Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence, justly tuned major scale, Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the

diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps ...

proposed by

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...

, and corresponding with modern 5-limit

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and c ...

.Chisholm, Hugh (1911).
The Encyclopædia Britannica
', Vol.28, p. 961. The Encyclopædia Britannica Company. This tuning was declared by Zarlino to be the only tuning that could be reasonably sung, it was also supported by

Giuseppe Tartini Giuseppe Tartini (8 April 1692 – 26 February 1770) was an Italian composer and violinist of the Baroque era born in the Republic of Venice. Tartini was a prolific composer, composing over a hundred of pieces for the violin with the majority of ...

, and is equivalent to Indian Gandhar tuning which features exactly the same intervals. It is produced through a

tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency propo ...

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

Structure

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:

Comparison with other diatonic scales

Ptolemy's intense diatonic scale can be constructed by lowering the pitches of

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: McG ...

's 3rd, 6th, and 7th degrees (in C, the notes E, A, and B) by the

syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125 ...

, 81:80. This scale may also be considered as derived from the just major chord (ratios 4:5:6, so a major third of 5:4 and fifth of 3:2), and the major chords a fifth below and a fifth above it: FAC–CEG–GBD. This perspective emphasize the central role of the tonic, dominant, and subdominant in the diatonic scale. In comparison to

, which has just 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),Johnston, Ben and Gilmore, Bob (2006). ''"Maximum clarity" and Other Writings on Music'', p. 100. . 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. Intervals between notes ( wolf intervals bolded): Note that D–F is a Pythagorean minor third or semiditone (32:27), its inversion F–D is a Pythagorean major sixth (27:16); D–A is a

wolf fifth In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth) Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...

(40:27), and its inversion A–D is a wolf fourth (27:20). All of these differ from their just counterparts by a

(81:80). More concisely, the triad built on the 2nd degree (D) is out-of-tune. F-B is the

tritone In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...

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

References

{{Scales 5-limit tuning and intervals Heptatonic scales Ptolemy