Just intonation, meantone and well temperaments

Definition

Just intonation builds intervals from ratios of small whole numbers taken from the harmonic series, including the prime number 5 (and sometimes 7). The major triad becomes the ratio 4:5:6, the minor triad 10:12:15: perfectly steady chords, with no beating.

Zarlino's just scale (5-limit)

The three main triads (C–E–G, F–A–C, G–B–D) are all 4:5:6. However, there are two different tones: the major tone 9/8 (203.91 cents: C–D, F–G, A–B) and the minor tone 10/9 (182.40 cents: D–E, G–A). The diatonic semitone is 16/15 (111.73 cents).

The problem: the syntonic comma

The fifth D–A measures 5/3 : 9/8 = 40/27 = 680.45 cents, flat by a syntonic comma (81/80 = 21.51 cents). The D minor triad is out of tune. In just intonation every note changes pitch depending on the chord: the D of the ii chord (D–F–A) must be 21.5 cents lower than the D of the V (G–B–D). Choirs, string quartets and horn sections do exactly this, adjusting notes in real time: major thirds are lowered by about 14 cents compared with the piano, minor thirds are raised by about 16, and the dominant seventh can drop toward 7/4 (−31 cents) in barbershop.

Meantone temperament

In the Renaissance and early Baroque, quarter-comma meantone was used: each fifth is narrowed by 1/4 of a syntonic comma (696.58 cents) so that four fifths produce a pure major third (C–G–D–A–E = 5/4). The tone measures 193.16 cents, the “mean” between 9/8 and 10/9 (hence the name). Strengths: 8 pure major thirds, very sweet triads in keys with few accidentals. Weaknesses: G# and Ab are different notes (G# is lower by about 41 cents); the fifth G#–Eb becomes a wolf of about 737.6 cents. For this reason some historical organs and harpsichords had split keys (D#/Eb and G#/Ab separate).

Well temperaments (circulating temperaments)

Between the late 17th and the 18th century, temperaments spread in which all keys are usable but with different characters: keys near C have almost pure thirds, distant ones have wide, “tense” thirds.

• Werckmeister III (1691): four fifths narrowed by 1/4 of a Pythagorean comma, the others pure.
• Kirnberger III (1779) and Vallotti (mid-18th century): variants still widely used today in early music.

This is the root of the theory of the affects of keys (Mattheson, Schubart): C major “pure”, Eb “heroic”, F# minor “gloomy”. With 12-TET these differences disappear, but they remain in tradition because of the timbre of instruments (open strings, wind keys).

Just intonation in today's music

• A cappella and barbershop: the harmonic seventh 7/4 creates the famous “ring” (the overtone that seems like a fifth voice).
• Synths and DAWs: Surge XT, Pigments, Serum 2 and many others read Scala files (.scl/.tun); plugins such as MTS-ESP exist that dynamically retune chords in just intonation.
• Composers: Harry Partch (43-note scale per octave), La Monte Young (The Well-Tuned Piano), Ben Johnston (quartets in extended just intonation), Jacob Collier (controlled microtonal drift).
• Non-Western music: Indian shruti (22 per octave, in theory) are conceived as ratios; see the cards on ragas and maqam.

Listening example

Listen to a professional choir singing Allegri's “Miserere” and then the same harmony on the piano: in the choir the final major thirds “lock in”. In barbershop (e.g. The Buffalo Bills) dominant sevenths sound without beating thanks to 7/4.

Exercise

1. With a synth that reads .scl files, load a just tuning on C and play I–IV–V–I: listen to the purity. Then play ii–V–I and hear the out-of-tune D minor.
2. Record your own three-part vocal harmony and analyze it with Melodyne: measure in cents how much lower the major thirds are compared with the grid.
3. With two sine oscillators create a 7/4 seventh (e.g. 200 Hz and 350 Hz) and compare it with the tempered minor seventh (200 Hz and 356.4 Hz).

Related topics

Pythagorean tuning and comma; Equal temperament and cents; The harmonic series and timbre; Vocal harmonies and voicing for choir.