Huygens and 31-tone

A few days ago, we observed the 386th anniversary of the birth of the Dutch polymath Christiaan Huygens. Among his discoveries and inventions were Saturn’s moon Titan, the pendulum clock, and the division of the octave into 31 equal parts as an extension of quarter-comma meantone. Centuries later, another Dutchman, Adriaan Fokker, built an organ using this tuning.

The last of these is lesser known than the others, since we normally use twelve equally-tempered tones per octave today. Also, over a century before, Nicola Vicentino proposed his archicembalo, which was to be tuned in what was essentially 31 equal plus five additional notes in each octave.

Huygens found that 31-tone tuning approximated septimal intervals better than other tunings. The seventh harmonic is often thought of as a flat minor seventh, but he found that in his tuning, it really approximated the augmented sixth. Likewise, 7/6 would be an augmented second, and the 7/5 tritone the augmented fourth.

It is also an example of miracle temperament, discovered by George Secor in 1974. What Huygens missed is that 31-tone also well approximates the 11th harmonic, which would be a perfect fourth raised by a 38.71-cent diesis, or an augmented fourth lowered by the same. Other miracle equal temperaments are 41 and 72.

I have been using a 41-tone subset of 72-equal, using 31 pitch classes; these can be heard in the fourth movement of my First Symphony and one of my early microtonal compositions, "The Waterloo Rag". These are given below using my shape note notation, with alternates for some pitch classes beamed together (these are there to avoid ‘wolves’), with measurements in 72-tone and the just intonation ratios that they represent:

I’ve found that these could also be used for tuning Arabic maqams, with some modifications. For quarter-tone music, A double flat is equivalent to G half sharp, and G double sharp to A half flat, and so on. The fifths, 700 cents, are more in tune than the 696.77-cent fifths of 31-equal. Also, in augmented-second scales such as Hijaz, the minor second is slightly sharp and the major third slightly flat, unlike as in conventional 24-tone tuning.

(I started on G, instead of C or another note, because Yigah, the lowest pitch of the two-octave traditional maqam scale, is a low G, and Partch based his 43-tone just scale on G.)