Ptolemy's intense diatonic scale

Diatonic scale on C, equal tempered  Play  and Ptolemy's intense or just  Play .

Ptolemy's intense diatonic scale, also known as Ptolemaic Sequence,^[1] justly tuned major scale,^[2][3][4] or syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy,^[5] declared by 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).^[6] This is called Ptolemy's intense diatonic tetrachord, as opposed to Ptolemy's soft diatonic tetrachord, formed by 21/20, 10/9 and 8/7 intervals.^[7]

Note	Name	C		D		E		F		G		A		B		C
	Solfege	Do		Re		Mi		Fa		Sol		La		Ti		Do
	Ratio	1/1		9/8		5/4		4/3		3/2		5/3		15/8		2/1
	Harmonic	24		27		30		32		36		40		45		48
	Cents	0		204		386		498		702		884		1088		1200
Step	Name		T		t		s		T		t		T		s
	Ratio		9/8		10/9		16/15		9/8		10/9		9/8		16/15
	Cents		204		182		112		204		182		204		112

Pythagorean diatonic scale on C  Play . + indicates the syntonic comma.

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

1. Partch, Harry (1979). Genesis of a Music, p.165&73. ISBN 978-0-306-80106-8.
2. Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, p.172-73. ISBN 978-0-19-816505-7.
3. Wright, David (2009). Mathematics and Music, p.140-41. ISBN 978-0-8218-4873-9.
4. 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.
5. see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39. (Contains Harmonics by Claudius Ptolemy.)
6. Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p.961. The Encyclopædia Britannica Company.
7. Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 9
8. Johnston, Ben and Gilmore, Bob (2006). "Maximum clarity" and Other Writings on Music, p.100. ISBN 978-0-252-03098-7.