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Western Tunings

Why a piano cannot keep every fifth and every third pure, and the five compromises musicians actually used, in the order they appear. Each word is defined the first time it is needed. The table that lines the notes up comes after the story.

You can hear them on the tuning workbench. The Sa in the bar there is the note this page calls C. Every number printed here stays at C = 261.63 Hz, the C next to the A that tuners set at 440, so the tables do not change when the bar does.

1. The problem: pure fifths do not close the octave

A keyboard has twelve notes and then the same notes again, an octave higher. The ear also wants certain gaps to be pure. Those two demands do not fit.

An octave is the same note, twice as high. Its ratio is 2/1: the upper note vibrates twice while the lower one vibrates once. A fifth is the next most stable gap, from C up to G on a piano. Its pure ratio is 3/2. A major third is the gap from C up to E, four piano steps, and its pure ratio is 5/4, the fifth harmonic of a string brought down into the octave. A cent measures a gap. 100 cents is one step of a modern piano, and an octave is 1200 cents. The word is modern: Alexander Ellis added it to his 1875 translation of Helmholtz. This page uses it so tunings from different centuries can be compared. The musicians in the story did not count cents.

Twelve fifths are not seven octaves. A tuner does not measure every key from C. He tunes the one gap he can hear lock, a pure fifth: C until G is pure, then G until D, then D until A, and on through the twelve names. Twelve, because that is how many names the keyboard has, and the twelfth fifth ought to be the C he started from. Each fifth climbs, so he keeps dropping the note by an octave. An octave drop does not change the name, only which octave it sits in. Twelve fifths climb about seven octaves. If they were exactly seven, the last note would be that C. They are a little more:

(3/2)^12 = 531441 / 4096 = 129.7463...
2^7 = 128

The fifths overshoot by 531441 / 524288, which is 23.46 cents, about a quarter of a piano step. That leftover is the Pythagorean comma. If every fifth stays pure, the circle does not close. One fifth has to be wrong. That sour fifth is called a wolf, because the beating howls.

Four fifths are not a pure third. The box above was about getting back to C. This one is about E. In a C major chord, C–E–G, the E that does not beat against C is the pure major third, 5/4. The tuner from the box above never aims at that E. His path is C to G, G to D, D to A, A to E: four fifths, and the fourth one lands on a note he also calls E. Four fifths climb a little more than two octaves, so he drops two octaves, which again does not change the name. That fifths-built E is not the one the chord wants:

Four fifths, brought down two octaves: (3/2)^4 / 4 = 81/64 = 407.82 cents
The pure major third: 5/4 = 386.31 cents

The fifths land 81/80 sharper, which is 21.51 cents. That leftover is the syntonic comma. On the Eastern page, later writers use this same fraction for Bharata's one-śruti unit. Bharata did not write 81/80.

A singer can put E wherever the chord wants it. A keyboard has one key for E. That key cannot be both the E reached by four pure fifths, about 408 cents, and the pure third, about 386 cents. It also cannot keep every fifth pure and still land on the same C. One of those pure gaps has to be narrowed or widened. A temperament is that choice: which gap to spoil, and by how much, so the twelve notes fit in one octave and the instrument can change key.

Every tuning below is one such choice. The order is the order of the years: fifths only, then pure thirds, then a keyboard that keeps the thirds and accepts a wolf, then a tuning in which every key works but they do not all sound alike, then twelve equal steps. The table that sets them on one keyboard comes after that.

2. How each compromise was built

Fifths only, then pure thirds, then meantone, then Werckmeister, then twelve equal steps. Werckmeister's full chart is on Other tunings. He is not a column in the table that follows this section.

1. Pythagorean tuning (6th century BCE, written down by about 300 BCE)

Who used it, and why

A monochord is one string and a movable bridge. Stop the string at half its length and the sounding part is an octave, 2/1. Stop it so two thirds still sound and you have a fifth, 3/2. Three quarters gives a fourth, 4/3. The school of Pythagoras of Samos, in the 6th century BCE, treated these as the intervals that mattered. The written proof we still have is the Sectio Canonis, the division of the monochord, from about 300 BCE and attributed to Euclid. The ratios in this tuning are built only from 2 and 3. The pure third, which needs a 5, is not in the set.

From about 500 to 1400 this is the tuning of European chant and of the early music for several voices at Notre Dame in Paris. In a stone church a pure fifth rings clean. The third this tuning produces, 81/64, beats fast, so composers of that period treat thirds as harsh and end phrases on fifths and octaves.

How the notes are made

Start at C and go up by pure fifths, 3/2, which is 701.96 cents. Whenever the note climbs past the octave, divide by 2 to bring it back. Eleven of the fifths can be pure. The twelfth, the one that should have closed the circle, is the wolf: about 678 cents, roughly 23 cents narrow of a pure fifth.

The Pythagorean Chain (Centered on D):

Eb ↔ Bb ↔ F ↔ C ↔ G ↔ D ↔ A ↔ E ↔ B ↔ F# ↔ C# ↔ G#

Eight fifths up from C reach G♯: G, D, A, E, B, F♯, C♯, G♯.
Three fifths down from C reach E♭: F, B♭, E♭.
Twelve notes, and then the circle fails. The last step, from G♯ to E♭, is the wolf fifth, 678.49 cents, about 23 cents narrower than 3/2.

The twelve notes, one at a time

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).
Note Generation Step Exact Ratio Interval Cents (¢) Freq (Hz) vs 12-TET Acoustic Quality
C4Origin (1/1)1 / 1Tonic0.0 ¢261.63 Hz0.0 ¢Perfect fundamental unison
C#4+7 Fifths up2187 / 2048Chromatic Semitone90.2 ¢275.60 Hz-9.8 ¢Narrow, driving leading tone
D4+2 Fifths up9 / 8Major Tone203.9 ¢294.33 Hz+3.9 ¢Pure major second; clean melodic step
Eb4-3 Fifths down32 / 27Minor 3rd (Semiditone)294.1 ¢310.08 Hz-5.9 ¢Narrow minor third; sombre, sharp beating
E4+4 Fifths up81 / 64Major 3rd (Ditone)407.8 ¢331.12 Hz+7.8 ¢Extremely wide (+21.5 ¢ vs 5/4); 16 Hz rattle in triad
F4-1 Fifth down4 / 3Perfect 4th498.0 ¢348.83 Hz-2.0 ¢Acoustically pure fourth (beatless)
F#4+6 Fifths up729 / 512Tritone611.7 ¢372.51 Hz+11.7 ¢Sharp, piercing augmented fourth
G4+1 Fifth up3 / 2Perfect 5th702.0 ¢392.44 Hz+2.0 ¢Acoustically pure fifth (beatless)
G#4+8 Fifths up6561 / 4096Augmented 5th792.2 ¢413.31 Hz-7.8 ¢Border of the circle
A4+3 Fifths up27 / 16Major 6th905.9 ¢441.49 Hz+5.9 ¢Wide Pythagorean major sixth
Bb4-2 Fifths down16 / 9Minor 7th996.1 ¢465.11 Hz-3.9 ¢Standard Pythagorean minor seventh
B4+5 Fifths up243 / 128Major 7th1109.8 ¢496.68 Hz+9.8 ¢Sharp leading tone to C5
C5Octave2 / 1Octave1200.0 ¢523.25 Hz0.0 ¢Pure octave (2:1)
WOLF G#4 to Eb5 262144 / 177147 Wolf Fifth 678.5 ¢ — -21.5 ¢ Catastrophic flat fifth (flat by 23.5 ¢ comma); howling discord

2. Just intonation: keep the pure third

Who wanted it, and why

A triad is three notes at once: a root, a third, and a fifth, such as C–E–G. By the 1400s, composers wanted that third to sound sweet. John Dunstable (c. 1390–1453) brought a style full of thirds and sixths from England to the continent. Du Fay, Ockeghem, and Josquin built pieces on the major triad.

Pythagorean E is 81/64, 407.82 cents. The pure third, 5/4, is 386.31 cents. Singers were already leaning toward the pure one, because it matches a harmonic of the voice and does not beat. In 1482 Bartolomé Ramos de Pareja, in Musica Practica, wrote 5/4 for the major third and 6/5 for the minor third. In 1558 Gioseffo Zarlino, in Venice, took the numbers 1 through 6 as the set a consonance may use. A pure major triad is then 4:5:6, which is C, a third at 5/4, and a fifth at 3/2.

5-limit means the ratios may use the prime 5, not only 2 and 3. Ptolemy, about 150 CE, had already written a scale with 5/4. The Renaissance is when that third becomes the one a choir is expected to sing. A keyboard cannot follow a choir that adjusts. That is the crisis the next tuning tries to solve.

Three pure triads

Build the scale from three chords, each in the proportion 4:5:6. Tonic means the home chord, C–E–G. Subdominant means the chord on the fourth, F–A–C. Dominant means the chord on the fifth, G–B–D.

  • Tonic Triad (C - E - G): C(4) : E(5) : G(6) → E = 5/4 (386.3 ¢), G = 6/4 = 3/2 (702.0 ¢).
  • Subdominant Triad (F - A - C): F(4) : A(5) : C(6) → Since C = 1, F = 4/6 = 2/3 (transposed up an octave = 4/3, 498.0 ¢), and A = (4/3) * (5/4) = 5/3 (884.4 ¢).
  • Dominant Triad (G - B - D): G(4) : B(5) : D(6) → G = 3/2. Therefore, B = (3/2) * (5/4) = 15/8 (1088.3 ¢), and D = (3/2) * (6/4) = 9/4 (transposed down an octave = 9/8, 203.9 ¢).

The twelve notes, one at a time

  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, the same fraction as komal Ni (कोमल नि) on the Eastern page, 9/5.).
  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).
Note Triad Origin Ratio Interval Cents (¢) Freq (Hz) vs 12-TET Acoustic Quality
C4Tonic Root1 / 1Tonic0.0 ¢261.63 Hz0.0 ¢Pure reference anchor
C#4 / DbInversion of 15/816 / 15Diatonic Semitone111.7 ¢279.07 Hz+11.7 ¢Wide, sweet major semitone
D4Dominant 5th9 / 8Major Tone (T)203.9 ¢294.33 Hz+3.9 ¢Pure major whole tone
Eb4Minor 3rd on C6 / 5Minor 3rd315.6 ¢313.95 Hz+15.6 ¢Pure minor third (beatless)
E4Tonic 3rd5 / 4Major 3rd386.3 ¢327.03 Hz-13.7 ¢Pure natural third: zero beating, completely serene
F4Subdominant Root4 / 3Perfect 4th498.0 ¢348.83 Hz-2.0 ¢Pure subdominant anchor
F#4Major 3rd on D45 / 32Augmented 4th590.2 ¢367.92 Hz-9.8 ¢Pure harmonic tritone
G4Tonic 5th / Dom Root3 / 2Perfect 5th702.0 ¢392.44 Hz+2.0 ¢Pure beatless fifth
Ab4Inversion of 5/48 / 5Minor 6th813.7 ¢418.60 Hz+13.7 ¢Pure minor sixth
A4Subdominant 3rd5 / 3Major 6th884.4 ¢436.05 Hz-15.6 ¢Pure major sixth (21.5 ¢ flatter than Pyth 27/16!)
Bb4Harmonic 7th9 / 5Minor 7th1017.6 ¢470.93 Hz+17.6 ¢Pure resonant minor 7th (Indian Komal Ni)
B4Dominant 3rd15 / 8Major 7th1088.3 ¢490.55 Hz-11.7 ¢Pure leading tone in G major
C5Octave2 / 1Octave1200.0 ¢523.25 Hz0.0 ¢Pure octave (2:1)
WOLF D4 to A4 Triad 40 / 27Defective 5th 680.5 ¢ — -19.5 ¢ Flat by 21.5 ¢ Syntonic Comma; D minor triad is unplayable!

Why a keyboard cannot keep them

Why Just Intonation Failed on Keyboards:
  1. Two Different Whole Tones: The distance C to D is a Major Tone (9/8 = 203.9 cents), but D to E is a Minor Tone (10/9 = 182.4 cents). The difference is exactly the Syntonic Comma: (9/8) / (10/9) = 81/80 = 21.51 cents!
  2. The D-Minor Wolf: The fifth above D should be A. But D is 9/8 (203.9 ¢) and A is 5/3 (884.4 ¢). The interval D to A is: (5/3) / (9/8) = 40/27 = 680.45 cents. This is 21.51 cents flat of a pure fifth! Playing a simple ii-V-I cadence (Dm - G7 - C) produces an agonizing out-of-tune howl.
  3. Pitch Drift (Syntonic Pump): If a choir sings a harmonic progression that shifts between major and minor tones, their overall pitch drops by 21.5 cents with every cycle. For fixed-key instruments like organs and harpsichords, pure Just Intonation was a logistical impossibility.

3. Quarter-comma meantone (16th and 17th centuries)

Who needed it, and why

A choir can nudge a note. An organ cannot. By the 1500s, organs and harpsichords were being asked to play the pure thirds that singers wanted, on keys that do not move. Just intonation gives those thirds in one key and then breaks the fifth somewhere else, and it also makes two different sizes of whole step, so one key cannot serve as both Ds.

In 1523 Pietro Aaron, in Toscanello de la musica, described the tuning organ builders were already using. In 1577 Francisco de Salinas, in De musica libri septem, wrote the arithmetic. Narrow every fifth by a quarter of the syntonic comma, about 5.4 cents. Four of those narrowed fifths land on a pure 5/4 third. The whole step is the average of the two just whole steps, 9/8 and 10/9, which is why it is called the mean tone.

That is the usual keyboard tuning for about two centuries, the one Byrd, Frescobaldi, Sweelinck, Schütz, and Buxtehude wrote for. The price is at the end of the chain: one fifth, the wolf, is much too wide, and keys with many sharps or flats are not used.

The arithmetic

Four pure fifths miss the pure third by one syntonic comma, 21.51 cents. Salinas takes that error off the fifths, a quarter from each.

In Pythagorean tuning, four stacked fifths generate the major third:
(3/2)^4 / 4 = 81/64 = 407.82 cents
The error against the pure natural third (5/4 = 386.31 cents) is the Syntonic Comma:
81/80 = 21.51 cents

Salinas reasoned: if 4 fifths produce an error of 1 comma, simply flatten (temper) each fifth by exactly 1/4 of the syntonic comma:
1/4 Comma = 21.506 / 4 = 5.377 cents
Quarter-Comma Tempered Fifth Q = 701.955 - 5.377 = 696.578 cents (ratio = 5^(1/4) ≈ 1.49534878)

When four such tempered fifths are stacked:
4 * 696.578 - 2400 = 2786.312 - 2400 = 386.314 cents = EXACT PURE 5/4 MAJOR THIRD!

Furthermore, the whole step is the exact geometric mean of the major tone (9/8) and minor tone (10/9):
Mean Tone = 386.314 / 2 = 193.157 cents (ratio = sqrt(5)/2 ≈ 1.118034). Hence the name: Mean-Tone!

The twelve notes, one at a time

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 ¢).
Note Fifth Step Ratio Formula Interval Cents (¢) Freq (Hz) vs 12-TET Acoustic Quality
C401.00000Tonic0.0 ¢261.63 Hz0.0 ¢Tonic center
C#4+7 Q5^(7/4) / 8Chromatic Semitone76.1 ¢273.35 Hz-24.0 ¢Very narrow semitone; pure third to A
D4+2 Qsqrt(5) / 2Mean Tone193.2 ¢292.45 Hz-6.8 ¢Exact mean of 9/8 and 10/9
Eb4-3 Q4 / 5^(3/4)Minor 3rd310.3 ¢312.92 Hz+10.3 ¢Pure third below G
E4+4 Q5 / 4Major 3rd386.3 ¢327.03 Hz-13.7 ¢Pure 5/4 third: heavenly stillness, zero beats
F4-1 Q2 / 5^(1/4)Perfect 4th503.4 ¢349.91 Hz+3.4 ¢Slightly wide fourth; pure third to A
F#4+6 Q5*sqrt(5) / 8Tritone579.5 ¢365.64 Hz-20.5 ¢Pure third to D
G4+1 Q5^(1/4)Tempered 5th696.6 ¢391.21 Hz-3.4 ¢Tempered fifth (-5.4 ¢; very slow, gentle beat)
G#4+8 Q25 / 16Augmented 5th772.6 ¢408.74 Hz-27.4 ¢Pure third to E; sharp limit of keyboard
A4+3 Q5^(3/4) / 2Major 6th889.7 ¢437.38 Hz-10.3 ¢Pure third to F
Bb4-2 Q4 / sqrt(5)Minor 7th1006.8 ¢468.01 Hz+6.8 ¢Pure third below D
B4+5 Q5^(5/4) / 4Major 7th1082.9 ¢489.00 Hz-17.1 ¢Pure third to G
C5Octave2.00000Octave1200.0 ¢523.25 Hz0.0 ¢Pure octave
WOLF G#4 to Eb5 Interval Gap Wolf Fifth 737.6 ¢ — +37.6 ¢ Massively sharp (+35.7 ¢ vs pure 3/2, +41 ¢ vs Q); grotesque screeching

The wolf that remains

Because eleven fifths were narrowed by 5.38 cents each, the total contraction was 11 * 5.38 = 59.18 cents. The remaining gap between G# and Eb was forced to absorb all this contraction plus the Pythagorean comma:
Wolf Fifth = 701.96 + (11 * 5.377) - 23.46 = 737.63 cents!
This wolf is about 738 cents, far wider than a pure fifth of 702. An A♭ major chord, which needs that G♯, clashes. Some Italian and German organs were built with split black keys, a separate lever for D♯ and for E♭, so the wolf could be avoided. Keys with many sharps or flats were simply not used.

4. Werckmeister III (1691)

Meantone keeps a pure major third and pays for it with one fifth that is too wide to use, so some keys are unusable. A well temperament spends the Pythagorean comma across several fifths instead of dumping it into one wolf, so every key can be played. They do not all sound the same. That is what circulating means: you can move from key to key without retuning, and the keys still have different thirds.

Andreas Werckmeister's third recipe, in Musicalische Temperatur (1691), narrows four fifths (C–G, G–D, D–A, and B–F♯) by a quarter of the Pythagorean comma and leaves the other fifths pure. C major's third sits near 390 cents. A key that rides the narrowed fifths is sharper.

J. S. Bach's Well-Tempered Clavier (1722 and 1742) is a prelude and fugue in each of the 12 major keys and each of the 12 minor keys, for a circulating temperament of this kind. Which of Werckmeister's recipes Bach used is not settled. The twelve-note chart is on Other tunings. The cents below are the ones the western workbench plays when Well is selected.

NoteCentsNoteCents
C0.0F♯588.3
C♯90.2G696.1
D192.2G♯792.2
E♭294.1A888.3
E390.2B♭996.1
F498.0B1092.2

5. Twelve equal steps

Who needed it, and why

A lute or guitar fret is one straight bar under all the strings. If the fret is in the right place for a pure third on one string, it is in the wrong place for the next string, which is tuned a fourth away. The frets only line up if every step is the same size.

In 1581 Vincenzo Galilei, a lutenist and Galileo's father, proposed a practical fret spacing of 18:17, about 99 cents, in Dialogo della musica antica et della moderna. The exact step is the twelfth root of two: multiply the frequency by the same number twelve times and you have doubled it, one octave. In 1584 Zhu Zaiyu (朱載堉), in Ming China, calculated that number to 25 decimal places in Lüxue Xinshuo and cast twelve pitch pipes to it. Simon Stevin calculated the same root in Europe around 1585–1605. Whether Zhu's figure travelled west is debated. Marin Mersenne published equal string lengths in 1636. Keyboard players kept well temperament for another two centuries, because the equal third is the coarse one: 400 cents, about 14 cents sharp of the pure 386.

Equal temperament becomes the ordinary piano tuning in the 1800s, when one standard has to serve a factory-built piano, a valve brass instrument, and a composer who changes key in the middle of a piece. Liszt, Chopin, Wagner, and Mahler write as if all twelve steps are the same size. In this tuning they are.

The twelfth root of two

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

Frequency Ratio per Semitone: r = 2^(1/12) ≈ 1.059463094359...
Cents per Semitone: 1200 / 12 = Exactly 100.0000 cents
Frequency of Semitone k: f(k) = f(0) * 2^(k/12)

The 23.46-cent Pythagorean comma is divided strictly equally among all 12 fifths:
Tempered Fifth = 701.955 - (23.460 / 12) = 701.955 - 1.955 = Exactly 700.00 cents.

The twelve notes, one at a time

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.
Note Exponent k Power Ratio 2^(k/12) Interval Cents (¢) Freq (Hz) vs Pure Just Acoustic Character
C401.000000Tonic0.0 ¢261.63 Hz0.0 ¢Universal reference pitch
C#411.059463Minor 2nd100.0 ¢277.18 Hz-11.7 ¢Uniform semitone step
D421.122462Major 2nd200.0 ¢293.66 Hz-3.9 ¢Uniform whole tone step
Eb431.189207Minor 3rd300.0 ¢311.13 Hz-15.6 ¢Significantly flat vs natural 6/5 (315.6 ¢)
E441.259921Major 3rd400.0 ¢329.63 Hz+13.7 ¢Audibly sharp vs pure 5/4; beats at 10.3 Hz in C4-E4
F451.334840Perfect 4th500.0 ¢349.23 Hz+2.0 ¢Very clean fourth (+1.96 ¢)
F#461.414214Tritone600.0 ¢369.99 Hz+9.8 ¢Exact square root of 2 (octave center)
G471.498307Perfect 5th700.0 ¢391.99 Hz-2.0 ¢Nearly pure fifth (-1.96 ¢; slow 0.9 Hz wave)
Ab481.587401Minor 6th800.0 ¢415.30 Hz-13.7 ¢Inversion of sharp major third
A491.681793Major 6th900.0 ¢440.00 Hz+15.6 ¢Global Concert Pitch Standard (ISO 16)
Bb4101.781797Minor 7th1000.0 ¢466.16 Hz-17.6 ¢Noticeably flat vs pure 9/5 harmonic seventh
B4111.887749Major 7th1100.0 ¢493.88 Hz+11.7 ¢Sharp leading tone
C5122.000000Octave1200.0 ¢523.25 Hz0.0 ¢Exact pure octave (2:1)

What equal steps cost

Every key is now the same size. C major and F♯ major have the same thirds and the same fifths. The difference a player used to hear between keys is gone. Helmholtz, in 1863, called this a compromise that trains the ear to ignore the beating of imperfect thirds.

A harmonium brings those twelve equal reeds into Hindustani music. A śruti there is not one equal slice: Bharata's 22 are unequal gaps of 2, 3, and 4, and a rāga still moves a note inside them. One reed cannot be both of Todi's komal Re and Bhairav's, and a reed cannot make a mīṇḍ (मींड), the glide between notes. Tagore called the harmonium the bane of Indian music and would not have it at Santiniketan. From 1940 to 1971 All India Radio kept it off classical broadcasts. The longer sentence sometimes printed under his name is a paraphrase. The rest of that story is on Eastern tunings.

3. The four lists, side by side

Pythagorean, just, meantone, and twelve equal steps, with C at 261.63 Hz. A cent was defined in the first section. Werckmeister is not a column: his tuning is the one just above, and the full chart is on Other tunings. The last column is the nearest name on the Eastern page. Śuddha (शुद्ध) means plain, komal (कोमल) means lowered, tīvra (तीव्र) means raised. Where a black key has two spellings, the column prints the note the chain of fifths reaches. Pythagorean C♯ is the narrow step, about 90 cents. The wider step, about 112 cents, is a different spelling, and it is the one just intonation uses for D♭.

Note Interval Pythagorean (¢) Just (5-Limit) (¢) Meantone (¢) 12-TET (¢) Eastern Swara
C4 Tonic (Unison) 0.0 ¢ (261.63 Hz) 0.0 ¢ (261.63 Hz) 0.0 ¢ (261.63 Hz) 0.0 ¢ (261.63 Hz) Sa (सा)
C# / Db Minor 2nd 90.2 ¢ (275.6 Hz) 111.7 ¢ (279.1 Hz) 76.0 ¢ (273.3 Hz) 100.0 ¢ (277.18 Hz) Komal Re (रे)
D4 Major 2nd 203.9 ¢ (294.33 Hz) 203.9 ¢ (294.33 Hz) 193.2 ¢ (292.5 Hz) 200.0 ¢ (293.66 Hz) Śuddha Re (रे)
Eb / D# Minor 3rd 294.1 ¢ (310.08 Hz) 315.6 ¢ (313.95 Hz) 310.3 ¢ (312.9 Hz) 300.0 ¢ (311.13 Hz) Komal Ga (ग)
E4 Major 3rd 407.8 ¢ (331.12 Hz) 386.3 ¢ (327.03 Hz) 386.3 ¢ (327.03 Hz) 400.0 ¢ (329.63 Hz) Śuddha Ga (ग)
F4 Perfect 4th 498.0 ¢ (348.83 Hz) 498.0 ¢ (348.83 Hz) 503.4 ¢ (349.9 Hz) 500.0 ¢ (349.23 Hz) Śuddha Ma (म)
F# / Gb Tritone 611.7 ¢ (372.5 Hz) 590.2 ¢ (367.9 Hz) 579.5 ¢ (365.6 Hz) 600.0 ¢ (369.99 Hz) Tīvra Ma (म॑)
G4 Perfect 5th 702.0 ¢ (392.44 Hz) 702.0 ¢ (392.44 Hz) 696.6 ¢ (391.2 Hz) 700.0 ¢ (391.99 Hz) Pa (प)
Ab / G# Minor 6th 792.2 ¢ (413.3 Hz) 813.7 ¢ (418.6 Hz) 772.6 ¢ (408.7 Hz) 800.0 ¢ (415.30 Hz) Komal Dha (ध)
A4 Major 6th 905.9 ¢ (441.49 Hz) 884.4 ¢ (436.05 Hz) 889.7 ¢ (437.4 Hz) 900.0 ¢ (440.00 Hz) Śuddha Dha (ध)
Bb / A# Minor 7th 996.1 ¢ (465.1 Hz) 1017.6 ¢ (470.9 Hz) 1006.8 ¢ (468.0 Hz) 1000.0 ¢ (466.16 Hz) Komal Ni (नि)
B4 Major 7th 1109.8 ¢ (496.68 Hz) 1088.3 ¢ (490.55 Hz) 1082.9 ¢ (489.0 Hz) 1100.0 ¢ (493.88 Hz) Śuddha Ni (नि)
C5 Octave 1200.0 ¢ (523.25 Hz) 1200.0 ¢ (523.25 Hz) 1200.0 ¢ (523.25 Hz) 1200.0 ¢ (523.25 Hz) Tāra Sa (सा)

4. The books this page used

In the same year order as the story.

Primary Historical Treatises

  1. Euclid (attributed): Sectio Canonis (The Division of the Monochord, c. 300 BCE).
    Earliest surviving mathematical demonstration that six pure whole tones (9/8)^6 exceed the octave (2/1) by the Pythagorean comma (531441 / 524288 = 23.46 cents). English translation and commentary by Andrew Barker in Greek Musical Writings II: Harmonic and Acoustic Theory (Cambridge University Press, 1989).
  2. Claudius Ptolemy: Harmonica (Ἁρμονικά, c. 150 CE).
    Book III sets out his intense diatonic, with steps 16/15, 9/8, and 10/9, which includes the pure major third 5/4. Translation by Jon Solomon, Ptolemy's Harmonics (Brill, 2000).
  3. Bartolomé Ramos de Pareja: Musica Practica (Bologna, 1482).
    He proposed 5/4 for the major third and 6/5 for the minor third. Edited and translated by Clement A. Miller (American Institute of Musicology, 1993).
  4. Pietro Aaron: Toscanello de la musica (Venice, 1523).
    Written instructions for the meantone tuning organ builders were already using.
  5. Gioseffo Zarlino: Le Istitutioni Harmoniche (Venice, 1558).
    Book III. He takes the numbers 1 through 6 as the set a consonance may use, and the major triad as 4:5:6.
  6. Francisco de Salinas: De musica libri septem (Salamanca, 1577).
    The arithmetic of meantone: four fifths, each narrowed by a quarter of the syntonic comma, land on a pure major third, 5/4.
  7. Zhu Zaiyu (朱載堉): Lüxue Xinshuo (1584).
    He calculated 2^(1/12) to 25 decimal places and cast twelve pitch pipes to that ratio. Whether the figure reached Europe is debated.
  8. Simon Stevin: Vande Spiegheling der Singconst (c. 1605).
    He calculated the same twelfth root in Europe. Marin Mersenne published equal string lengths in 1636.
  9. Andreas Werckmeister: Musicalische Temperatur (Quedlinburg, 1691).
    His third temperament narrows four fifths by a quarter of the Pythagorean comma. Every key can be played. They do not all sound the same.
  10. Johann Sebastian Bach: Das Wohltemperierte Clavier (1722 and 1742, BWV 846–893).
    A prelude and fugue in each of the twelve major keys and each of the twelve minor keys, for a circulating temperament. Which recipe Bach used is not settled.

Acoustics & Modern Scholarship

  1. Hermann von Helmholtz: Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (1863).
    Translated into English as On the Sensations of Tone as a Physiological Basis for the Theory of Music by Alexander J. Ellis (Longmans, Green, and Co., 1875; Dover reprint, 1954). The foundational masterwork of musical acoustics, containing Ellis's invention and definition of the modern logarithmic cent (1200 cents per octave).
  2. J. Murray Barbour: Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951; Dover Publications, 2004).
    The definitive modern academic history of Western tuning systems, calculating cent values and comma distributions across all major historical temperaments.
  3. Mark Lindley: Lutes, Viols and Temperaments (Cambridge University Press, 1984) and "Temperaments" in The New Grove Dictionary of Music and Musicians.
    Authoritative modern reference on Renaissance and Baroque performance practice and tuning.
  4. Ross W. Duffin: How Equal Temperament Ruined Harmony (and Why You Should Care) (W. W. Norton & Company, 2007).
    Engaging historical analysis of the musical trade-offs of equal temperament vs. historical temperaments.