Ptolemy's intense diatonic scale

Summary

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 scalesEdit

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.

ReferencesEdit

  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.)
  6. ^ a b 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.