Music Theory for Conworlders: An Example: 11-tone Equal Temperament

Looking at something that is very close to 12-tone equal temperament in one sense (very similar size for the semitone), yet very different in other senses (complete lack of perfect fifths and fourths, very sharp major thirds and minor thirds, ...) might be useful.

11-tone equal temperament's lack of fifths might seem to be a rather challenging thing, given the prevalence of fifths in the music we're used to. Let us compare the intervals of 11 and 12. In the cents given, I will underline repeating sequences. 12-edo to the left, 11-edo to the right.

minor second: 100 cents, 109.09... cents
major seconds 200 cents, 218.18... cents
minor third: 300 cents, 327.27... cents
major third: 400 cents, 436.36... cents
perfect fourth: 500 cents, 545.45... cents

Here already we're seeing a pretty large divergence already - the perfect fourth is already almost as close to the tritone as it is to the perfect fourth. We compare:

tritone: 600 cents; 545.45... cents? 654.54... cents?

Therefore, we'll consider the next half of the octave to be flat versions of intervals - rather than sharp versions as we did in the first half:

perfect fifth: 700 cents, 654.54... cents
minor sixth: 800 cents, 763.63... cents
major sixth: 900 cents, 872.72... cents
minor seventh: 1000 cents, 981.81... cents
major seventh: 1100 cents, 1090.90...

So, our main "weird" place is the fourths, fifths, and tritones - none of them seem to have very good intervals to correspond to. Of course the thirds as well are pretty far off - only the two kinds of seconds really do rather well (but even then, they're somewhat off). Here's our familiar composition (the mixolydian version) with all intervals replaced by their closest "neighbour" in the other tuning.

This is not particularly beautiful, is it? We will come back to this composition with some reworkings later on. For now, we'll try and consider scales, chords and chord-scale relationships. When we come back to the piece a bit further down, we'll have "better" chords, and maybe have figured out what particular rotation of what particular scale works best (or at least better) for the composition.

The scale I was using in that piece was, in cents, like this:

0, 218, 436, 545, 655, 872, 981

or in terms of step size:

LLssLsL

or in terms of the distribution of step-sizes between the notes: C _L͡  D _L͡  E_sF_sG _L͡  A_sB _L͡  C. L have ties on them to illustrate that they are 'bigger' than s. If we were to use piano keyboard design to illustrate this scale, it would be like this:

The keys at both ends are C. 

We find that the scale we're now using is not a MOS. (It is a MODMOS, however - a scale that is a 'distortion' of a MOS, given certain restrictions on what kinds of distortions we may use. Many MODMOSes are very useful - and some MOSes might not be! However, let's try and see if a MOS is a good tool for this right now.). By changing one note, we do get a MOS, the results being here:

Now the scale is LsLsLsL. Slightly more of a 'minor' tonality there. We alter the one tone we need altered (viz. we lower one note - basically, our E becomes an Eb). We're now playing on this somewhat more "evenly distributed" keyboard:

Notice again, how the keys at the edge are C:s.

In terms of cents, this is 0, 218, 327, 545, 655, 872, 981. The scale we're now using has a name, the history of which I don't quite know. It's orgone temperament.

Anyways, we may notice the chords are not particularly pretty there - it sounds somewhat off all the way through. We shall go and attempt to see if we could find something even more off - that does sound somewhat good or at least has its own logic, or something else along those lines in the next post. For this, we'll need to look at several different things - chords as approximations of parts of the overtone series, smooth voice movement from chord to chord, and how chords relate to the scales in which they reside.