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, and corresponding with modern

5-limit Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...

just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...

.Chisholm, Hugh (1911).
The Encyclopædia Britannica
', Vol.28, p. 961. The Encyclopædia Britannica Company. This tuning was declared by

Zarlino Gioseffo Zarlino (31 January or 22 March 1517 – 4 February 1590) was an Italian music theorist and composer of the Renaissance. He made a large contribution to the theory of counterpoint as well as to musical tuning. Life and career Zarlino w ...

to be the only tuning that could be reasonably sung, it was also supported by Giuseppe Tartini, and is equivalent to Indian Gandhar tuning which features exactly the same intervals. It is produced through a tetrachord consisting of a greater tone (9:8),

lesser tone In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent staff positions ( ...

(10:9), and

just diatonic semitone A semitone, also called a half step or a half tone, is the smallest interval (music), musical interval commonly used in Western tonal music, and it is considered the most Consonance and dissonance#Dissonance, dissonant when sounded harmonically ...

(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: Mc ...

's 3rd, 6th, and 7th degrees (in C, the notes E, A, and B) by the syntonic comma, 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 In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two common ...

or semiditone (32:27), its inversion F–D is a Pythagorean major sixth (27:16); D–A is a wolf fifth (40:27), and its inversion A–D is a wolf fourth (27:20). All of these differ from their just counterparts by a syntonic comma (81:80). More concisely, the triad built on the 2nd degree (D) is out-of-tune. F-B is the tritone (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