Ptolemy's intense diatonic scale: Difference between revisions
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Lowering the pitches of [[Pythagorean tuning]]'s notes E |
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. |
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Intervals between notes ([[Wolf interval |wolf intervals]] bolded): |
Intervals between notes ([[Wolf interval |wolf intervals]] bolded): |
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Revision as of 23:10, 9 August 2019
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 | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | |||||||||
| 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 | ||||||||||
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 | 15/9 | 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 | 18/15 | 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 |
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, 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.
Sources
- ^ 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.)
- ^ a b 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.