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 also supported by Giuseppe Tartini.[7]

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]

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.[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).

This scale may also be considered as derived from the major chord, and the major chords on top and bottom of it: FAC-CEG-GBD.


  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. 1 2 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.
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