Definition
Pythagorean tuning builds all the notes by stacking pure fifths (3:2 ratio) and bringing them back into the octave (dividing by 2). It is the theoretical system of ancient Greece (attributed to the Pythagorean school) and of medieval theory: it uses only the prime numbers 2 and 3.
Building the scale
Starting from F and going up by fifths: F – C – G – D – A – E – B. Brought back into the octave of C:
| Note | Ratio | Cents |
|---|---|---|
| C | 1/1 | 0 |
| D | 9/8 | 203.91 |
| E | 81/64 | 407.82 |
| F | 4/3 | 498.04 |
| G | 3/2 | 701.96 |
| A | 27/16 | 905.87 |
| B | 243/128 | 1109.78 |
| C | 2/1 | 1200 |
It contains only two kinds of step: the 9/8 tone (203.91 cents) and the limma (diatonic semitone) 256/243 (90.22 cents). The chromatic semitone, called the apotome, is 2187/2048 (113.69 cents): in the Pythagorean system F# is higher than Gb. Many violinists, for expressiveness, play high leading tones close to the tonic, a logic similar to the Pythagorean one.
The Pythagorean comma
Twelve pure fifths do not come back exactly to seven octaves: (3/2)^12 / 2^7 = 531441/524288 ≈ 1.01364, that is 23.46 cents. This discrepancy is the Pythagorean comma. Consequences:
- The spiral of fifths does not close: B# does not coincide with C.
- If you tune 11 pure fifths, the twelfth (usually G#–Eb) comes out 23.46 cents flat (678.49 cents): this is the wolf fifth, unusable because it beats violently.
Equal temperament distributes the comma by narrowing each fifth by 1/12 of a comma (≈1.96 cents).
Strengths and weaknesses
- Strengths: perfect fifths and fourths, melodies with tense, “bright” leading tones; ideal for monophony, Gregorian chant and medieval organum built on parallel fifths and fourths.
- Weaknesses: the major third 81/64 (407.82 cents) is wider by 21.51 cents than the pure third 5/4: the difference is the syntonic comma 81/80. Major triads sound harsh and beating. It is no accident that in medieval theory thirds were treated as imperfect consonances or dissonances, and cadences ended on empty fifths or unisons.
From the syntonic comma to just intonation
When polyphony began to use full triads (Dunstable, Dufay, then Zarlino with the Istitutioni harmoniche of 1558), the Pythagorean third became a problem. Hence the just systems and the meantone temperaments, which sacrifice the purity of the fifths to obtain pure thirds (see the dedicated card).
Pythagorean tuning today
- George Russell's Lydian Chromatic Concept justifies the Lydian scale with six stacked fifths on the tonic (C G D A E B F#): a Pythagorean-style line of reasoning.
- Violin, viola and cello are tuned in pure fifths: the open strings produce Pythagorean intervals, and this is one of the reasons why quartets adapt the intonation of thirds to the context.
- Turkish theory (the Arel-Ezgi-Uzdilek system) is based on Pythagorean intervals and on Holder's comma (1/53 of an octave, ≈22.6 cents).
Musical example
Gregorian chant and Perotin's organum (“Viderunt omnes”, late 12th century) are conceived for pure fifths and fourths: heard in Pythagorean tuning they sound more stable and resonant than on the piano.
Exercise
- In a synth that supports .scl files (free Surge XT, Pigments, Serum 2), load a Pythagorean tuning on C and play C–E–G: compare it with 12-TET and with just intonation.
- Calculate the Pythagorean E starting from C = 261.63 Hz (× 81/64 = 331.1 Hz) and compare it with tempered E (329.63 Hz) and just E (327.0 Hz).
- Play 12 pure fifths starting from Ab with an oscillator and listen to the final discrepancy.
Related topics
Equal temperament and cents; Just intonation and historical temperaments; Lydian Chromatic Concept; Consonance and dissonance.