As mentioned above, Pythagorean tuning defines all notes and intervals of a scale from a series of pure fifths with a ratio of 3:2. Thus it is not only a mathematically elegant system, but also one of the easiest to tune by ear. To derive a complete chromatic scale of the kind common on keyboards by around 1300, we take a series of 11 perfect fifths:

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

The one potential flaw of this system is that the fourth or
fifth between the extreme notes of the series, Eb-G#, will be out
of tune: in the colorful language of intonation, a "wolf"
interval. This complication arises because 12 perfect fifths do
not round off to precisely an even octave, but exceed it by a small
ratio known as a *Pythagorean comma* (see
Section 4). Happily, since Eb and G#
rarely get used together in medieval harmony, this is hardly a
practical problem.

Since all intervals have integer (whole number) ratios based on
the powers of two and three, Pythagorean tuning is a form of *just
intonation* (see Section 5).

More specifically, it is a form of just intonation based on the numbers 3 and 9. Thus we get just or ideally blending fifths (3:2), fourths (4:3), major seconds (9:8), and minor sevenths (16:9).

In fact, Pythagorean tuning is described in the medieval sources
as being based on four numbers: 12:9:8:6. Jacobus of Liege (c.
1325) describes a "quadrichord" with four strings having
these lengths: we get an octave (12:6) between the outer notes,
two fifths (12:8, 9:6), two fourths (12:9, 8:6), and a *tonus*
or major second between the two middle notes (9:8).

Other intervals can be derived from these, and the result in a medieval context is, by the 13th century, a subtle spectrum of interval tensions in practice and theory.

The following table shows how the standard intervals of Pythagorean tuning except the pure unison (1:1) and octave (2:1) are derived primarily from superimposed fifths (3:2), thus having ratios which are powers of 3:2, or secondarily from the differences between these primary intervals and the octave. We show the 13 usual intervals of medieval music from unison to octave as listed by Anonymous I around 1290, and by Jacobus of Liege around 1325. (On some other intervals generated in tuning a complete chromatic scale, see Section 4.2.2.)

Pythagorean intervals and their derivations |
|||

Interval | Ratio | Derivation | Cents* |

Unison | 1:1 | Unison 1:1 | 0.00 |

Minor Second | 256:243 | Octave - M7 | 90.22 |

Major Second | 9:8 | (3:2)^2 | 203.91 |

Minor Third | 32:27 | Octave - M6 | 294.13 |

Major Third | 81:64 | (3:2)^4 | 407.82 |

Fourth | 4:3 | Octave - 5 | 498.04 |

Augmented Fourth | 729:512 | (3:2)^6 | 611.73 |

Fifth | 3:2 | (3:2)^1 | 701.96 |

Minor Sixth | 128:81 | Octave - M3 | 792.18 |

Major Sixth | 27:16 | (3:2)^3 | 905.87 |

Minor Seventh | 16:9 | Octave - M2 | 996.09 |

Major Seventh | 243:128 | (3:2)^5 | 1109.78 |

Octave | 2:1 | Octave 2:1 | 1200.00 |

* For an explanation of cents, see Section 4.2. | |||

To Section 3 - Pythagorean tuning and Gothic polyphony.

To Section 4 - Pythagorean tuning in more detail.

Margo Schulter