Questions: Microtonal Systems and Harmonic Implications

Question 1

A composer trained in 12-TET wants to compose in 31-ET by 'stretching' their usual harmonic vocabulary to fit the new tuning. What is the fundamental problem with this approach?

• A. 31-ET has fewer consonant intervals than 12-TET, so the vocabulary would need to shrink, not stretch
• B. In 31-ET, enharmonic equivalents like C# and Db are genuinely distinct pitches, so the 12-TET system of equivalences breaks down and the harmonic grammar must be rebuilt from scratch
• C. 31-ET cannot approximate any just-intonation intervals, making harmonic analysis impossible
• D. Nothing is fundamentally wrong; 31-ET is essentially 12-TET with higher resolution

Answer: (B)

In 12-TET, C# and Db are the same pitch — enharmonic equivalence is a structural feature of the system. In 31-ET they are separated by a diesis (~41 cents), making them genuinely distinct. This is not a small adjustment; it means interval classes, set equivalences, and even the concept of 'tritone substitution' all depend on the specific modular arithmetic of 31 rather than 12. The theoretical apparatus — transposition, inversion, prime forms — must be rebuilt around mod-31 arithmetic, not adapted from mod-12.

Question 2

The Bohlen-Pierce scale is built on what structural foundation that sets it apart from most Western microtonal systems?

• A. It divides the perfect fifth (3:2) into equal logarithmic steps
• B. It divides the tritave (3:1 frequency ratio) into 13 equal steps, replacing the octave as the equivalence interval
• C. It is derived from the overtone series of a specific instrument's timbre
• D. It uses 19 equal steps instead of 12, producing a closer approximation to just intervals

Answer: (B)

Most tuning systems — 12-TET, 19-ET, 31-ET — divide the octave (2:1). Bohlen-Pierce instead takes the tritave (3:1, a frequency ratio of three to one) as its equivalence interval and divides it into 13 equal logarithmic steps. This produces a scale with no conventional octave equivalence but strong consonances built on 3:5:7 ratios. It is not a variation of octave-based tuning but a fundamentally different structural choice.

Question 3

Analyzing a microtonal piece requires rebuilding theoretical concepts — interval classes, set equivalences, transposition operators — specific to that system's step count.

• T. True
• F. False

Answer: True

In 12-TET, the interval class of a tritone is 6 (out of 12 possible pitch classes), and the transposition operator T_n cycles through 12 steps. In 19-ET, every interval has a different class number out of 19, and what was a 'tritone' (6 semitones in 12-TET) has no direct equivalent. Set-class equivalences, inversional symmetry, and prime forms are all relative to the modular structure of the specific system. You cannot import 12-TET pitch-class theory into 19-ET or 31-ET without rebuilding it.

Question 4

A microtonal piece composed in 24-TET (quarter-tone tuning) can be analyzed using standard 12-TET pitch-class theory, since 12-TET is simply a subset of 24-TET.

• T. True
• F. False

Answer: False

While 12-TET pitches do appear within 24-TET (every other step), the harmonic structure of a 24-TET composition includes intervals that have no 12-TET equivalents (quarter tones), and the equivalence classes, consonance hierarchies, and voice-leading norms are defined within the 24-step system. Using 12-TET theory to analyze a work that exploits quarter-tone intervals misses what is structurally essential to those intervals — it would be like analyzing functional tonal harmony with only pentatonic theory.

Question 5

Why does moving from 12-TET to a different equal-temperament system (such as 31-ET) require rebuilding music-theoretical concepts from scratch, rather than simply adjusting interval sizes?

Model answer: In any n-ET system, the theoretical concepts — interval classes, inversional equivalence, set classes, transposition operators — are defined relative to the modular arithmetic of n pitch classes. In 12-TET, mod-12 arithmetic defines which intervals are equivalent, what counts as an inversion, and how set classes are catalogued. In 31-ET, this becomes mod-31 arithmetic: intervals that were equivalent in 12-TET (like the tritone and its inversion) are now distinct, new approximations to just intervals (like the harmonic seventh) become available, and the entire structure of consonance and equivalence reorganizes. The grammar is not stretched — it is rewritten.

An analogy: switching from a 12-hour clock to a 31-hour clock doesn't just change the spacing between hours — it changes which times are equivalent (mod 12 vs mod 31) and therefore the entire structure of time-keeping relationships.