Ptolemy's intense diatonic scale

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 also supported by Giuseppe Tartini.[7] It is equivalent to Indian Gandhar tuning which features exactly the same intervals.

Diatonic scale on C, equal tempered

Play and Ptolemy's intense or just

Play .

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.[8] 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:

Note	Name	C		D		E		F		G		A		B		C
	Solfege	Do		Re		Mi		Fa		Sol		La		Ti		Do
	Ratio from C	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

Note	Name	A		B		C		D		E		F		G		A
	Ratio from A	1:1		9:8		6:5		4:3		3:2		8:5		9:5		2:1
	Harmonic of Fundamental B♭	120		135		144		160		180		192		216		240
	Cents	0		204		316		498		702		814		1018		1200
Step	Name		T		s		t		T		s		T		t
	Ratio		9:8		16:15		10:9		9:8		16:15		9:8		10:9
	Cents		204		112		182		204		112		204		182

Comparison with other diatonic scales

Lowering the pitches of Pythagorean tuning's notes E, A, and B by the syntonic comma, 81/80, to give a just intonation, changes it to Ptolemy's intense diatonic scale.

Intervals between notes (wolf intervals bolded):

	C	D	E	F	G	A	B	C'	D'	E'	F'	G'	A'	B'	C"
C	1	9/8	5/4	4/3	3/2	5/3	15/8	2	9/4	5/2	8/3	3	10/3	15/4	4
D	8/9	1	10/9	32/27	4/3	40/27	5/3	16/9	2	20/9	64/27	8/3	80/27	30/9	32/9
E	4/5	9/10	1	16/15	6/5	4/3	3/2	8/5	9/5	2	32/15	12/5	8/3	3	16/5
F	3/4	27/32	15/16	1	9/8	5/4	45/32	3/2	27/16	15/8	2	9/4	5/2	45/16	3
G	2/3	3/4	5/6	8/9	1	10/9	5/4	4/3	3/2	5/3	16/9	2	20/9	5/2	8/3
A	3/5	27/40	3/4	4/5	9/10	1	9/8	6/5	27/20	3/2	8/5	9/5	2	9/4	12/5
B	8/15	9/15	2/3	32/45	4/5	8/9	1	16/15	6/5	4/3	64/45	8/5	16/9	2	32/15
C'	1/2	9/16	5/8	2/3	3/4	5/6	15/16	1	9/8	5/4	4/3	3/2	5/3	15/8	2

Pythagorean diatonic scale on C

Play . Johnston's notation; + 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.[9]

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).

F-B is the tritone (more precisely, the augmented fourth), here 45/32.

This scale may also be considered as derived from the major chord, and the major chords above and below it: FAC–CEG–GBD.

References

Partch, Harry (1979). Genesis of a Music, pp. 165, 173. ISBN 978-0-306-80106-8.
Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, pp. 172–73. ISBN 978-0-19-816505-7.
Wright, David (2009). Mathematics and Music, pp. 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.)
Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p. 961. The Encyclopædia Britannica Company.
Dr. Crotch (October 1, 1861). "On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.", The Musical Times, p. 115.
Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 9
Johnston, Ben and Gilmore, Bob (2006). "Maximum clarity" and Other Writings on Music, p. 100. ISBN 978-0-252-03098-7.

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[1] Partch, Harry (1979). Genesis of a Music, pp. 165, 173. ISBN 978-0-306-80106-8.

[2] Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, pp. 172–73. ISBN 978-0-19-816505-7.

[3] Wright, David (2009). Mathematics and Music, pp. 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.)

[EB-6] Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p. 961. The Encyclopædia Britannica Company.

[7] Dr. Crotch (October 1, 1861). "On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.", The Musical Times, p. 115.

[8] Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 9

[9] Johnston, Ben and Gilmore, Bob (2006). "Maximum clarity" and Other Writings on Music, p. 100. ISBN 978-0-252-03098-7.