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 a heptatonic scale, heptatonic (seven-note) 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 eith ...
proposed by
Ptolemy
Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
, and corresponding with modern
5-limit just intonation
In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
.
[Chisholm, Hugh (1911). ]
The Encyclopædia Britannica
', Vol.28, p. 961. The Encyclopædia Britannica Company. While Ptolemy is famous for this version of just intonation, it is important to realize this was only one of several genera of just, diatonic intonations he describes. He also describes
7-limit "soft" diatonics and an
11-limit "even" diatonic.
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 Pirano in the Republic of Venice (now Piran, Slovenia). Tartini was a prolific composer, composing over a hundred pieces for the ...
, and is equivalent to
Indian Gandhar tuning which features exactly the same intervals.
It is produced through a
tetrachord
In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
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 determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...
's 3rd, 6th, and 7th
degrees (in C, the notes E, A, and B) by the
syntonic comma
In music theory
Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
, 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 emphasizes the central role of the tonic, dominant, and subdominant in the diatonic scale.
In comparison to
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...
, 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),
[
]
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 (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
In music theory
Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
(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 interval (music), musical interval spanning three adjacent Major second, 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 ...
(more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.
References
{{Scales
5-limit tuning and intervals
Heptatonic scales
Ptolemy