# Pythagorean Tuning

## 2. Basic concepts

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