Ptolemy's intense diatonic
Ptolemy's intense diatonic scale (also known as Zarlino, or simply the just diatonic scale) is a heptatonic 5-limit JI scale with the nicetone step pattern. In Ben Johnston's notation, the scale is the basic uninflected diatonic scale. It is a chiral scale, meaning it comes in two variations, which are called right and left-handed. Both versions contain a 40/27 wolf fifth.
It can be constructed as a Fokker block in multiple ways, making it a wakalix. Another derivation is through a tetrachord consisting of 16/15, 9/8 and 10/9.
It can also be derived from tuning the tonic, dominant and subdominant chords as 5-limit just major chords (4:5:6): FAC – CEG – GBD.
Modes
The major mode of the right-handed version gives the usual 5-limit major diatonic: 1/1 – 9/8 – 5/4 – 4/3 – 3/2 – 5/3 – 15/8 – 2/1.
| Mode name | Intervals | ||||||
|---|---|---|---|---|---|---|---|
| Lydian | 9/8 | 5/4 | 45/32 | 3/2 | 27/16 | 15/8 | 2/1 |
| Ionian (major) | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 |
| Mixolydian | 10/9 | 5/4 | 4/3 | 3/2 | 5/3 | 16/9 | 2/1 |
| Dorian | 10/9 | 32/27 | 4/3 | 40/27 | 5/3 | 16/9 | 2/1 |
| Aeolian (minor) | 9/8 | 6/5 | 27/20 | 3/2 | 8/5 | 9/5 | 2/1 |
| Phrygian | 16/15 | 6/5 | 4/3 | 3/2 | 8/5 | 9/5 | 2/1 |
| Locrian | 16/15 | 6/5 | 4/3 | 64/45 | 8/5 | 16/9 | 2/1 |
The left-handed version is most commonly encountered in its minor mode, which gives the usual 5-limit minor diatonic: 1/1 – 9/8 – 6/5 – 4/3 – 3/2 – 8/5 – 9/5 – 2/1. In the major mode, it only differs from the right-handed by having a 10/9 instead of 9/8 on the second degree. Reversing the steps of the right-handed major scale gives the left-handed phrygian mode.
| Mode name | Intervals | ||||||
|---|---|---|---|---|---|---|---|
| Lydian | 9/8 | 5/4 | 45/32 | 3/2 | 5/3 | 15/8 | 2/1 |
| Ionian (major) | 10/9 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 |
| Mixolydian | 10/9 | 5/4 | 4/3 | 40/27 | 5/3 | 16/9 | 2/1 |
| Dorian | 9/8 | 6/5 | 27/20 | 3/2 | 27/16 | 9/5 | 2/1 |
| Aeolian (minor) | 9/8 | 6/5 | 4/3 | 3/2 | 8/5 | 9/5 | 2/1 |
| Phrygian | 16/15 | 6/5 | 4/3 | 3/2 | 8/5 | 16/9 | 2/1 |
| Locrian | 16/15 | 32/27 | 4/3 | 64/45 | 8/5 | 16/9 | 2/1 |
Scala files
Both are given in the major mode.
! zarlino-RH.scl Ptolemy's Intense Diatonic Systonon, also Zarlino's scale ! 7 ! 9/8 5/4 4/3 3/2 5/3 15/8 2/1
! zarlino-LH.scl ! Ptolemy's Intense Diatonic Systonon, also Zarlino's scale 7 ! 10/9 5/4 4/3 3/2 5/3 15/8 2/1
Fokker blocks
The scale can be found in three different Fokker blocks, using the chromas 25/24, 81/80 and 135/128.
- Fokker block 1
Chroma basis: {81/80, 135/128}
Mode 5|1, dome 2|4 (LlsLlLs)
L = 9/8, l = 10/9, s = 16/15
- Fokker block 2
Chroma basis: {25/24, 81/80}
Mode 5|1, dome 4|2 (LlsLlLs)
L = 9/8, l = 10/9, S = 27/25, s = 16/15
- Fokker block 3
Chroma basis: {25/24, 135/128}
Mode 4|2, dome 4|2 (lSslSls)
L = 75/64, l = 9/8, S = 10/9, s = 16/15
See also
- Nicetone – regular 3L 2m 2s scale
- Superzarlino
- mavchrome6, dimeanmav2