Ptolemy's intense diatonic scale
Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence,[1] justly tuned major scale,[2][3][4] Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a
This tuning was declared by
It is produced through a
Structure
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 | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ⓘ | ||||||||
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 scales
Ptolemy's intense diatonic scale can be constructed by lowering the pitches of Pythagorean tuning's 3rd, 6th, and 7th degrees (in C, the notes E, A, and B) by the syntonic comma, 81:80. This scale may also be considered as derived from the just major chord (ratios 4:5:6, so a major third of 5:4 and fifth of 3:2), and the major chords a fifth below and a fifth above it: FAC–CEG–GBD. This perspective emphasizes the central role of the tonic, dominant, and subdominant in the diatonic scale.
In comparison to Pythagorean tuning, which only uses 3:2 perfect fifths (and fourths), the Ptolemaic provides just thirds (and sixths), both major and minor (5:4 and 6:5; sixths 8:5 and 5:3), which are smoother and more easily tuned than Pythagorean thirds (81:64 and 32:27),[9] with one minor third (and one major sixth) left at the Pythagorean interval, at the cost of replacing one fifth (and one fourth) with a wolf interval.
Intervals between notes (wolf intervals bolded):
C | D | E | F | G | A | B | C′ | D′ | E′ | F′ | G′ | A′ | B′ | C″ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
C | 1:1 | 9:8 | 5:4 | 4:3 | 3:2 | 5:3 | 15:8 | 2:1 | 9:4 | 5:2 | 8:3 | 3:1 | 10:3 | 15:4 | 4:1 |
D | 8:9 | 1:1 | 10:9 | 32:27 | 4:3 | 40:27 | 5:3 | 16:9 | 2:1 | 20:9 | 64:27 | 8:3 | 80:27 | 10:3 | 32:9 |
E | 4:5 | 9:10 | 1:1 | 16:15 | 6:5 | 4:3 | 3:2 | 8:5 | 9:5 | 2:1 | 32:15 | 12:5 | 8:3 | 3:1 | 16:5 |
F | 3:4 | 27:32 | 15:16 | 1:1 | 9:8 | 5:4 | 45:32 | 3:2 | 27:16 | 15:8 | 2:1 | 9:4 | 5:2 | 45:16 | 3:1 |
G | 2:3 | 3:4 | 5:6 | 8:9 | 1:1 | 10:9 | 5:4 | 4:3 | 3:2 | 5:3 | 16:9 | 2:1 | 20:9 | 5:2 | 8:3 |
A | 3:5 | 27:40 | 3:4 | 4:5 | 9:10 | 1:1 | 9:8 | 6:5 | 27:20 | 3:2 | 8:5 | 9:5 | 2:1 | 9:4 | 12:5 |
B | 8:15 | 9:15 | 2:3 | 32:45 | 4:5 | 8:9 | 1:1 | 16:15 | 6:5 | 4:3 | 64:45 | 8:5 | 16:9 | 2:1 | 32:15 |
C′ | 1:2 | 9:16 | 5:8 | 2:3 | 3:4 | 5:6 | 15:16 | 1:1 | 9:8 | 5:4 | 4:3 | 3:2 | 5:3 | 15:8 | 2:1 |
Note that D–F is a
F-B is the tritone (more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.
References
- 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 0-945996-04-7Chapter 2, Page 9
- ^
Johnston, Ben; Gilmore, Bob (2006). 'Maximum Clarity' and Other Writings on Music. p. 100. ISBN 978-0-252-03098-7.