Ptolemy's intense diatonic: Difference between revisions

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Add mode tables, clarifications, and HEJI scale graphics
 
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{{interwiki
| de = Durtonleiter
| en = Ptolemy's intense diatonic
}}
{{Wikipedia|Ptolemy's intense diatonic scale}}
{{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}}.
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>
<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
! x^7-11*x^5-10*x^4+21*x^3+30*x^2+9*x
!  
!  
! 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
7
7
!
!
Line 25: Line 79:
2/1
2/1
</pre>
</pre>
<pre>
<pre>
! zarlino-LH.scl
! 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
! x^7-11*x^5-10*x^4+21*x^3+30*x^2+9*x
7
7
!
!
Line 40: Line 94:
2/1
2/1
</pre>
</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
; Fokker block 1
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* [[Nicetone]] – regular 3L 2m 2s scale
* [[Nicetone]] – regular 3L 2m 2s scale
* [[Superzarlino]]
* [[Superzarlino]]
* [[mavchrome6]], [[dimeanmav2]]


[[Category:7-tone scales]]
[[Category:7-tone scales]]

Latest revision as of 19:53, 19 September 2026

English Wikipedia has an article on:
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

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