Ground's intro to tuning diversity

NOTE: Audio examples are coming soon.

If you've been a musician for long, you should be familiar with the 12 notes:

A A#/Bb B C C#/Db D D#/Eb E F F#/Gb G G#/Ab (A)

If you're anything like me, you took them for granted for years. After all, why wouldn't you? No one has told you how these were chosen, that's just how music is. But the truth is, these notes are a centuries-long compromise based on lots of math and stylistic evolution. There are endless other valid ways to do it, some of which may sound a little odd at first, but it's just because you've only been exposed to one tuning system for most of your life. In order to fight this monopoly on music, it's important to step out of your comfort zone.

Diatonic tunings

Let's take a look at the piano keyboard. 12 keys per octave, and the steps between them are all the same size. We call that 12edo, short for 12 equal divisions of the octave. There are 7 white keys and 5 black keys. For reasons that aren't important to explain now, 7 and 5 are very important numbers to our diatonic musical practice, with the major and minor scales and the circle of fifths. So we can add 7 to 12edo and get 19edo. With these extra notes, suddenly sharps and flats aren't the same anymore:

A A# Bb B B#/Cb C C# Db D D# Eb E E#/Fb F F# Gb G G# Ab (A)

The diatonic scale works exactly as you'd expect. Because sharps and flats have been notated separately as a historical vestige, you could play any sheet music in 19edo if you wanted, but it sounds a little different. It's a bit softer and mellower, and the fifth is flat. It may sound jarring, or it may sound even better than 12edo. And because of these extra notes, there's a new scale related to diatonic that you've probably never heard before. This is semiquartal, and instead of being mellower, it's harsher and has two extra small steps. It's full of this interval called the semifourth. As the name implies, it's half of a perfect fourth, and it's also halfway between a whole tone and a minor third. Just this adds so many new possibilities to fit intervals together to achieve new musical moods.

Now instead of adding 7 to 12edo, let's add 5. This gives us 17edo, and it's sort of a mirror image of 19edo:

A Bb A# B C Db C# D Eb D# E F Gb F# G Ab G# (A)

In 19edo, A# is below Bb; in 12edo, they're equal; and in 17edo, they're reversed with A# above Bb. 17edo is sort of a mirror image of 19edo. The diatonic scale is now harsher and more articulate, and the fifth is sharp, but there's a new scale called mosh that's softer. Mosh is full of neutral intervals such as the neutral third, halfway between a major and a minor third.

You can keep finding new tunings like this just by adding 7 and 5:

• 19 + 7 = 26edo,
• 19 + 7 + 5 = 31edo,
• 17 + 7 + 5 = 29edo,
• 17 + 5 = 22edo.

All of these tunings are also great, but you can also do more interesting things that will truly test your comfort zone.

The weird ones

Let's take a look at a tuning that's just 3 times 5, 15edo:

A ^A vB/vC B/C ^B/^C vD D ^D vE/vF E/F ^E/^F vG G ^G vA (A)

B and C are the same note, and so are E and F.

Being only a multiple of 5 does weird things. The fifth is very sharp, to the point that the semitones disappear and reading from 12edo sheet music doesn't sound right. It's possible to notate the result in many ways, but I'm using ups and downs notation with the little arrows altering the pitch by one step of the edo. Despite how alien this tuning is, the new chords actually sound pretty good. Instead of being based on a diatonic scale, it has several new ones including blackwood and pine, which comes from a temperament called Porcupine. Instead of having whole tones and semitones like the other tunings discussed so far, there are actually two different intervals that take the function of a whole tone: the neutral second, halfway between a standard whole tone and semitone, and the semifourth, halfway between a standard whole tone and minor third. It's called a semifourth because it's half of a perfect fourth.

So we've discussed adding 7 and 5, but what about the other numbers? 6 ((7+5)/2), 8 (5+3), and 9 (7+2) can also be added to create even more unusual tunings. Let's look at 7 plus 9, 16edo:

A Ab/B# B Bb C# C Cb/D# D Db/E# E Eb F# F Fb/G# G Gb/A# A

This tuning is antidiatonic, which means that the fifth is so flat that it flips the sharps and flats, which now change the pitch in opposite directions. The antidiatonic scale is made up of 2 large and 5 small steps rather than the 5 large and 2 small steps in the diatonic scale. You can actually read 12edo sheet music in this tuning, and it will warp all the major intervals to minor and vice versa. It sounds pretty weird, but I actually love how gritty and dark it is, and how it's absolutely full of neutral seconds.

It's possible to split the large steps into a neutral second and an even smaller step, resulting in a 9-note scale called armotonic. I like to notate this with a version of my own notation that extends the circle of fifths for 8 and 9-note scales. It adds two new letters in this case, XFCGDAEBY. Let's also fix the sharps and flats:

A A#/Bb B X X#/Cb C C#/Db D D#/Eb E E#/Yb Y F F#/Gb G G#/Ab (A)

So if making the fifth super flat flips major and minor intervals, what happens if you make it super sharp? Enter 5 plus 8, 13edo:

A G#/Db C B A#/Eb D C#/Gb F E D#/Ab G F#/Cb Bb A

There's no commonly accepted way to notate this one. Everything works out with a normal circle of fifths as long as you let Bb go on a white key. It could also be replaced with X as in my notation, or B and Bb could be replaced with H and B as in German notation.

13edo is based on the oneirotonic scale, which has 8 notes rather than the diatonic 7. Of course, it's not really that simple, but there's a reason this scale is probably my favorite absolutely baffling piece of music theory. You may notice that C is lower than B and F is lower than E, and the sharps and flats are all very extreme. It may be prudent to abandon standard note names altogether and try something like this:

1 1#/2b 2 3 3#/4b 4 4#/5b 5 6 6#/6b 7 7#/8b 8 1

Oneirotonic is made up of pretty normal whole tones and semitones, but chords are completely different. If you want something completely from another world, this is your scale.

Conclusion

These are just a few examples meant to show the range of ways tuning can change how music works with only a few extra notes. Getting deeper into tuning theory allows for better subtlety and control over compositional intent.