A Pythagorean tuning of the diatonic scale works well for monophonic music. However, once Western art music began to utilize the triad as a fundamental harmonic sonority, the 81:64 major third produced by the Pythagorean tuning process became a problem for many ears.

Fig. 1. A comparison of the tuning of a major triad on C4 produced
by a Pythagorean, just (from the harmonic series),
and 12TET (piano) tuning of the diatonic scale on C. (mov)

The major third found in the harmonic series is 5:4 (386 cents). In Fig. 2a you will find it between C₄ and E₄ (red bracket). By comparison, there are 408 cents in a 81:64 Pythagorean major third. By definition, the 12TET major third is 400 cents.

A just tuning attempts to use the intervals found in the harmonic series as its basis. The intervals found in the lower reaches of the harmonic series are called natural intervals. These intervals produce significantly less beating and roughness than higher order ratios because of the way the harmonic series produced by each tone involved in the interval overlaps. Thus a 5:4 just major third is said to be purer than a 81:64 Pythagorean major third.

Fig. 2a: First 10 harmonics of the harmonic series based on C₂.
Fig. 2b: A just tuning of the diatonic scale.

Playback:

Fig. 3: A chart showing the size of each adjacent scale step in a Just diatonic scale on C₄ in cents.

US Standard Pitch Name	C4		D4		E4		F4		G4		A4		B4		C5
Frequency ratio above C4	1:1		9:8		5:4		4:3		3:2		5:3		15:8		2:1
No. of cents above C4	0		204		386		498		702		884		1,088		1,200
Size of step (in cents)		204		182		112		204		182		204		112
3 step sizes: 1. major tone = 9:8 or 204 cents; 2. Minor tone = 10:9 or 182 cents; diatonic semitone 16:15 or 112 cents

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¹The "unique prime factorization theorem" can be used to prove the validity of this conjecture. For a clear and concise summary of its mathematical proof consult Blackwood, Easley, The Structure of Recognizable Diatonic Tunings, (Princeton, NJ: Princeton University Press, 1985), 8-11.
²The Pythagorean semitone was also called the limma (left over), as it was calculated by the Greeks to be the difference (or amount left over) between a fourth and two whole tones. See Willi Apel, ed., Harvard Dictionary of Music (Cambridge: Harvard University Press, 1969), pp. 709-10.

Further Study

See Bibliography	On-line Sources
Campbell/Greated, 170-71. Partch, 398-406.	BAIN The Overtone Series Soundscape Production's JICalc, a Hypercard stack that can be used to explore a variety of tunings and temperaments. Margo Schulter's FAQ on Pythagorean Tuning