Ptolemy's intense diatonic scale



Ptolemy's intense diatonic scale, also known as Ptolemaic sequence, justly tuned major scale, or syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, declared by Zarlino to be the only tuning that could be reasonably sung, and corresponding with modern just intonation. It is also supported by Giuseppe Tartini.

It is produced through a tetrachord consisting of a greater tone (9:8), lesser tone (10:9), and just diatonic semitone (16:15). This is called Ptolemy's intense diatonic tetrachord, as opposed to Ptolemy's soft diatonic tetrachord, formed by 21/20, 10:9 and 8:7 intervals.

Lowering the pitches of Pythagorean tuning's notes E and A and B by the syntonic comma, 81/80, to give a just intonation, changes it to Ptolemy's intense diatonic scale.

Intervals between notes (wolf intervals bolded):

In comparison to Pythagorean tuning, while both provide just perfect fourths and fifths, the Ptolemaic provides just thirds which are smoother and more easily tuned.

Note that D–F is a Pythagorean minor third (32:27), D–A is a defective fifth (40:27), F–D is a Pythagorean major sixth (27:16), and A–D is a defective fourth (27:20). All of these differ from their just counterparts by a syntonic comma (81:80).

F-B is the tritone, here 45/32.

This scale may also be considered as derived from the major chord, and the major chords above and below it: FAC–CEG–GBD.