Ptolemy's intense diatonic scale

[6] This is called Ptolemy's intense diatonic tetrachord (or "tense"), as opposed to Ptolemy's soft diatonic tetrachord (or "relaxed"), which is formed by 21:20, 10:9 and 8:7 intervals.

[8] The structure of the intense diatonic scale is shown in the tables below, where T is for greater tone, t is for lesser tone and s is for semitone: Ptolemy's intense diatonic scale can be constructed by lowering the pitches of Pythagorean tuning's 3rd, 6th, and 7th degrees (in C, the notes E, A, and B) by the syntonic comma, 81:80.

This perspective emphasizes the central role of the tonic, dominant, and subdominant in the diatonic scale.

In comparison to Pythagorean tuning, which only uses 3:2 perfect fifths (and fourths), the Ptolemaic provides just thirds (and sixths), both major and minor (5:4 and 6:5; sixths 8:5 and 5:3), which are smoother and more easily tuned than Pythagorean thirds (81:64 and 32:27) and Pythagorean sixths (27:16 and 128/81),[9] with one minor third (and one major sixth) left at the Pythagorean interval, at the cost of replacing one fifth (and one fourth) with a wolf interval.

F-B is the tritone (more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.