Western Tunings

1. Pythagorean Tuning (c. 500 BCE)

A. Historical Genesis: The Sacred Tetractys & Monochord Chant

Pythagorean tuning arose in ancient Greece in the 6th century BCE through the philosophical school of Pythagoras of Samos, later formalized in the Euclidean treatise Sectio Canonis (c. 300 BCE). Pythagoras investigated string acoustics using the monochord—a single string stretched over a movable wooden bridge. By dividing the string into simple integer lengths, he discovered that stopping the string at 1/2 produced the octave (2:1), stopping at 2/3 produced the fifth (3:2), and stopping at 3/4 produced the fourth (4:3).

To the Pythagoreans, this was divine: the numbers 1, 2, 3, and 4 comprised the sacred Tetractys, summing to 10 (the symbol of cosmic harmony and the "Music of the Spheres"). Consequently, Pythagorean acoustics strictly prohibited any prime numbers beyond 2 and 3.

During the European Middle Ages (c. 500–1400 CE), Pythagorean tuning reigned supreme. Medieval Christian plainchant (Gregorian chant) and early Gothic polyphony (such as the organum of Léonin and Pérotin at Notre Dame Cathedral, Paris) were sung in vast stone sanctuaries. In these reverberant acoustics, pure fourths (4:3) and pure fifths (3:2) rang like celestial bells. Thirds and sixths were considered harsh dissonances (discordiae) because their Pythagorean versions beat rapidly. Cadences resolved exclusively to bare, open fifths and octaves.

B. Mathematical Mechanism: The Chain of Pure Fifths

Every note in the Pythagorean system is generated by stacking pure fifths of ratio 3/2 (701.96 cents), and reducing octaves by dividing by 2 whenever the pitch exceeds the octave boundary.

C. Step-by-Step Derivation of All 12 Notes

Starting from reference C4 = 261.63 Hz (0.0 cents, ratio 1/1):

1. G4 (Fifth 1 Up): Multiply by 3/2 = 3/2 (1.50000). Cents: 1200 * log2(3/2) = 701.96 ¢. Freq: 261.63 * 1.5 = 392.44 Hz.
2. D4 (Fifth 2 Up): (3/2) * (3/2) = 9/4. Exceeds octave; divide by 2 = 9/8 (1.12500). Cents: 203.91 ¢. Freq: 261.63 * (9/8) = 294.33 Hz. This is the pure Major Tone (Epogdoon).
3. A4 (Fifth 3 Up): (9/8) * (3/2) = 27/16 (1.68750). Cents: 905.87 ¢. Freq: 261.63 * (27/16) = 441.49 Hz. Pythagorean Major Sixth.
4. E4 (Fifth 4 Up): (27/16) * (3/2) = 81/32. Divide by 2 = 81/64 (1.26563). Cents: 407.82 ¢. Freq: 261.63 * (81/64) = 331.12 Hz. The Ditone! A full 21.51 ¢ sharper than pure 5/4 (386.31 ¢), creating an acoustic rattle of ~16 beats/sec.
5. B4 (Fifth 5 Up): (81/64) * (3/2) = 243/128 (1.89844). Cents: 1109.78 ¢. Freq: 261.63 * (243/128) = 496.68 Hz. Major Seventh.
6. F#4 (Fifth 6 Up): (243/128) * (3/2) = 729/256. Divide by 2 = 729/512 (1.42383). Cents: 611.73 ¢. Freq: 261.63 * (729/512) = 372.51 Hz. Pythagorean Tritone.
7. C#4 (Fifth 7 Up): (729/512) * (3/2) = 2187/1024. Divide by 2 = 2187/2048 (1.06787). Cents: 90.22 ¢. Freq: 261.63 * (2187/2048) = 275.60 Hz. Chromatic Semitone (Apotome).
8. G#4 (Fifth 8 Up): (2187/2048) * (3/2) = 6561/4096 (1.60181). Cents: 792.18 ¢. Freq: 261.63 * (6561/4096) = 413.31 Hz. Augmented Fifth.
9. F4 (Fifth 1 Down): 1 / (3/2) = 2/3. Octave up: (2/3) * 2 = 4/3 (1.33333). Cents: 498.05 ¢. Freq: 261.63 * (4/3) = 348.83 Hz. Pure Fourth.
10. Bb4 (Fifth 2 Down): (4/3) / (3/2) = 8/9. Octave up: (8/9) * 2 = 16/9 (1.77778). Cents: 996.09 ¢. Freq: 261.63 * (16/9) = 465.11 Hz. Pythagorean Minor Seventh.
11. Eb4 (Fifth 3 Down): (16/9) / (3/2) = 32/27. Already in octave: 32/27 (1.18519). Cents: 294.13 ¢. Freq: 261.63 * (32/27) = 310.08 Hz. Pythagorean Minor Third (Semiditone).
12. Ab4 (Fifth 4 Down, Alternative to G#): (32/27) / (3/2) = 64/81. Octave up: 128/81 (1.58025). Cents: 792.18 ¢ if G# (apotome from G) or 813.69 ¢ if Ab (limma from G).

2. Just Intonation (5-Limit Ptolemaic, Renaissance)

A. Historical Genesis: The Renaissance Polyphonic Revolution

By the 15th century, Western music underwent an aesthetic revolution. English composers like John Dunstable (c. 1390–1453) brought the contenance angloise ("English guise") to continental Europe—a luscious choral style saturated with parallel sweet thirds and sixths. Franco-Flemish masters like Guillaume Du Fay, Johannes Ockeghem, and Josquin des Prez transformed the major triad into the cornerstone of European art music.

Under Pythagorean tuning, these triads were acoustically unusable: the ditone third (81/64 = 407.82 cents) clashed with vocal harmonics, producing a rough, unpleasant beating. Theorists realized that human vocalists intuitively flattened their thirds toward the natural 5th harmonic of the overtone series: ratio 5/4 (386.31 cents).

In 1482, the Spanish mathematician and theorist Bartolomé Ramos de Pareja published Musica Practica in Bologna, boldly abandoning the sacred Pythagorean monochord divisions and proposing 5/4 for the major third and 6/5 for the minor third. In 1558, Gioseffo Zarlino of Venice formalized this in Le Istitutioni Harmoniche by expanding the ancient Tetractys (1–4) into the Senario (numbers 1 to 6). For Zarlino, any interval formed by ratios of the numbers 1 through 6 was natural, divine, and consonantal. The ideal major triad was defined by the harmonic proportion 4 : 5 : 6 (1.000 : 1.250 : 1.500).

B. Mathematical Mechanism: The Three Pure Harmonic Triads

5-Limit Just Intonation constructs the entire diatonic scale around the three fundamental harmonic pillars of tonal music: the Tonic (C), the Subdominant (F), and the Dominant (G). Each triad is tuned to the exact pure proportion 4 : 5 : 6:

C. Step-by-Step Derivation of All 12 Notes

1. C4: Tonic fundamental = 1/1 (0.0 ¢, 261.63 Hz).
2. C#4 / Db4: Diatonic semitone below D or minor second above C: 16/15 (inversion of major seventh 15/8): 2 / (15/8) = 16/15 (1.06667) → 111.73 ¢ → 279.07 Hz.
3. D4: Dominant fifth of G: (3/2 * 3/2) / 2 = 9/8 (1.12500) → 203.91 ¢ → 294.33 Hz. (Major tone).
4. Eb4: Pure minor third from C: 6/5 (1.20000) → 315.64 ¢ → 313.95 Hz.
5. E4: Pure major third of the tonic triad: 5/4 (1.25000) → 386.31 ¢ → 327.03 Hz. Completely beatless!
6. F4: Subdominant root (pure fourth below C octave): 2 / (3/2) = 4/3 (1.33333) → 498.05 ¢ → 348.83 Hz.
7. F#4: Pure major third above D: (9/8) * (5/4) = 45/32 (1.40625) → 590.22 ¢ → 367.92 Hz.
8. G4: Pure fifth of the tonic triad: 3/2 (1.50000) → 701.96 ¢ → 392.44 Hz.
9. Ab4: Pure minor sixth (inversion of pure 5/4 third): 2 / (5/4) = 8/5 (1.60000) → 813.69 ¢ → 418.60 Hz.
10. A4: Pure major third of the subdominant triad: (4/3) * (5/4) = 5/3 (1.66667) → 884.36 ¢ → 436.05 Hz.
11. Bb4: Harmonic minor seventh: (3/2) * (6/5) / (2) ... or 9/5 = 9/5 (1.80000) → 1017.60 ¢ → 470.93 Hz. (Acoustically pure minor 7th, identical to Indian Komal Ni!).
12. B4: Pure major third of the dominant triad: (3/2) * (5/4) = 15/8 (1.87500) → 1088.27 ¢ → 490.55 Hz.
13. C5: Octave = 2/1 (1200.0 ¢, 523.25 Hz).

D. The Fatal Dilemma of Just Intonation

3. Quarter-Comma Meantone Temperament (16th–17th Century)

A. Historical Genesis: The Keyboard Compromise

As pipe organs, harpsichords, and virginals became central to European sacred and secular music in the 16th century, musicians demanded an answer to the Just Intonation crisis. A keyboard cannot dynamically alter pitch like a choir. How could an organist play pure major thirds without suffering the 40/27 wolf on D or having two different keys for D?

In 1523, the Venetian theorist Pietro Aaron published Toscanello de la musica, describing a brilliant tuning practice already used by master organ builders. In 1577, the blind Spanish organist and theorist Francisco de Salinas published the complete mathematical formulation in De musica libri septem at Salamanca: Quarter-Comma Meantone Temperament.

Meantone was the undisputed sovereign tuning of European keyboard music for over two centuries, shaping the masterpieces of William Byrd, Girolamo Frescobaldi, Jan Pieterszoon Sweelinck, Heinrich Schütz, and Dieterich Buxtehude.

B. Mathematical Mechanism: Taming the Syntonic Comma

Salinas identified the exact acoustic geometry:

C. Step-by-Step Derivation of All 12 Notes

Meantone tunes a chain of 11 quarter-comma fifths (Q = 696.578 cents) traditionally centered on D or C, running from Eb up to G#:

1. C4: Origin = 0.00 ¢ (261.63 Hz).
2. G4 (+1 Fifth): 0 + 696.58 = 696.58 ¢ → 391.21 Hz. (Narrowed by 5.4 ¢).
3. D4 (+2 Fifths): 696.58 + 696.58 - 1200 = 193.16 ¢ → 292.45 Hz. (The Mean Tone!).
4. A4 (+3 Fifths): 193.16 + 696.58 = 889.74 ¢ → 437.38 Hz.
5. E4 (+4 Fifths): 889.74 + 696.58 - 1200 = 386.31 ¢ → 327.03 Hz. Exact pure 5/4 third! Zero beating.
6. B4 (+5 Fifths): 386.31 + 696.58 = 1082.89 ¢ → 489.00 Hz. Pure major third above G (696.58 + 386.31 = 1082.89 ¢).
7. F#4 (+6 Fifths): 1082.89 + 696.58 - 1200 = 579.47 ¢ → 365.64 Hz. Pure major third above D (193.16 + 386.31 = 579.47 ¢).
8. C#4 (+7 Fifths): 579.47 + 696.58 - 1200 = 76.05 ¢ → 273.35 Hz. Pure major third above A (889.74 + 386.31 - 1200 = 76.05 ¢).
9. G#4 (+8 Fifths): 76.05 + 696.58 = 772.63 ¢ → 408.74 Hz. Pure major third above E (386.31 + 386.31 = 772.63 ¢).
10. F4 (-1 Fifth down): 1200 - 696.58 = 503.42 ¢ → 349.91 Hz. Pure major third below A (889.74 - 386.31 = 503.42 ¢).
11. Bb4 (-2 Fifths down): 503.42 - 696.58 + 1200 = 1006.84 ¢ → 468.01 Hz. Pure major third below D (193.16 + 1200 - 386.31 = 1006.84 ¢).
12. Eb4 (-3 Fifths down): 1006.84 - 696.58 = 310.26 ¢ → 312.92 Hz. Pure major third below G (696.58 - 386.31 = 310.26 ¢).

D. The Price of Meantone: The Great Wolf & Key Confinement

4. Well Temperament (Werckmeister III, 1691)

A. Historical Genesis: The Dawn of Total Modulation & Bach

By the late 17th century, High Baroque composers rebelled against the prison of meantone. Opera, cantata, and organ music demanded adventurous, dramatic modulations: shifting from C minor to Ab major, or plunging into F# minor to depict grief, death, and resurrection. Organists could not retune massive multi-pipe cathedral organs mid-performance.

In 1691, the organist, composer, and acoustic theorist Andreas Werckmeister published Musicalische Temperatur in Quedlinburg, Germany. Werckmeister proposed a revolutionary concept: the circulating temperament (Wohltemperiert, or "Well Temperament").

Rather than dumping all comma errors into a single catastrophic wolf fifth, Werckmeister distributed the 23.46-cent Pythagorean comma across several fifths in unequal proportions. The result was historic: the wolf was completely banished, all 24 major and minor keys became fully playable without retuning, yet each key retained a distinct emotional personality or "key color" (Affekt).

In 1722 (and again in 1742), Johann Sebastian Bach crowned this acoustic victory with his masterwork, Das Wohltemperierte Clavier (The Well-Tempered Clavier, BWV 846–893). Bach wrote 24 Preludes and Fugues in Book I and another 24 in Book II—one pair for every single major and minor key from C major to B minor—to demonstrate that music could now voyage across the entire universe of harmony.

B. Mathematical Mechanism: The Four Tempered Fifths of Werckmeister III

Among several temperaments devised by Werckmeister, his "Number III" (often designated Werckmeister I in modern German catalogues) is the most celebrated:

C. Step-by-Step Derivation of All 12 Notes

Tracing the circle of fifths step-by-step from C4 = 0.00 ¢ (261.63 Hz):

1. C4: Tonic reference = 0.00 ¢ → 261.63 Hz.
2. G4 (Tempered Fifth 1): 0 + 696.09 = 696.09 ¢ → 391.10 Hz. (Narrowed by 5.9 ¢).
3. D4 (Tempered Fifth 2): 696.09 + 696.09 - 1200 = 192.18 ¢ → 292.28 Hz. (Narrowed by 5.9 ¢).
4. A4 (Tempered Fifth 3): 192.18 + 696.09 = 888.27 ¢ → 437.01 Hz. (Narrowed by 5.9 ¢).
5. E4 (Pure Fifth from A): 888.27 + 701.96 - 1200 = 390.22 ¢ → 327.77 Hz. A brilliant result! Because C-G, G-D, and D-A were all tempered, the major third C-E is 390.22 ¢—only 3.9 cents wider than pure 5/4 (386.31 ¢). C major sounds sweet, peaceful, and almost beatless!
6. B4 (Pure Fifth from E): 390.22 + 701.96 = 1092.18 ¢ → 491.64 Hz.
7. F#4 (Tempered Fifth 4): 1092.18 + 696.09 - 1200 = 588.27 ¢ → 367.51 Hz. (Final tempered fifth).
8. C#4 (Pure Fifth from F#): 588.27 + 701.96 - 1200 = 90.22 ¢ → 275.60 Hz.
9. G#4 (Pure Fifth from C#): 90.22 + 701.96 = 792.18 ¢ → 413.31 Hz.
10. Eb4 / D#4 (Pure Fifth from G#): 792.18 + 701.96 - 1200 = 294.13 ¢ → 310.08 Hz.
11. Bb4 (Pure Fifth from Eb): 294.13 + 701.96 = 996.09 ¢ → 465.11 Hz.
12. F4 (Pure Fifth from Bb): 996.09 + 701.96 - 1200 = 498.05 ¢ → 348.83 Hz.
13. Check: F to C: 498.05 + 701.96 = 1200.00 ¢. An exact pure fifth of 701.96 cents closes the circle!

D. The Aesthetics of Key Color (Affektenlehre)

5. 12-Tone Equal Temperament (12-TET)

A. Historical Genesis: Straight Frets, Abacuses, and the Industrial Era

While keyboard players cherished the subtle key colors of well temperament, players of fretted instruments—the lute, viol, guitar, and the Chinese pipa—faced a stubborn mechanical obstacle. On a lute or guitar, frets are tied straight across the wooden neck beneath all six or twelve strings. If a fret is positioned for a pure major third on one string, it simultaneously ruins the notes on adjacent strings tuned to fourths!

In 1581, Vincenzo Galilei (lutenist, theorist of the Florentine Camerata, and father of the astronomer Galileo Galilei) published Dialogo della musica antica et della moderna. Vincenzo argued fiercely against Zarlino's pure fifths, proposing that lutes must use an equal semitone ratio of 18:17 (about 98.95 cents per fret) so that frets could run perpendicular across all strings without unisons clashing.

The exact mathematical calculation of equal temperament requires taking the twelfth root of two (2^(1/12)). In 1584, during the Ming Dynasty in China, Prince Zhu Zaiyu (朱載堉) accomplished this historic feat in his monumental treatise Lüxue Xinshuo ("A New Theory of the Science of Pitch Pipes"). Using an enormous 81-position counting abacus, Zhu calculated the ratio of equal temperament to 25 decimal places:
2^(1/12) ≈ 1.059463094359295264561825
Zhu cast a set of 12 precisely tuned bronze pitch pipes, becoming the first human in history to achieve exact acoustic equal temperament.

In Europe, the mathematician Simon Stevin calculated 2^(1/12) around 1605 (published in Vande Spiegheling der Singconst), followed by Marin Mersenne in Harmonie Universelle (1636). Yet European keyboard players resisted equal temperament for over two centuries because its thirds were noticeably coarse.

Equal temperament's ultimate victory was driven by the 19th-century Industrial Revolution:

B. Mathematical Mechanism: The Twelfth Root of Two

In 12-TET, the octave is divided into 12 mathematically identical geometric steps:

C. Step-by-Step Derivation of All 12 Notes

Calculated from standard reference C4 = 261.6256 Hz (0.0 cents):

1. C4 (k=0): 261.6256 * 2^(0/12) = 0.00 ¢ → 261.63 Hz. Ratio 1.000000.
2. C#4 (k=1): 261.6256 * 2^(1/12) = 100.00 ¢ → 277.18 Hz. Ratio 1.059463. (+9.8 ¢ vs Pyth, -11.7 ¢ vs Just, +24.0 ¢ vs Meantone).
3. D4 (k=2): 261.6256 * 2^(2/12) = 200.00 ¢ → 293.66 Hz. Ratio 1.122462. (-3.9 ¢ vs Pyth/Just, +6.8 ¢ vs Meantone).
4. Eb4 (k=3): 261.6256 * 2^(3/12) = 300.00 ¢ → 311.13 Hz. Ratio 1.189207. (+5.9 ¢ vs Pyth, -15.6 ¢ vs Just, -10.3 ¢ vs Meantone).
5. E4 (k=4): 261.6256 * 2^(4/12) = 400.00 ¢ → 329.63 Hz. Ratio 1.259921. +13.69 ¢ sharper than pure 5/4 third! Causes ~10.3 beats/second acoustic rattle in C-E.
6. F4 (k=5): 261.6256 * 2^(5/12) = 500.00 ¢ → 349.23 Hz. Ratio 1.334840. (+2.0 ¢ vs Pyth/Just, -3.4 ¢ vs Meantone).
7. F#4 (k=6): 261.6256 * 2^(6/12) = 600.00 ¢ → 369.99 Hz. Ratio sqrt(2) ≈ 1.414214. Exact midpoint of the octave.
8. G4 (k=7): 261.6256 * 2^(7/12) = 700.00 ¢ → 391.99 Hz. Ratio 1.498307. 1.96 ¢ flat of pure 3/2 fifth (slow, subtle beat of ~0.9 Hz, practically imperceptible).
9. Ab4 (k=8): 261.6256 * 2^(8/12) = 800.00 ¢ → 415.30 Hz. Ratio 1.587401. (-13.7 ¢ vs Just, +27.4 ¢ vs Meantone).
10. A4 (k=9): 261.6256 * 2^(9/12) = 900.00 ¢ → 440.00 Hz. Ratio 1.681793. Concert pitch anchor!
11. Bb4 (k=10): 261.6256 * 2^(10/12) = 1000.00 ¢ → 466.16 Hz. Ratio 1.781797. (-17.6 ¢ vs Just 9/5).
12. B4 (k=11): 261.6256 * 2^(11/12) = 1100.00 ¢ → 493.88 Hz. Ratio 1.887749. (+11.7 ¢ vs Just 15/8).
13. C5 (k=12): 261.6256 * 2^(12/12) = 1200.00 ¢ → 523.25 Hz. Ratio 2.000000. Pure octave.

D. The Double-Edged Sword of 12-TET & The Harmonium Clash in India