Pythagorean Tuning is probably the earliest organised form of tuning, credited to the Greek philosopher and mathematician Pythagoras, though no writings survive from that time to substantiate this. There is documentary evidence for use of Pythagorean Tuning in the 9th century and it was the predominant method of tuning organs and keyboard instruments through to the 15th century.

The method tunes using the interval of the 5th, formed by the simplest ratio 3/2 (702 cents). The 5th is an easy interval to use since it produces very pronounced 'beats' when slightly out-of-tune, which become slower as the note approaches the perfect ratio. Using this method it is possible to tune a succession of 5ths C-G-D-A etc. and, by moving down an octave at appropriate points, it is fairly simple to tune each of the notes of the scale.

The finished effect can be seen to have each 5th at 702 cents with one rather significant exception. Unfortunately, starting at C and moving up 12 perfect 5ths, we reach B# at 8424 cents (702x12) which should be the enharmonic of C, 7 octaves higher at 8400 cents (1200x8) but clearly it is 24 cents sharp. This gap is known as the Pythagorean Comma and shows up in our circle of 5ths where one of the intervals G# to Eb, has to be 24 cents narrower. the so-called 'wolf interval'.

Looked at another way, the Pythagorean Eb forms a pure 5th in Eb-Bb but is 24 cents flat of the note D# we would desire for the pure 5th G#-D#. Likewise G# is perfectly in tune for the 5th C#-G# but is 24 cents too sharp for a pure 5th with Eb (Ab-Eb). Thus, on a 12 note Pythagorean keyboard tuned with a chain of 5ths from Eb to G#, the 5ths Ab-Eb and G#-D# are not available, although the notes Eb(D#) and A(G#) are both very useful in other 14th century sonorities. Fortunately, Ab-Eb and G#-D# doesn't occur often in pieces from this era so the problem is mostly academic when playing music for which the tuning was intended.

For the purist, the Pythagorean Comma is 23.46 cents, based on a 5th of 701.95 cents but almost all of the literature uses 24 cents, based on a rounded up value of 702 cents for the 5th. If you place the mouse over the Eb you will see that it is actually 23.46 cents flat.

Even though Pythagorean Tuning contains wolf intervals, it may be seen that all of the keys from Bb to E behave identically allowing free modulation between them, which is one test of a 'good' tuning; though switching to the minor, modulation looks less attractive!

Looking at Pythagorean Tuning relative to Natural Tuning shows another very interesting effect. The Major 3rd and the Major 6th are significantly sharp, relative to the natural ratios of 5/4 and 5/3. Though this is a problem for modern music (Baroque onwards), where consonant major 3rd are expected, this was a positive advantage for medieval music, where tense or active 3rds asking to be resolved, were the order of the day. It is interesting to compare Pythagorean Tuning with today's Equal Temperament and note that the differences are not so great. The conclusion to be drawn from this is that, by accident, Equal Temperament is good base for playing medieval music

One last useful element to be drawn from the 'musical calculator' concerns the size of semitones. Looking at the pitches of Pythagorean Tuning, it will be seen that the scale is made up of a set of tones, each 204 cents wide and two 90 cent semitones to complete the 1200 cent scale. This is fine except that the tone is much larger than two semitones, so looking at the full chromatic scale, it may be seen that there have to be two types of semitone, one 90 cents wide and one 114 cents wide.

The ability to modulate to near keys, the perfect 4ths and 5ths and the sharpish major 3rds were all factors which made Pythagorean Tuning a good choice for medieval music. It was only in the 15th century as composers began to want more consonant 3rds that its use waned, in favour of Meantone Tuning.