Scales in 19-tone Equal Temperament
2016-03-31
According to "regular temperament theory", any equal temperament may be used as a tuning for one or more "rank-2 temperaments". In the case of 19-ET there are four such temperaments worth knowing: meantone, magic, sensi, and keemun.
If I write "meantone[7]", I mean a scale of 7 notes pulled from meantone temperament. It's the familiar diatonic scale, with step pattern LLsLLLs. In 19-ET, L = 3 steps and s = 2 steps.
This scale behaves similarly in any equal temperament that supports it. There's a pattern of major and minor triads on scale degrees 1-3-5: Maj min min Maj Maj min dim, in the Ionian mode.
~
What about magic temperament?
https://en.wikipedia.org/wiki/Magic_temperament
http://xenharmonic.wikispaces.com/Magic
Scales of 7/octave and 10/octave both work. The step patterns are
sLsLsLs = 1 5 1 5 1 5 1
for the 7-note scale and
sLssLsssLs = 1 4 1 1 4 1 1 1 4 1
for the 10-note scale. You can see that both these patterns sum to 19. Also, they interlock in the same way the pentatonic and diatonic scales do in 12-ET: the large step in the smaller scale is split in the bigger scale.
Here's some music in magic temperament by Graham Breed
http://x31eq.com/music/dingsheng.mp3
So that's magic[7] and magic[10].
~
For sensi there is sensi[8], which is
sLsLssLs = 2 3 2 3 2 2 3 2
These scales contain chords that alternate through their modes, much like the major and minor triads alternate through the modes of the diatonic scale. That's what makes them so cool.
Dustin Schallert shows how to harmonize sensi[8]
https://www.youtube.com/watch?v=y2S2qOLHlg8
how to build up a groove
https://www.youtube.com/watch?v=KiEoq_kRUHA
and finally, how to add a melody
https://www.youtube.com/watch?v=AUxsiltacNQ
~
Last but not least, keemun[7]
LsLsLsL = 4 1 4 1 4 1 4
~
FAQ
Q: So magic[7] would be C C# E E# G# Ab B# C?
A: In chain-of-fifths notation, that's exactly right.
Magic scales are made of chains of thirds (approximate 5:4). If you stick to 19, you can use chain-of-fifths notation, since every note in 19 is reachable that way (not true for all equal temperaments... 24 has two chains of fifths which never meet). It can produce some weird-looking notation though (try magic[10]...).
There's a lot of debate about the best way to notate these. My preferred method is to start with letter "A" and just go up the scale
A B C D E F G (and H etc, as needed)
and then use sharps and flats according to the chain of generators, whatever the generator is (major third in the case of magic).
I also simply run the scale up consecutive lines and spaces of the staff. That way, a given chord pattern will look the same anywhere on the staff, just as major and minor triads do in standard notation.
With my method only one pair of accidentals is needed, i.e. sharp and flat, so you don't need any special fonts. Others prefer instead to write only diatonic notes (allowing no letters beyond "G") and then apply new accidentals to adjust them. That's how Sagittal notation works, if you're familiar with it.
You can read more about notation options here
http://lumma.org/music/theory/notation/