Ptolemy's intense diatonic

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Ptolemy's intense diatonic scale

The left and right handed variations of the just diatonic scale on the 5-limit lattice.

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

Right-handed just major scale on C, in HEJI notation.

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.

Modes of Zarlino (right-handed)

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

Left-handed just minor scale on C, in HEJI notation.

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.

Modes of Zarlino (left-handed)

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