Some of you may remember a post that I did a little while ago on temperaments, titled an illustration of the non-existence of God. If you haven’t read it then read it now, because this post might not make much sense without it. Since then, I have heard much more of the problems of temperaments, especially from David Howard of the University of York at the Institute of Acoustics Musical Acoustics Group (bit of a mouthful that) one day meeting in June. I then heard more from Graham Hair, Nick Bailey, Alex South and Bill Evans at the Galpin Society conference in July, specifically on 19 tone equal temperament.
When dealing with temperaments, there is a choice facing us. We can either have ‘pure’ intervals, meaning that they are made of whole ratios (see the previous post for an explanation), or we can have constant pitch. In other words, if we decide to keep intervals pure, the pitch of a particular note will necessarily change throughout a work. If we keep A=440Hz, then we cannot have pure intervals throughout. Playing ‘in tune’ is not possible.
However, can we make a better approximation to our pure intervals than 12 tone ET (which is the current standard). Last time I talked about Pythagorean, quarter comma and Just, which are all temperaments which allow playing in a limited number of keys. What if we want to play in any key?
The answer is to go away from the idea of twelve notes in a scale. Instead, why not have more? It turns out that if you have 19 notes in a scale, and divide the octave into 19 equal notes then this has certain advantages. This temperament is known as 19 tone equal temperament (henceforth 19TET). The names of the 19 tones are below, with the twelve notes of the chromatic scale in red.
C – C# – Db – D – D# – Eb – E – E# – F – F# – Gb – G – G# – Ab – A – A# – Bb – B – B# -C
Firstly, there is harmony. 19TET gives a much better approximation to the Just major and minor thirds than 12TET. It also gives a better augmented fourth, minor sixth and major sixth. The perfect fourth and perfect fifth is barely perceptibly different from 12TET. In return, the major and minor seconds and the major and minor sevenths are a worse approximation to Just than 12TET. However, the error between these intervals in 19TET and Just is within a cent of the error between the Just major third and the 12TET major third. So all in all, harmonically it is probably better than 12TET. But better is probably the wrong word. Different is the right word. To someone trained in 12TET, it would seem harsh and dissonant at times, but so would 12TET to someone trained in 19TET.
Secondly, there is the melody. 19TET has seven more notes in the octave than 12TET, which gives many more possibilities for interesting melodies and incidentally, interestingly dissonant harmony. You can also play in any key with the same harmony (which you can with 12TET but can’t with other temperaments). An interesting quirk of the system, due to the fact that 19 is a prime number, means that for all intervals, it takes 19 steps to get back to unison as you increase in regular steps. If that sentence doesn’t make sense, recall that the circle of fifths in 12TET has 12 steps. The circle of augmented fourths in 12TET has only 2. In 19TET the circle of any interval has 19 steps, meaning that all sorts of harmony are possible.
So, all seems well and interesting. But not much to do with melodeons. How can we design a system to cope with the vagaries of 19TET when we are used to 12TET? My initial attempt is below. The notes in red refer to the tones which we would recognise from 12TET. The others are somewhere in between!
Unfortunately, it isn’t a temperament which is at particularly well adapted to a bisonoric layout, but I’ve done my best. This version (I’ve drawn several) has push and pull separated by three ‘semitones’, which corresponds to a whole tone in 12TET. The rows are separated by five ‘semitones’, which corresponds to a minor third.
The Loomes Chromatic and the Atzarin do not change fingering as you go up the octave, whereas the melodeon does. This layout changes fingering every two octaves – the bellows direction reverses (unless you happen to have an unlimited number of rows). Mostly the fingering is fairly sensible, except for a row change every octave. All 12 notes of our chromatic scale are available, and the other notes of the 19 note scale are available. In particular, the ‘reverse diagonal’ gives a 19 tone chromatic scale with no repetitions, which is quite pretty. The distance between octave buttons is slightly larger than on the melodeon, but this is probably not a game-changer.
Let me know what you think. It’s quite an interesting problem and I suspect that it isn’t one that I’ll be able to really get to grips with without an instrument in front of me. If you can think of a better way of presenting the layout then please say so in the comments!
To finish, here is a piece of music in 19TET. It is a work written in some sort of Well Temperament (a topic for another day), so there are some interesting dissonances here and there. Enjoy.