Ptolemy's intense diatonic scale

Ptolemy's intense or syntonous diatonic scale, or syntonic diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy[1], declared by Zarlino to be the only tuning that could be reasonably sung, and corresponding with modern just intonation.[2]

It is produced through a tetrachord consisting of a greater tone (8/9), lesser tone (9/10), and diatonic semitone (15/16).[2] Thus Ptolemy's intense diatonic scale, Ptolemaic Sequence,[3] or the justly tuned diatonic major scale[4][5][6]:

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
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

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.[7]

Sources

  1. ^ see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39.  (Contains Harmonics by Claudius Ptolemy.)
  2. ^ a b Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p.961. The Encyclopædia Britannica Company.
  3. ^ Partch, Harry (1979). Genesis of a Music, p.165&73. ISBN 9780306801068.
  4. ^ Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, p.172-73. ISBN 9780198165057.
  5. ^ Wright, David (2009). Mathematics and Music, p.140-41. ISBN 9780821848739.
  6. ^ Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p.78. ISBN 9780252030987.
  7. ^ Johnston, Ben and Gilmore, Bob (2006). "Maximum clarity" and Other Writings on Music, p.100. ISBN 9780252030987.