Certain divisions of the octave are natural because they represent the points at which multiples of the perfect fifth coincide with multiples of the octave. For example, 12.
12 perfect fifths = 12 x 701.955 cents = 8423.46 cents
7 octaves = 7 x 1200 cents = 8400 cents
8400 and 8423 are pretty close, so if you fudge the fifths a little, you can divide the octave into 12 steps and get both fifths and octaves.
Like wise, 19:
19 perfect fifths = 19 x 701.955 cents = 13337 cents
11 octaves = 11 x 1200 cents = 13200
13337 and 13200 are, proportionately pretty close, so 19 ends up being one of the natural divisions of the octave if you want perfect fifths. So do 31, 34, and, more spectacularly, 53.
The Colorado-based guitarist Neil Haverstick plays, and has recorded with, a 19-tone guitar, on his impressive discs, The Gate and Acoustic Stick. The influence of tuning on Haverstick’s blues playing is fun to listen to; he sometimes has to extend the rhythm of blues phrases to fit in all his chromatic pitches. He also plays a 34-tone-to-the-octave guitar. Fretted instruments such as guitars and lutes have the oldest history of playing equal temperaments, since you have to have equal temperament if the frets are going to go straight across the fretboard. Just intonation on guitar (or any other unequal temperament) requires jagged frets that shift up and down for each string. Since at least the 16th century it’s been considered easier just to tune guitars in equal.
The 16th-century theorist Nicola Vicentino invented a 31-tone-to-the-octave harpsichord, the keys divided between two manuals and with some of the black keys split. He claimed that with his 31-tone scale one could play melodies from the Hebrew and Arabic worlds and the Slavic and Germanic countries without distorting them into the Italian scale. He understood that tuning is ultimately a multicultural issue, and that standardized, invariant tuning was a means of oppression used against foreign musicians from allegedly inferior cultures. Incidentally, in 31-tone equal temperament the perfect fifth is a slightly flat 696.77 cents, almost exactly the same size as the meantone fifth Vincentino was used to. [Ed. Note: The 31-tone system has inspired a great many new music composers both in the Netherlands and the United States — American tricesimoprimalists include Joel Mandelbaum, and Jon Catler who in the 1980s led a rock band called J. C. and the Microtones.]
A 53-tone equal temperament has sometimes been held up as a dream tuning. In 53-tone, each pair of adjacent pitches is separated by 22.64 cents. The major third in this scale is playable as 384.9 cents (instead of an optimum 386.3); the minor third is 316.98 cents (instead of 315.6); and the perfect fifth is 701.886 cents (instead of 701.955). All of the (five-limit) intervals of European music can be played within a half of a percent accuracy in 53-tone equal temperament. Inspired by such realizations, an Englishman named T. Perronet Thompson built a 53-pitch organ in the 1850s, its keyboard a Dr. Seussian fantasy of split keys, curved keys, different colored-keys, and knobs sticking up through other keys. Estimated arrival time moving from a C major chord to a G major chord is probably four minutes, as the organist cogitates on where the right keys are, but it was a wonderful idea.
A couple of other modern experiments with equal temperaments should be mentioned. One is Easley Blackwood‘s remarkable series of 12 Microtonal Etudes, each written in a different equal temperament from 13 to 24 pitches to the octave. Blackwood invented his own different notation for each division of the octave; the score is published, and the CD is available on Cedille. The electronic sounds are a little cheesy, and it’s a little disappointing that, instead of treating you to the most unusual intervals, he concentrates wherever possible on intervals found in 12-pitch tuning; for instance, 15-pitch equal contains the same major thirds as 12-pitch. But the tunings themselves are all the weirder for not being grounded in any natural acoustic basis, and they’ll stretch your ears.
Wendy Carlos has also worked with equal-tempered scales not based on the octave, so that you get different pitches from octave to octave. For example, if you have a scale of 35-cent increments, you’ll have a pitch at 1190 cents and 1225, but not at 1200. I haven’t had an opportunity to hear her results, but her Web page – which seems to say virtually nothing about her tunings — has some brief samples of her music. Recently, at the last Festival of Microtonal Music that Johnny Reinhard organizes, I heard Skip LaPlante’s Music for Homemade Instruments group-sing a happy little tune in 13-tone equal temperament. And they really did it. There may be no natural acoustical basis for a 13- division of the octave, but it can be sung, and it blows your mind to hear it.