The Red-Blue Scale and Superpyth

22 Equal Temperament is rich in scales and harmonies. However one thing it does not contain, is the same diatonic scale which exists in 12 ET. What do I mean exactly? Well see, in 12 ET we stack seven fifths to produce the "diatonic scale" which serious tuning dudes call "meantone". It's called such because in 5-limit just intonation, there are two sizes of whole tones: a major and minor whole tone which have the ratios 9/8 and 10/9.   To the untrained ear, these sound VERY similar, there isn't much of a width difference between the two. If you played a JUST major scale then you would get a major whole tone from C to D but a minor whole tone from D to E. This sounds very good but it also has the draw backs of creating wolf intervals when you move around to a different root than C.

NOTE: Wolves are what tuning guys call intervals which were considered "very dissonant" for the day. Most common is wolf-fifths which are nearly a quarter tone flat from a nice 3/2.




So the dudes back in the say saw that in order to play in a much simpler manner, the difference between the two could be made the same producing a MEAN between the two tones.
THUS MEANTONE WAS BORN!!

Now obviously, if your into just intonation then these two probably have quite a different character to you. Even myself, who isn't into differences this tiny, finds them to sound a bit different. I actually prefer 10/9 as a dyad.  This isn't an ultra technical guide and I'm not even qualified to teach such things so I will skim over the other technical ideas of meantone.  However, meantone is the basic diatonic scale, 10/9 is equated with 9/8.

In 22 ET though, it's not the case. In 22 we have something different but similar at the same time. If we stack seven perfect fifths in 22 ET then we get a different temperament called "superpyth".  While meantone equates 9/8 with 10/9, superpyth equates 8/7 with 9/8.  8/7 is a whole tone which is quite a bit wider than 9/8.  You could think of it as a "super wholetone".

What this does is build a scale which has the same virtual structure PROPORTIONALLY as the diatonic scale yet has a completely different set of intervals. While the major scale is 2 2 1 2 2 2 1, this scale is 4 4 1 4 4 4 1.  Notice how they both have the structure of L L s L L L s but the values of the step sizes are different?  Superpyth has really wide large steps and tiny small steps making it sound altogether new. At least to me it does. \O__O/

The Red-Blue Scale 

Because superpyth is a TEMPERAMENT name, it doesn't specify how far we are moving out in a chain of fifths.  It's like you technically could say 12 ET is one big meantone[12] scale, in fact in a tuning such as 31, meantone[12] is a subset. Point is to avoid confusion, I have named the seven tone superpyth scale the "Red-Blue" scale.  Why you may ask?  Well it's kind of like how we call the major scale, the MAJOR scale and minor scale the MINOR scale, it's based on the quality of the thirds. If you have read the basic quick and dirty guide to 22 ET WHICH YOU SHOULD HAVE READ FIRST!  Then you will know what I mean by red and blue thirds.    

I could have named it the "Blue scale" but that is WAY too confusing with the Blues scale. The red scale also would have worked but I then just decided, well it has both types of thirds in it so I'll just call it "Red-Blue."  In addition, Red-blue scale sounds more unique than just Red-scale which is a term in photography.  

Modes of the Red-Blue Scale 

Because Red-Blue scale is proportionally the same as the diatonic, mode names can remain the same. 

MODE NAMES

Lydian - F G A B C D E F  or 4 4 4 1 4 4 1     

Mixolydian - G A B C D E F G or 4 4 1 4 4 1 4 

Aeolian - A B C D E F G A or 4 1 4 4 1 4 4 

Locrian - B C D E F G A B or 1 4 4 1 4 4 4 

Ionian - C D E F G A B C or 4 4 1 4 4 4 1 

Dorian - D E F G A B C D or 4 1 4 4 4 1 4
Phrygian - E F G A B C D E or 1 4 4 4 1 4 4  

Lydian, Mixolydian, Ionian have red triads on the root.
Aeolian, Dorian, and Phrygian have blue triads on the root.
Locrian has a blue third on the root but the root triad also has a yellow fifth on it rather than a white fifth.

You can play any diatonic music from 12 ET on the red-blue scale but it will probably sound wonky and very different. One particular difference between the Red-blue scale and the diatonic scale of 12 ET is that Blue triads which are more similar to minor are MORE harmonious and stable than Red triads which are less harmonious and less stable. So in essence, if you choose to think of Red as major and Blue as minor then they are basically reversed in this scale.

Superpyth[12] 

We can go beyond the normal Red-blue scale to a larger scale of superpyth[12] which I call the "Supertonic scale." It contains twelve notes in the scale. Unlike 12 EDO, this scale is not evenly spaced but rather very uneven, it is a larger sort of diatonic scale and has the potential for music which goes way beyond simple diatonic music.  

The scale's basic structure is 3 1 3 1 3 1 1 3 1 3 1 1 

That's enough for now. 

Reading Music in 22 Equal Temperament

Make sure you read the beginner guide first because it addresses many of the concepts here.  You will also want to read the page on basic 22 ET scale systems.

Since 22 ET is not a standard anywhere in the world, we can pretty much do whatever we want. However, to make things as easy as possible to those who know how to read 12 ET scores, we will simply make notation an extension.  If you were confused about accidentals in the previous quick and dirty guide, then this article will go into more detail about how they work.

First thing to know is 22 ET is not 12 ET, notation works a little funky in 22 ET compared to 24 EDO which just makes new accidentals for the quarter tones and everything else remains the same. In 22 ET, things change a bit more.  I'm gonna attempt to break this down so it all makes sense hopefully.

1. The Staff Naturals correspond to the Red-Blue Scale  

If you read the 22 ET scale systems page, you will recall that the normal diatonic scale doesn't really exist in 22 ET but rather a sort of variation called superpyth[7] exists which we named the "Red-Blue" scale.  In this sense to recap, the Red-Blue scale is also built from a chain of fifths like the typical diatonic scale yet it sounds very different.  The small steps are all blue seconds (1 step of 22) while the large steps are all red seconds (4 steps of 22). Since we know that red and blue sounds are novel, that means that this scale in general sounds very odd even though it has a reminiscent structure to the diatonic scale.   

Well this Red-Blue seven tone scale makes up the notes of the 22 ET basic staff. The structure of 12 ET major scale is 2 2 1 2 2 2 1 which translates to L L s L L L s so in 22 ET it's the same proportional structure of L L s L L L s except it becomes 4 4 1 4 4 4 1 from C.    Keep in mind though that Red Ionian is on C not Red Lydian! Point is we're trying to make notation easy so there is no need to place the Red Lydian scale on C just because I happen to consider it the most tonal of all the Red modes. So think of it if you want as the "Red Ionian" being on C not Red Lydian. 

Fig 1: The Staff corresponding to notes C D E F G A B C which is 4 4 1 4 4 4 1

Basic Interval Logic 

In order to internalize this system of 22 ET notation, you should be familiar with the basic four-color logic.  Here are the basic principles: 

1) Blue intervals are 1 step below Green, Green are 1 step below yellow, yellow are 1 step below red all of the SAME category of seconds, thirds, sixths, and sevenths.  

This means that if I use a v accidental on what would normally be red then it's yellow, if I use a v accidental on what is normally green, it becomes blue ect...  Just make sure you know the order bgyr which can be remembered as  "Boring Girls Yawn Ridiculously".   

2)  Anything 1 step below blue is Red of the category below it ONLY IF the below category it's a third, second, sixth or seventh.  Otherwise, it's a perfect unison or fifth.  

So if I flatten a blue third then I know it'll be a Red second automatically. If I flatten a Blue seventh, Red Sixth, if I flatten a blue sixth, then it must be a perfect fifth.  Get it?  

3)  Anything 1 step sharp of a fifth or fourth is a WIDE fifth or fourth, anything 1-step flat of a fifth is a NARROW fifth.  

4) Be aware of the Aug4/Dim5 which exists between the Wide fourth and Narrow Fifth.  

It may seem like alot to remember but it's more about getting a grasp at thinking in this way. It makes reading and interpreting MUCH easier.  

Accidentals 

The normal accidentals # and b correspond to raising and lowering a pitch by THREE steps. C# is three steps above C natural. Gb is three steps below G. Got it?   One step intervals use v and ^ which look a bit different scores depending on which variation is being used as the font.  G^ is one step above G natural and Bv is one step below B.  

The somewhat tricky part is the two step and four step accidentals because it requires a combination of the four symbols.  If we take a sharp sign and combine it with a down sign then we get this #v "Up-Sharp"which is the sign used for raising a pitch by two steps of 22. The flat version is b^ pronounced "Down-Flat".   

If we want four step accidentals we simply switch v with ^ respectively, #^ or "up-sharp" indicates an alteration of four steps of 22.  C#^ is four steps higher than C.  Gbv is four steps lower than G.

Double sharps and flats correspond to six steps of 22 ET, i.e two times three = six.  Gx is six steps higher than G natural.  Fbb is six steps lower than F natural.  

We can again, combine ^ and v with x and b to get five and even seven step intervals. 

Gx^ is seven steps higher than G natural,  Gxv is five steps higher than G natural.  Bbbv is seven steps lower than B. and so on. 

Here's a nice table to help with this: 

1 step:  v or ^ 

2 steps:  b^ or #v 

3 steps:  b or # 

4 steps:  bv or #^

5 steps:  bb^ or xv  

6 steps:  bb or x 

7 steps:  bbv or xv 

8 steps:  bbb^ or x#^ 

9 steps:  bbb or x# 

You get the idea, pray to the cat gods that you do not ever need the eight and nine steps lol. In general, the idea here is nesting accidentals like this. You're taking one value and altering it with another, in other words, you are essentially applying ROMAN NUMERAL logic to accidentals. 

#v being two steps is the same logic as IV being four.  5-1 = 4,  # with a v = 2.  

Here's all 22 ET intervals from C, excuse the crappy image quality:

Last words

In general the most difficult aspect of this is getting your head around different combinations. It's pretty easy to read from a root of C, but reading from say a root of F#v can get tricky and you'll see why stuff like bbv pops up occassionally.  

Of course this doesn't cover key signatures but that's a whole different animal for another post.

Have fun!