Ptolemy's intense diatonic scale, also known as Ptolemaic Sequence, [1 ] , justly tuned major scale [2 ] [3 ] or [4 ] syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, declared by [5 ] Zarlino to be the only tuning that could be reasonably sung, and corresponding with modern just intonation. [6 ]
It is produced through a
tetrachord 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, as opposed to Ptolemy's soft diatonic tetrachord, formed by [6 ] 21/20, 10/9 and 16/15 intervals. [7 ]
In comparison to
Pythagorean tuning, while both provide just perfect fourths and fifths, the Ptolemaic provides just thirds which are smoother and more easily tuned. [8 ]
Note that D-F is a Pythagorean minor third (32/27), D-A is a
defective fifth (40/27), F-D is a Pythagorean major sixth (27/16), and A-D is a defective fourth (27/20). All of these differ from their just counterparts by a syntonic comma (81/80).
Sources [ edit ]
^ Partch, Harry (1979). , p.165&73. Genesis of a Music ISBN 978-0-306-80106-8.
^ Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, p.172-73. ISBN 978-0-19-816505-7.
^ Wright, David (2009). Mathematics and Music, p.140-41. ISBN 978-0-8218-4873-9.
^ Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p.78. ISBN 978-0-252-03098-7.
^ see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39. (Contains Harmonics by Claudius Ptolemy.)
^ a b Chisholm, Hugh (1911). , Vol.28, p.961. The Encyclopædia Britannica Company. The Encyclopædia Britannica
^ Chalmers, John H. Jr. (1993). . Hanover, NH: Frog Peak Music. Divisions of the Tetrachord ISBN 0-945996-04-7 Chapter 2, Page 8
^ Johnston, Ben and Gilmore, Bob (2006). "Maximum clarity" and Other Writings on Music, p.100. ISBN 978-0-252-03098-7.