Case Study 11.1: The 19-Tone Equal Temperament Scale — What Happens When You Double the Notes?

Introduction: Dividing the Octave Differently

Twelve-tone equal temperament has become so dominant in Western music that most students encounter it as a given — the way music "is," rather than one particular choice among many. But twelve is not the only number of equal divisions that produces a musically useful scale. Mathematicians and musicians have explored dozens of alternatives, and one of the most revealing is nineteen-tone equal temperament (19-TET): an octave divided into nineteen equal steps rather than twelve.

Nineteen-TET is not a theoretical curiosity. It has been used by serious composers, built into physical instruments, and analyzed extensively by music theorists. Studying 19-TET illuminates something important: by changing one number — the count of equal steps — what becomes consonant, what becomes dissonant, and what becomes structurally possible all shift in systematic, physics-determined ways.

The Mathematics of 19-TET

In 12-TET, each semitone represents a frequency ratio of 2^(1/12) ≈ 1.0595. In 19-TET, each step represents 2^(1/19) ≈ 1.0375. The steps are smaller — roughly 63 cents each, compared to 100 cents in 12-TET.

The key question for any equal temperament system is: how well does it approximate the simple integer ratios of the harmonic series? In 12-TET, the best approximations are:

  • Perfect fifth (3:2): 12-TET fifth = 700 cents; just fifth = 702 cents → error of 2 cents
  • Major third (5:4): 12-TET major third = 400 cents; just major third = 386 cents → error of 14 cents
  • Minor third (6:5): 12-TET minor third = 300 cents; just minor third = 316 cents → error of 16 cents

In 19-TET (where each step = 63.2 cents): - Perfect fifth: 11 steps × 63.2 = 695 cents; just fifth = 702 cents → error of 7 cents (worse than 12-TET) - Major third: 6 steps × 63.2 = 379 cents; just major third = 386 cents → error of 7 cents (better than 12-TET!) - Minor third: 5 steps × 63.2 = 316 cents; just minor third = 316 cents → error of 0 cents (essentially perfect!)

This is the central revelation of 19-TET: it trades a slightly worse fifth for dramatically better thirds. The major third error drops from 14 cents (12-TET) to 7 cents (19-TET), and the minor third is nearly perfect. For music centered on tertian harmony — harmony built from thirds, which is virtually all of Western tonal music — 19-TET is acoustically superior to 12-TET.

Unique Harmonic Properties

The improvement in thirds is not the only distinctive feature of 19-TET. Several other properties make it acoustically interesting:

The minor third is nearly pure. At essentially 0 cents error, the 19-TET minor third (ratio 6:5) is as acoustically pure as an equal temperament can achieve for that interval. Minor chords in 19-TET have a clarity and resonance that 12-TET cannot match.

Enharmonic equivalents split apart. In 12-TET, C# and D♭ are the same pitch (enharmonic equivalents). In 19-TET, they are different pitches — C# is two steps (126 cents) above C, while D♭ is one step (63 cents) below D. This separation is not a problem; it's a feature. The distinctions between enharmonic notes that just intonation maintains — and that 12-TET collapses — are preserved in 19-TET. This makes 19-TET capable of much finer harmonic distinctions.

The tritone splits differently. In 12-TET, the tritone is exactly half an octave (6 semitones out of 12). In 19-TET, the augmented fourth (11 steps = 695 cents) and diminished fifth (8 steps = 505 cents... wait, that's not right — in 19-TET, 9 steps = 568 cents and 10 steps = 632 cents) are different-sized intervals. The symmetry of the tritone breaks, giving composers different tools for handling tension and resolution.

Diatonic modes work naturally. The 19-TET version of the major scale uses a different step pattern from 12-TET, but all seven traditional modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) are available and usable with the same relationships between them that Western musicians expect.

Composers Who've Used 19-TET

Joel Mandelbaum wrote his doctoral dissertation on multiple divisions of the octave in 1961, became one of the most important theorists of 19-TET, and composed extensively in the system. His work demonstrated that 19-TET is not merely theoretical — it can support complex, emotionally coherent music in Western harmonic idioms.

Easley Blackwood created a landmark series of recordings called Microtonal Études (1980) that systematically explored equal temperaments from 13-TET through 24-TET. His études in 19-TET are among the most celebrated, demonstrating how familiar harmonic progressions sound when thirds are purer and the fifth is slightly narrower. Many listeners find the 19-TET major chord more beautiful than its 12-TET equivalent precisely because the thirds are cleaner.

Marc Jones and various experimental guitarists have built 19-TET guitars — standard guitar bodies refretted with 19 frets per octave instead of 12. The frets are closer together (requiring some adjustment in technique) but the instrument is fully playable. Recordings from these instruments provide the most accessible introduction to 19-TET for guitarists.

The xenharmonic community — a loose network of microtonalists working online — has produced substantial bodies of music in 19-TET using digital audio workstations and microtonal MIDI controllers. Composers like Kite Giedraitis and others have advocated specifically for 19-TET as a practical upgrade to 12-TET.

What 19-TET Reveals About 12-TET

The existence of 19-TET as a viable, acoustically defensible tuning system raises uncomfortable questions about 12-TET's dominance.

12-TET's major weakness is its thirds. The 14-cent error in the major third is audible to trained listeners and is the reason that a cappella choirs, string quartets, and barbershop ensembles instinctively adjust away from equal temperament when performing without keyboard instruments. The thirds simply sound better when pure. 19-TET solves this problem — at the cost of slightly worse fifths.

12-TET won for historical and practical reasons, not purely acoustic ones. Keyboard instruments became the center of Western music during the Baroque and Classical periods. Keyboard instruments need equal temperament (or some fixed temperament) because their pitches cannot be adjusted in real time. A 12-note keyboard is more practical than a 19-note keyboard — physically simpler, easier to learn. These practical advantages drove 12-TET's adoption, not its acoustic superiority over all alternatives.

Physics doesn't uniquely determine the "right" number. The fact that both 12 and 19 produce musically coherent systems, each with different trade-offs, demonstrates that there is no single "natural" scale determined by physics. Physics constrains the space of possibilities; multiple viable solutions exist within that space. The choice between them is partly aesthetic and partly historical accident.

Could 19-TET replace 12-TET? Probably not in standard acoustic instruments — the retrofitting cost would be enormous, and the transition period would be musically chaotic. But in electronic music, where any tuning is a software parameter, 19-TET is already used by adventurous producers and composers. As music production becomes increasingly software-based, the arbitrary lock-in to 12-TET weakens.

Discussion Questions

  1. The trade-off problem. 19-TET has better thirds but worse fifths than 12-TET. If you were a composer who primarily wrote for string quartet (which can adjust tuning in real time), would you care about this trade-off? What if you wrote primarily for piano? How does your choice of instrument affect your relationship to tuning systems?

  2. Historical contingency. The text argues that 12-TET became dominant partly for practical reasons (keyboard instrument design) rather than purely acoustic reasons. Does this mean 12-TET is "wrong"? Or does it mean that practical considerations are a legitimate part of evaluating a tuning system? What standards should we use to judge tuning systems?

  3. The xenharmonic community. Online communities of microtonalists have composed extensively in 19-TET, 31-TET, 53-TET, and other systems. Their music is largely unknown to mainstream audiences. Is this an example of physics-informed musical creativity being ignored by culture? Or is it evidence that most listeners can't hear the difference — which would suggest acoustic accuracy matters less than other musical qualities?

  4. Perceptual limits. Some research suggests that the 14-cent error of the 12-TET major third is below the threshold of perception for many casual listeners but above it for trained musicians. If most listeners can't hear the difference, does the acoustic argument for 19-TET matter? What does your answer reveal about your values regarding music — is it primarily a physical phenomenon or a perceptual/cultural one?