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.

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10 comments

  1. Hi Jeffery. So nice to read the thoughts of someone learning microtonal equal temperaments. Many people using 72 equal believe it *does* offer more consonant intervals, such as the pure third (383 cents in 72 ET, just 3 cents smaller than 5/4) and the pure seventh (967 cents in 72, 2 cents smaller than 7/4). This is why Ezra Sims uses 72, for example. (Not my own personal reason, but I thought I’d point it out.) In fact, for the majority of trained professional musicians, who cannot discern a difference of just 2 or 3 cents, you could say that these small temperaments *allow* composers to write in JI – that a 5/4 third and a third just a few cents away from it are in essence the same thing.

    Also, I’m curious about what you mean by rote learning for microtonal equal temperaments. All of this interests me!

    -Julia

  2. Julia – I did neglect to mention the possibilities of ET systems approximating JI intervals! Thanks for bringing it up. 72ET approximates very (and probably imperceptibly) well up to 11-limit JI systems, which is part of its great flexibility.

    By rote learning I just mean playing the intervals or the entire part back to me in some way – depending on the complexity of the part, it can mean either preparing or using a full computer realization of the work or working with a retuned midi keyboard to play back pitches freely within the scale. I’ve done both, though the latter requires writing out a kind of keyboard tablature to be able to play the part when there are a lot of notes (as I had to when I was learning Barstow).

    • I see. I guess the same method would be helpful for complex JI music, as well. Whereas with simpler textures in JI you can kind of just feel your way to the pure thirds and sevenths.

      The Barstow vocal part (the intoning voice part) is more a matter of spoken inflections than just tuning, though – am I right? A kind of sprechstimme. (Against his ensemble of tuned instruments.)

      You might like to ask Avery Griffin how he worked on the songs Ezra Sims wrote for NotaRiotous (in 72 JI, but a complex texture). It’s all very exciting.

      I sure wish I could hear Ekmeles.

      • The original score of Barstow just has 12-note notation, which you’re supposed to to make fit with the ensemble tunings. I performed Ben Johnston’s string quartet arrangement which has his JI accidentals on every note of the vocal part, so though it is still “intoned” and not sung, I learned the actual pitches.

        Maybe we can make a trip up your way some time soon! We’d love to sing for you.

        • That would be great. BTW, I’m not sure if I’ve told you that I teach a course in microtonal (72 equal tempered) ear training, performance and composition in Boston at NEC. I’m working on a new textbook now, in fact. That’s why your comments on learning music in microtonal ETs interest me – to compare with my students’ comments (not to mention my own experiences). Well, we’ll have plenty of chance to discuss it more in the fall, I guess.

          • Wow – I wish I had a class like that! I’ve been forced to fend for myself. Yes we’ll definitely talk more about your work. Thanks for your input!

  3. A few tiny (possible) typos.

    “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-. ”
    – you mean 31c+, not 31c-

    “24ET introduces the quarter tone, 36 the sixth tone, 48 the eighth tone, and 72 the twelveth tone.”
    -twelfth, not twelveth

    “19ET has been used as a better compromise for true JI intervals in tonal music than 13ET”
    – what does 13 have to do with it? Do you mean 12?

    Thanks for introducing me to some new names, btw. Keep up the great work with your blog.

    • Thanks for spotting those, Philipp! And regarding the keyboard tunings, that’s an intriguing point. I’ve always been interested in the relationship between fixed-pitch instruments and vocal tuning practice, and how they influence one another theoretically and practically.

  4. In addition to the harmonic affordances of 19et and 31et it’s at least interesting to also note that their popularity (if that’s what you’d call it) come from their convenient adaptability vis a vis the traditional keyboard layout.

    In 12et, whole steps have one intervening note, and half steps have zero.

    In 19et, whole steps have two intervening notes and half steps have one (C-C#-Db-D; E-E# (or Fb) -F)

    In 31et, whole steps have four intervening notes and half steps just two (C, C+, C#, Db, D-, D; E-E#-Fb-F)

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