Ptolemy's intense diatonic scale

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Ptolemy's intense diatonic scale, also called the tense diatonic scale or syntonous diatonic scale, is a tuning for the diatonic scale based on just intonation (5-limit). It is one of several diatonic tunings Ptolemy describes; he also notes a softer diatonic and an 11‑limit diatonic. This intense version was praised as singable and was supported by Zarlino and Tartini. It is closely related to Indian Gandhar tuning, sharing the same interval pattern.

The scale comes from a small tetrachord built from two whole steps and a small semitone (roughly 9:8, 10:9, and 16:15). This is called Ptolemy's intense tetrachord. To make the full scale, the 3rd, 6th, and 7th notes (in C: E, A, B) are lowered a tiny amount by the syntonic comma, so the thirds and sixths sound closer to simple 5:4 and 6:5 ratios.

In practice the scale can be seen as built from the just major chord (C–E–G) and the major chords a fifth above and below: F–A–C and G–B–D. This keeps the focus on the tonic, dominant, and subdominant.

Compared with Pythagorean tuning, Ptolemy's intense diatonic scale uses some true just intervals for thirds and sixths (5:4, 6:5, and related 8:5 and 5:3), which are smoother to tune. But to fit the full diatonic set, one fifth and one fourth are not perfectly in tune, producing a couple of wolf intervals. For example, D–F, D–A, F–B, and B–F are not perfectly in tune.

Overall, Ptolemy's intense diatonic scale is a compact, singable 5‑limit tuning of the major scale that emphasizes the tonic, dominant, and subdominant, and it shares its interval pattern with Gandhar tuning.