EQUAL-TEMPERED MAJOR THIRDS ARE DISCOMFITING

On most pianos, a major third is 14 cents wider than a just third (400 cents vs. 5:4 ≈ 386 cents).

Sometimes I walk up to a piano and play a major triad, cold. And it doesn’t sound right. I wish my acoustic piano had a joystick, or a pitch wheel, or that it could produce vibrato, or that it was responsive to changes in my embouchure, so these thirds would sound less jarring.

There’s something unusual about 12-TET. The idea of equal-tempered tones may be hundreds of years old. But a robust, systematic working out of twelve equally-tempered chromatic tones as a foundation for all musical performance is an anomaly in the history of human musical endeavor.

It’s tempting to look at tones in 20th-century popular music as a reaction against an equal-tempered system. There are blue thirds, blue fifths, and other “bent” intervals. A concert piano is equal-tempered, but blues and jazz pianists often play major and minor thirds together, or use harmonies built on fourths that elide third-based harmonies. Keyboard technology allows all sorts of work-arounds (rotary systems in organs, pitch wheels, joysticks, or expression pads in MIDI-based instruments) that muddy otherwise equal-tempered intervals.

In the Euroclassical tradition, one common argument against 12-TET is that it obliterates the differences in character between keys. If you listen to Bach or Chopin keyboard works on unequal-tempered instruments, you will hear the difference right away—especially as you get into keys with lots of sharps or flats.

There’s also a mathematical argument to be made against 12-TET. Traditionally, one reason Western tuning systems have tempered (widened or narrowed) certain intervals is to maintain the purity of others. To give an example, narrowing the perfect fifth slightly allowed Renaissance musicians to access pure thirds. In 12-TET, though, a half-step is defined exactly as the twelfth root of two (12√2), an irrational number. Larger intervals, which are made up of two or more irrational half-steps, are likewise irrational . So,

minor second = 2^1/12 = 1.05946309436…
major second = 2^2/12 = 1.122462…
minor third = 2^3/12 = 1.189207…
major third = 2^4/12 = 1.259921...

until…

perfect octave = 2^12/12 = 2^1 = 2

This means that in 12-TET, virtually every interval is mathematically irrational. In fact, only octaves and unisons can be expressed mathematically as a rational number.

This is a far cry from earlier tuning systems, like Pythagorean tuning, where a minor second might be expressed mathematically as 256:243. An unwieldly number (also 20 cents lower than a pure minor second), but still rational.