo are quite easy in the sense that like 12-edo they're meantone temperaments, meaning a major third (i.e. the approximation of 54) has the same size as two whole steps (i.e. the approximation of 98). Obviously, this is not the case in just intonation, and not in 22-edo and 34-edo either. Particularly in 22-edo, the ditone comes out as notably wider than the major third, which may result in unexpected asymmetries in melodies, which can be a difficulty but also an opportunity for the composer.

In meantone tunings it's generally quite straightforward to render music in standard notation, because the intervals can be read off and each interval has a clear correspondence. Typically, e.g. E♭ be different from D♯. In non-meantone tunings, even two E♭ notes may be different, depending on context (basically, whether they're approached by a ajor third or by two whole steps). Via modulations (look up comma pump) that can even happen even in meantone as well.

Unlike 12-edo, the higher EDOs also approximate JI ratios with prime factors higher than 5, for example 31-edo includes a Barbershop seventh 1.7489, very close to 74 = 1.75. This may or may not be exploited when composing music for those tunings.

• "Obviously, this is not the case in just intonation": but there are meantone temperaments where the major third is (or is very close to) a just third, and these temperaments are in fact much better for playing diatonic harmonies (on a keyboard) than is just intonation. – phoog Nov 23 '20 at 7:41
• 5-limit tuning includes minor thirds (6/5) as well, yes? I know 19-edo has an exceptionally just minor third: 1.2001 (+0.15 cents). – Karl Knechtel Nov 23 '20 at 15:56
• @phoog True, but the compromise you need to make, in order to make a temperament meantone, entails making some intervals different from just even if others are equal to just. In just, two octaves, a major third and a syntonic comma equal four perfect fifths. In meantone, you temper the syntonic comma to 0, so if your major thirds are just, your fifths a 1/4 of a syn comma narrow (1/4-comma meantone, which is close to 31et); and... – Rosie F Nov 23 '20 at 17:32
• @KarlKnechtel ..if your minor thirds are just, your fifths and major thirds are 1/3 of a syn comma narrow (1/3-comma meantone, close to 19et, as you say). – Rosie F Nov 23 '20 at 17:32
• If you're discussing why meantone makes standard western music easier to deal with, you might as well mention that "the" whole tone actually is both 9/8 (the major tone) and 10/9 (the minor tone), which combine into the major third. The difference, which, tempered out, defines meantone, is of course the syntonic comma. In a non-meantone temperament such as 53-EDO, you may have to spell the JI major triad 4:5:6 as C-Fb-G, instead of C-E-G (if you go by the circle of fifths for note names), precisely because 4 fifths is not two octaves and a third. – obscurans Nov 24 '20 at 1:25

As others have pointed out, one difference between 12et and 22et is that 12et is meantone and 22et is not.

You ask for similarities. Here is one. 12et, 22et and 34et are diaschismatic. Diaschismatic temperaments temper out the diaschisma, an interval of frequency ratio 2048:2025. This is a narrow interval: only 19.55 cents, that is, about 1/5 of a semitone of 12et. In terms of more familiar intervals, the diaschisma is the amount by which (in just intonation) 4 perfect fifths and 2 major thirds fall short of 3 octaves.

What this means in terms of building scales is that you can have a scale with 12 pitches per octave, dividing the octave into 12 semitones, 10 of them large (L) and 2 of them small (s):

```  L     L   L     s    L   L     L   L     s    L      L    L
C   C♯↓   D   D♯↓   E↓   F   F♯↓   G   G♯↓   A↓   A♯↓↓   B↓   C'
C   D♭↑   D   E♭↑            G♭↑       A♭↑        B♭

```

Pitch notation is more complicated than usual because the usual notation suits meantone temperaments, but diaschismatic ones aren't meantone (except for 12et). Perfect fourths and fifths are notated as usual. So is the greater tone, which is the interval (e.g. C D) which you get by going up two fifths and down an octave. But a major third is smaller than two tones, and the difference is a syntonic comma. ↓ means lowered, and ↑ raised, by a syntonic comma. So a major third (e.g. C E↓) is 3L+s, and a minor third (e.g. E↓ G) is 3L.

Some more technical mathematical details: the L is the tempered version of both the lesser minor second 16:15 and greater augmented unison 135:128; the s is the tempered version of both the Pythagorean limma 256:243 and lesser augmented unison 25:24. In each case the same tempered interval (L or s) represents two different just intervals because the difference between the two just intervals is a diaschisma, which is exactly the interval the temperament tempers out. By using the above intervals as building-blocks, and carefully choosing which ones to combine, we can make larger intervals. For example,

• 2L can represent 16:15 + 135:128 = 9:8, the greater tone
• 3L can represent 16:15 + 135:128 + 16:15 = 6:5, the minor third
• 3L+s can represent 16:15 + 135:128 + 16:15 + 25:24 = 5:4, the major third

One can divide the octave into any set of equal intervals. Whether these are musical useful always needs study. In addition to those already mentioned, division of the octave into 53 equal steps was suggested by Jing Fang (78BC-47BC). Harry Partch used a 43-step division but the steps were unequal.

The ratio 31/53 is really close to a just perfect fifth so the 53-tone scale mimics Pythagorean tuning rather well.

• Where the last sentence says "the ratio 31/53" it ought to say "2^(31/53).” But the real value of 53-tone equal temperament over 12-tone equal temperament lies not in its better approximation of the fifth, which is already quite good in 12-tone equal temperament, but in its better approximation of the major third, so it can approximate not only Pythagorean tuning (which 12-tone equal temperament does) but also just intonation (which 12-tone equal temperament decidedly does not). – phoog Nov 23 '20 at 7:30
• Good point. I should have caught that. – ttw Nov 23 '20 at 15:05