Ptolemy's intense diatonic: Difference between revisions

Latest revision as of 19:53, 19 September 2026

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Ptolemy's intense diatonic scale

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.

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

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

@@ Line 1: / Line 1: @@
+{{interwiki
+| de = Durtonleiter
+| en = Ptolemy's intense diatonic
+}}
 {{Wikipedia|Ptolemy's intense diatonic scale}}
-'''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. It is also the basis of [[Ben Johnston's notation]].
+[[File:zarlino2.png|400px|thumb|right|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 [[9/8]], [[10/9]] and [[16/15]].
+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]]): {{dash|FAC, CEG, GBD}}.
+== Modes ==
+[[File:heji_just_major.png|400px|thumb|right|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: {{dash|1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}}.
+{| class="wikitable left-1" style="text-align:center"
+|+ Modes of Zarlino (right-handed)
+|-
+! Mode name
+! colspan=7 | 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
+|}
+[[File:heji_just_minor.png|400px|thumb|right|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: {{dash|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.
+{| class="wikitable left-1" style="text-align:center"
+|+ Modes of Zarlino (left-handed)
+|-
+! Mode name
+! colspan=7 | 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.
 <pre>
-! zarlino-RH.scl AKA dimeanmav1.scl
+! zarlino-RH.scl
-Ptolemy's Intense Diatonic Systonon, also Zarlino's scale, mavchrome7, Dwarf(&lt;7 11 16|), or Dicot-meantone-mavila wakalix 1
+Ptolemy's Intense Diatonic Systonon, also Zarlino's scale
 !
-! wakalix info:
-! dicot: -2 to 4; meantone: -1 to 5; mavila: -4 to 2
-! dicot: LssLsLs; meantone: LLsLLLs; mavila: sLssLss
-! same as zarlino, Dwarf(<7 11 16|), mavchrome7, synchrome7, synmav7, ptolemy, yves, indian-magramma, even7a
 !
@@ Line 25: / Line 80: @@
 </pre>
-The left-handed variation has 10/9 instead of 9/8 on the second degree. It is most commonly encountered in the 6th mode (with 5/3 as the root) which gives the usual 5-limit minor diatonic: {{dash|1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 9/5, 2/1}}.
 <pre>
 ! zarlino-LH.scl
 !
-Ptolemy's Intense Diatonic Systonon, also Zarlino's scale, mavchrome7, Dwarf(&lt;7 11 16|)
+Ptolemy's Intense Diatonic Systonon, also Zarlino's scale
 !
@@ Line 40: / Line 94: @@
 /1
 </pre>
+== Fokker blocks ==
+The scale can be found in three different [[Fokker block]]s, using the chromas [[25/24]], [[81/80]] and [[135/128]].
 ; Fokker block 1
@@ Line 68: / Line 125: @@
 * [[Nicetone]] – regular 3L 2m 2s scale
 * [[Superzarlino]]
+* [[mavchrome6]], [[dimeanmav2]]
 [[Category:7-tone scales]]

Ptolemy's intense diatonic: Difference between revisions

Latest revision as of 19:53, 19 September 2026

Contents

Modes

Scala files

Fokker blocks

See also

Navigation menu

Ptolemy's intense diatonic: Difference between revisions

Latest revision as of 19:53, 19 September 2026

Modes

Scala files

Fokker blocks

See also

Navigation menu

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