Ptolemy's intense diatonic scale

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Diatonic scale on C, equal tempered <phonos file="Diatonic scale on C.mid">Play</phonos> and Ptolemy's intense or just <phonos file="Just diatonic scale on C.mid">Play</phonos>.

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 <phonos file="Unison on C.mid">24</phonos> <phonos file="Major tone on C.mid">27</phonos> <phonos file="Just major third on C.mid">30</phonos> <phonos file="Just perfect fourth on C.mid">32</phonos> <phonos file="Just perfect fifth on C.mid">36</phonos> <phonos file="Just major sixth on C.mid">40</phonos> <phonos file="Just major seventh on C.mid">45</phonos> <phonos file="Perfect octave on C.mid">48</phonos>
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 <phonos file="Pythagorean diatonic scale on C.mid">Play</phonos>. + 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.

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 Lua error in package.lua at line 80: module 'strict' not found. (Contains Harmonics by Claudius Ptolemy.)
  6. ↑ 6.0 6.1 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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