Ptolemy's intense diatonic scale, also known as thePtolemaic sequence,[1]justly tuned major scale,[2][3][4]Ptolemy's tense diatonic scale, or thesyntonous (orsyntonic)diatonic scale, is atuning for thediatonic scale proposed byPtolemy,[5] and corresponding with modern5-limitjust intonation.[6] While Ptolemy is famous for this version of just intonation, it is important to realize this was only one of several genera of just, diatonic intonations he describes. He also describes7-limit "soft" diatonics and an11-limit "even" diatonic.
This tuning was declared byZarlino to be the only tuning that could be reasonably sung, it was also supported byGiuseppe Tartini,[7] and is equivalent toIndian Gandhar tuning which features exactly the same intervals.
It is produced through atetrachord consisting of agreater tone (9:8),lesser tone (10:9), andjust diatonic semitone (16:15).[6] This is called Ptolemy's intense diatonic tetrachord (or "tense"), as opposed to Ptolemy's soft diatonic tetrachord (or "relaxed"), which is formed by21:20, 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 | |||||||||
Ptolemy's intense diatonic scale can be constructed by lowering the pitches ofPythagorean tuning's 3rd, 6th, and 7thdegrees (in C, the notes E, A, and B) by thesyntonic 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 toPythagorean 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) and Pythagorean sixths (27:16 and 128/81),[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 aPythagorean minor third or semiditone (32:27), its inversion F–D is aPythagorean major sixth (27:16); D–A is awolf fifth (40:27), and its inversion A–D is a wolf fourth (27:20). All of these differ from their just counterparts by asyntonic comma (81:80). More concisely, the triad built on the 2nd degree (D) is out-of-tune.
F-B is thetritone (more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.