Jun 11

The mathematics of tuning

Since we’re about to embark on a season of non-standard tunings, ranging from columns of septimal JI intervals (thanks to Mr. Randy Gibson) to 31-note ET (a speculative historical journey with everyone’s favorite musical murderer, Don Carlo Gesualdo da Venosa), I thought it might be interesting to have a quick review of the mathematics of tuning.

We’ve recently discussed JI and ET tunings, and they differ in mathematical foundations. Absurdly briefly put: JI is based on ratios; ET is based on logarithms.

I highly recommend Kyle Gann’s Just Intonation Explained for a basic background on ratio tunings, and to be able to hear exactly what all those numbers mean.

And though I hesitate to link to it for obvious reasons, Wikipedia actually has a very clear writeup on the math of equal temperament! I’ve linked you past the rambling and somewhat questionable narrative section straight to the goods, starting with the Chinese discovery of the logarithmic solution to equal temperament.

May 11

Equal Temperament

Equal temperament or ET is the current tuning framework for most Western music. It is a kind of acoustical compromise, compared with the pure mathematical relationships of just intonation (JI). No intervals are ‘true’ in the system, but the equality of half-steps allows for free modulation to any key, ensuring that each would be as viable as any other. Pitches which in JI would be derived from the lowest primes are generally the best approximated pitches in ET: perfect fifths (3/2 in JI) are 2 cents low in ET, major thirds (5/4) are 14c+, and minor (7/4) sevenths are 31c+. In JI, all pitches are related intervalically to a fundamental (1/1); in ET, pitches are derived as equal logarithmic subdivisions of an interval, most usually the octave. (An interesting exception in this case is the Bohlen-Pierce system, which divides a perfect 12th into 13 equal steps) Thus, instead of the simple ratios involved in JI, the size of each ET halfstep is derived from the twelveth root of 2. Any ET division of the octave can be reached this way. For example, a 24 note scale’s smallest interval can be derived from the twenty-fourth root of 2.

The ET system used in most Western music is 12 note ET, also called 12ET. However, since the early twentieth century (and with a few notable exceptions, hundreds of years before), composers have worked in other equal subidvisions of the octave. 24ET introduces the quarter tone, 36 the sixth tone, 48 the eighth tone, and 72 the twelfth tone. These are the most common divisions, though there are many musics, composers and cultures who use different divisions (Klaus Huber, for example, in his later works, uses 18ET, creating an equal tempered scale of third tones). 19ET has been used as a better compromise for true JI intervals in tonal music than 12ET, differentiating between sharps and flats as differently tuned. 31ET is a system which was approximated by instrument makers and theorists in Italy in the 16th century via a kind of mean-tone temperament. It allows for diatonic (white note) chromatic (accidentals both sharp and flat) and enharmonic (double sharps and flats) genera, extending the range of possible harmonies greatly. Ekmeles will be experimenting in 31ET tuning in the performance of Gesualdo madrigals this Fall, as historical records indicate that Scipone Stella, a composer in Gesualdo’s court, built replicas of Vicentino’s 31ET keyboard instruments.

Non-exhaustive list of composers using ET microtones

Charles Ives (24ET), Alois Hába (24ET, 36ET, 72ET) Julián Carillo (18ET, 24ET, 30ET, 36ET, 42ET, 48ET, 54ET, 60ET, 66ET, 12ET, 78ET, 84ET, 90ET, 96ET [if you don’t know him, you should really check him out!]). James Dillon, Brian Ferneyhough, Liza Lim, and many other second modern or complexist composers make liberal use of ET microtones.

Learning ET microtones

Without the aid of rote learning, ET microtones can be exceptionally difficult to find. Acoustically, further divisions of 12ET rarely become more consonant, with the exception of 11th partial relationships which lie only a few cents away from a quarter tone. I reccommend the use of computer models, and have made use of several. I have occasionally used simple software synths for learning quarter tones. I reprogrammed a fine-tuning knob built into the synth to instead move only in gradations of 50c, and altered the pitches by hand on the fly. This is useful for melodic work, but makes harmonic hearing of quarter tones impossible. OpenMusic is an IRCAM-developed program made for computer assisted composition. A companion program, microplayer, can handle up to 72ET playback in multiple channels. To hear the score of an ET microtonal piece, I can’t just sit down and play it at the piano, so I enter it into OpenMusic, and can hear a completely accurate version of it, harmonically and melodically. When you have a limited amount of time to rehearse with an ensemble for a difficult piece, practicing with a computer model can allow you to devote that rehearsal time to music making, and not to panicking over whether you’re singing the right notes.