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{{Todo|cleanup|inline=1|text=Adopt zero-indexed intervals/degrees}}{{Infobox MOS
{{Infobox MOS
| Name = checkertonic
| Name = checkertonic
| Periods = 1
| Periods = 1
Line 8: Line 8:
| Pattern = LsLssLss
| Pattern = LsLssLss
}}
}}
{{MOS intro|Other Names=anti-oneirotonic}}


{{MOS intro|Other Names=anti-oneirotonic}}
== Name ==
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.


==Intervals==
== Scale properties ==
{{TAMNAMS use}}
{{TAMNAMS use}}
=== Intervals ===
{{MOS intervals}}
{{MOS intervals}}


== Notation ==
=== Generator chain ===
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
{{MOS genchain}}
 
The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.


== Theory ==
=== Modes ===
In contrast to oneirotonic ([[5L 3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
{{MOS mode degrees}}


=== Low harmonic entropy scales ===
==== Proposed mode names ====
There are two significant harmonic entropy minima with this MOS pattern:
The modes of checkertonic can be named after its sister mos [[5L 3s]] (oneirotonic). {{u|R-4981}} has also proposed names based on {{w|grand chess}} pieces.
 
{{MOS modes
* [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
| Table Headers=
* [[Meantone family #Squares|Squares]], in which the generator is also a 9/7, but two of them make an 18/11 and four of them make a 4/3, which is improper.
Anti-modes of 5L 3s $
 
Grand chess names<sup>[proposed]</sup>
==Tuning ranges==
| Table Entries=
===Simple tunings===
Anti-Sarnathian (sar-NA(H)TH-iən) $
{| class="wikitable right-2 right-3 right-4 sortable "
King $
|-
Anti-Hlanithian (lə-NITH-iən) $
! class="unsortable" |Degree
Queen $
! Size in [[11edo]] (basic)
Anti-Kadathian (kə-DA(H)TH-iən) $
!Size in [[14edo]] (hard)
Marshall $
!Size in [[19edo]] (soft)
Anti-Mnarian (mə-NA(I)R-iən) $
! class="unsortable" |Note name on J
Cardinal $
!#Gens up
Anti-Ultharian (ul-THA(I)R-iən) $
|-
Rook $
|min. chk2nd
Anti-Celephaïsian (kel-ə-FAY-zhən) $
|1\11, 109.1
Bishop $
| 1\14, 85.7
Anti-Illarnekian (ill-ar-NEK-iən) $
|2\19, 126.3
Knight $
|K
Anti-Dylathian (də-LA(H)TH-iən) $
| +3
Pawn $
|-
}}
|maj. chk2nd
|2\11, 218.2
|3\14, 257.1
|3\19, 189.5
|K&
|  -5
|-
|min. chk3rd
|2\11, 218.2
|2\14, 171.4
|4\19, 252.6
|L@
| +6
|-
| maj. chk3rd
|3\11, 327.3
| 4\14, 342.9
| 5\19, 315.8
|L
| -2
|-
|perf. chk4th
|4\11, 436.4
|5\14, 428.6
| 7\19, 442.1
| M
| +1
|-
|aug. chk4th
|5\11, 545.5
|7\14, 600.0
|8\19, 505.3
|M&
| -7
|-
|min. chk5th
|5\11, 545.5
|6\14, 514.3
|9\19, 568.4
|N
|  +4
|-
|maj. chk5th
|6\11, 656.6
|8\14, 685.7
|10\19, 631.6
|N&
|  -4
|-
|dim. chk6th
|6\11, 656.6
| 7\14, 600.0
|11\19, 694.7
|O@
| +7
|-
|perf. chk6th
|7\11, 763.6
|8\14, 771.4
|12\19, 757.9
| O
| -1
|-
|min. chk7th
|8\11, 872.7
|10\14, 857.1
|14\19, 884.2
| P
| +2
|-
|maj. chk7th
| 9\11, 981.8
|12\14, 1028.6
|15\19, 947.4
| P&
| -6
|-
|min. chk8th
|9\11, 981.8
|11\14, 942.9
|16\19, 1010.5
|Q@
| +5
|-
|maj. chk8th
|10\11, 1090.9
|13\14, 1114.3
|17\19, 1073.7
| Q
| -3
|}
 
===Parasoft===
Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).
 
Sortable table of major and minor intervals in parasoft checkertonic tunings:
 
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable" |Degree
!Size in [[19edo]] (soft)
!Size in [[27edo]] (supersoft)
!Size in [[46edo]]
! class="unsortable" | Note name on J
! class="unsortable" |Approximate ratios
!#Gens up
|-
|unison
|0\19, 0.00
|0\27, 0.00
|0\46, 0.00
|J
|1/1
|0
|-
|min. chk2nd
|2\19, 126.3
|3\27, 133.3
|5\46, 130.4
|K
|14/13
|  +3
|-
| maj. chk2nd
|3\19, 189.5
|4\27, 177.8
| 7\46, 182.6
|K&
|10/9
| -5
|-
|min. chk3rd
|4\19, 252.6
|6\27, 266.7
| 10\46, 260.9
|L@
|7/6
| +6
|-
|maj. chk3rd
|5\19, 315.8
|7\27, 311.1
| 12\46, 313.0
|L
|6/5
| -2
|-
|perf. chk4th
|7\19, 442.1
|10\27, 444.4
|17\46, 443.5
| M
|9/7, 13/10
| +1
|-
|aug. chk4th
|8\19, 505.3
|11\27, 488.9
|19\46, 495.7
|M&
|4/3
|  -7
|-
|min. chk5th
|9\19, 568.4
|13\27, 577.8
| 22\46, 573.9
|N
|7/5, 18/13
| +4
|-
|maj. chk5th
|10\19, 631.6
| 14\27, 622.2
|24\46, 626.1
|N&
|10/7, 13/9
| -4
|-
|dim. chk6th
|11\19, 694.7
|16\27, 711.1
|27\46, 704.3
| O@
|3/2
| +7
|-
|perf. chk6th
|12\19, 757.9
| 17\27, 755.6
|20\46, 756.5
|O
| 14/9, 20/13
| -1
|-
|min. chk7th
|14\19, 884.2
| 20\27, 888.9
|34\46, 887.0
|P
|5/3
| +2
|-
|maj. chk7th
|15\19, 947.4
|21\27, 933.3
|36\46, 939.1
|P&
|12/7
| -6
|-
|min. chk8th
|16\19, 1010.5
|23\27, 1022.2
|39\46, 1017.4
|Q@
|9/5
| +5
|-
|maj. chk8th
| 17\19, 1073.7
|24\27, 1066.7
|41\46, 1069.6
|Q
|13/7
| -3
|}
 
Tunings in this region have a regular temperament interpretation called [[sensi]].
 
==Modes==
{{MOS mode degrees}}
===Proposed Names===
The modes of checkertonic can be named after its sister MOS [[5L 3s]] (oneirotonic). [[User:R-4981|R-4981]] has also proposed names based on [[wikipedia:Grand_Chess|grand chess]] pieces.
{{MOS modes|Table Headers=Anti-modes of 5L 3s;Grand chess names<sup>[proposed]</sup>|Table Entries=Anti-Sarnathian (sar-NA(H)TH-iən);
King;
Anti-Hlanithian (lə-NITH-iən);
Queen;
Anti-Kadathian (kə-DA(H)TH-iən);
Marshall;
Anti-Mnarian (mə-NA(I)R-iən);
Cardinal;
Anti-Ultharian (ul-THA(I)R-iən);
Rook;
Anti-Celephaïsian (kel-ə-FAY-zhən);
Bishop;
Anti-Illarnekian (ill-ar-NEK-iən);
Knight;
Anti-Dylathian (də-LA(H)TH-iən);
Pawn;}}
The order of modes on the white keys JKLMNOPQ are:
The order of modes on the white keys JKLMNOPQ are:


Line 311: Line 61:


{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Scale degrees (on J, {{nowrap|sLssLsLs {{=}} JKLMNOPQ}})
|-
|-
|+Scale degrees (on J, sLssLsLs = JKLMNOPQ)
! [[UDP]]
|-
! Anti-modes of 5L 3s
![[UDP]]
! Chess-based names
!Anti-modes of 5L 3s
! Step pattern
!Chess-based names
!Step pattern
! 1
! 1
!2
! 2
!3
! 3
! 4
! 4
!5
! 5
!6
! 6
!7
! 7
!8
! 8
!(9)
! (9)
|-
|-
|<nowiki>7|0</nowiki>
| 7{{pipe}}0
|Anti-Sarnathian
| Anti-Sarnathian
|King
| King
|LsLssLss
| LsLssLss
| J
| J
|K&
| K&amp;
|L
| L
|M&
| M&amp;
|N&
| N&amp;
|O
| O
|P&
| P&amp;
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>6|1</nowiki>
| 6{{pipe}}1
|Anti-Hlanithian
| Anti-Hlanithian
|Queen
| Queen
|LssLsLss
| LssLsLss
|J
| J
|K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
|O
| O
|P&
| P&amp;
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>5|2</nowiki>
| 5{{pipe}}2
|Anti-Kadathian
| Anti-Kadathian
|Marshall
| Marshall
|LssLssLs
| LssLssLs
|J
| J
| K&
| K&amp;
|L
| L
|M
| M
|N&
| N&amp;
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>4|3</nowiki>
| 4{{pipe}}3
|Anti-Mnarian
| Anti-Mnarian
|Cardinal
| Cardinal
|sLsLssLs
| sLsLssLs
|J
| J
|K
| K
|L
| L
|M
| M
|N&
| N&amp;
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>3|4</nowiki>
| 3{{pipe}}4
| Anti-Ultharian
| Anti-Ultharian
|Rook
| Rook
|sLssLsLs
| sLssLsLs
|J
| J
|K
| K
|L
| L
|M
| M
| N
| N
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>2|5</nowiki>
| 2{{pipe}}5
|Anti-Celephaïsian
| Anti-Celephaïsian
|Bishop
| Bishop
|sLssLssL
| sLssLssL
|J
| J
|K
| K
|L
| L
|M
| M
|N
| N
| O
| O
|P
| P
|Q@
| Q@
|(J)
| (J)
|-
|-
|<nowiki>1|6</nowiki>
| 1{{pipe}}6
|Anti-Illarnekian
| Anti-Illarnekian
|Knight
| Knight
|ssLsLssL
| ssLsLssL
|J
| J
|K
| K
|L@
| L@
|M
| M
|N
| N
|O
| O
|P
| P
|Q@
| Q@
|(J)
| (J)
|-
|-
|<nowiki>0|7</nowiki>
| 0{{pipe}}7
|Anti-Dylathian
| Anti-Dylathian
|Pawn
| Pawn
|ssLssLsL
| ssLssLsL
|J
| J
|K
| K
|L@
| L@
|M
| M
|N
| N
|O@
| O@
|P
| P
|Q@
| Q@
|(J)
| (J)
|}
|}


==Temperaments==
== Notation ==
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
 
The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &amp;/@ = up/down by chroma.
 
== Theory ==
In contrast to oneirotonic ([[5L&nbsp;3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
 
=== Low harmonic entropy scales ===
There are two significant harmonic entropy minima with this MOS pattern:
 
* [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
* [[Meantone family #Squares|Squares]], in which the generator is also a 9/7, but two of them make an 18/11 and four of them make a 4/3, which is improper.
 
== Tuning ranges ==
=== Simple tunings ===
{{MOS tunings}}
 
=== Parasoft tunings ===
Parasoft tunings (step ratios 4:3 to 3:2) are associated with [[sensi]] tempermament.
{{MOS tunings|Step Ratios=Parasoft|JI Ratios=Subgroup: 2.3.5.7.13; Int Limit: 50; Tenney Height: 8; Complements Only: 1|Tolerance=10}}
 
== Temperaments ==
The major temperaments in this area are:
The major temperaments in this area are:
*[[Sensi]] (Parasoft checkertonic)
* [[Sensi]] (Parasoft checkertonic)
*[[Squares]] (Parahard checkertonic)
* [[Squares]] (Parahard checkertonic)


== Music ==
== Music ==
* [[Uncreative Name]], [https://www.youtube.com/watch?v=XZ3zB3EDKOM The Nachtlandian Somersault] (19edo)
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=XZ3zB3EDKOM ''The Nachtlandian Somersault''] (19edo)


==Scale tree==
== Scale tree ==
Generator ranges:
Generator ranges:
*Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
* Chroma-positive generator: 750{{c}} (5\8) to 800{{c}} (2\3)
* Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)
* Chroma-negative generator: 400{{c}} (1\3) to 450{{c}} (3\8)
 
{{MOS tuning spectrum
{| class="wikitable center-all"
| 7/5 = [[Sensi]] (optimal around here)
! colspan="6" |Generator
| 11/7 = [[Clyde]]
!Cents
| 13/8 = Golden [[sentry]] (759.4078{{c}})
! L
| 13/5 = Unnamed golden tuning (768.8815{{c}})
!s
| 11/4 = [[Hamity]]
!L/s
| 7/2 = [[Squares]] (optimal around here)
!Comments
| 6/1 = [[Roman]]↓, [[hocus]]↓
|-
}}
|5\8|| || || || || ||750.000 ||1||1||1.000 ||
|-
| || || || || ||27\43 || 753.488||6||5||1.200||
|-
| || || || ||22\35|| ||754.286||5|| 4|| 1.250 ||
|-
| ||  ||  || || ||39\62 || 754.839||9||7||1.286||
|-
|  ||  || || 17\27|| || ||755.556||4||3 || 1.333||
|-
| ||  ||  || || ||46\73||756.164||11 || 8||1.375||
|-
| || ||  ||  || 29\46|| ||756.522||7|| 5||1.400||[[Sensi]] is in this region
|-
| || || || || ||41\65||756.923||10 ||7||1.429||
|-
|  ||  || 12\19|| || || ||757.895||3||2|| 1.500||
|-
|  || ||  ||  ||  ||43\68||758.824 ||11|| 7||1.571 ||[[Clyde]]
|-
| ||  ||  ||  ||31\49|| ||759.184|| 8|| 5|| 1.600||
|-
| || || || || ||50\79 || 759.494||13|| 8||1.625||Golden checkertonic/[[sentry]] (759.4078¢)
|-
| || || ||19\30|| || ||760.000||5|| 3|| 1.667||
|-
| || || || || ||45\71||760.563||12|| 7|| 1.714||
|-
|  ||  ||  ||  ||26\41|| || 760.976||7||4||1.750||
|-
| || || || || ||33\52||761.538 || 9|| 5|| 1.800||
|-
| ||7\11|| ||  ||  || ||763.636 ||2||1 || 2.000|| Basic checkertonic <br>(Generators smaller than this are proper)
|-
|  || || ||  || ||30\47||765.957||9||4 ||2.250 ||
|-
| || || || ||23\36|| ||766.667||7||3|| 2.333||
|-
| || ||  ||  || ||39\61||767.213||12||5|| 2.400||
|-
| ||  || || 16\25|| || ||768.000||5||2 ||2.500||
|-
| || || ||  ||  ||41\64|| 768.750||13 || 5||2.600||Unnamed golden tuning (768.8815¢)
|-
| || || || ||25\39|| ||769.231||8||3|| 2.667||
|-
| || || || || ||34\53||769.811||11||4||2.750||[[Hamity]]
|-
| || ||9\14|| ||  || || 771.429 ||3||1||3.000||
|-
| || || || || ||29\45||773.333 || 10||3||3.333 ||
|-
| ||  || || ||20\31|| ||774.194||7||2||3.500||[[Squares]] is in this region
|-
| ||  || || || ||31\48 ||775.000||11|| 3|| 3.667||
|-
|  ||  ||  || 11\17|| || ||776.471 || 4||1||4.000||
|-
| ||  ||  || || ||24\37||778.378||9|| 2|| 4.500||
|-
| || || || ||13\20||  || 780.000||5||1||5.000||
|-
| || || || || ||15\23||782.609 || 6|| 1||6.000||[[Roman]]↓, [[Hocus]]↓
|-
| 2\3|| || ||  ||  || ||800.000||1||0||→ inf||
|}
 
[[Category:8-tone scales]]
[[Category:checkertonic]]

Latest revision as of 22:25, 27 April 2025

↖ 2L 4s ↑ 3L 4s 4L 4s ↗
← 2L 5s 3L 5s 4L 5s →
↙ 2L 6s ↓ 3L 6s 4L 6s ↘ 
┌╥┬╥┬┬╥┬┬┐
│║│║││║│││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLss
ssLssLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\8 to 2\3 (750.0 ¢ to 800.0 ¢)
Dark 1\3 to 3\8 (400.0 ¢ to 450.0 ¢)
TAMNAMS information
Name checkertonic
Prefix check-
Abbrev. chk
Related MOS scales
Parent 3L 2s
Sister 5L 3s
Daughters 8L 3s, 3L 8s
Neutralized 6L 2s
2-Flought 11L 5s, 3L 13s
Equal tunings
Equalized (L:s = 1:1) 5\8 (750.0 ¢)
Supersoft (L:s = 4:3) 17\27 (755.6 ¢)
Soft (L:s = 3:2) 12\19 (757.9 ¢)
Semisoft (L:s = 5:3) 19\30 (760.0 ¢)
Basic (L:s = 2:1) 7\11 (763.6 ¢)
Semihard (L:s = 5:2) 16\25 (768.0 ¢)
Hard (L:s = 3:1) 9\14 (771.4 ¢)
Superhard (L:s = 4:1) 11\17 (776.5 ¢)
Collapsed (L:s = 1:0) 2\3 (800.0 ¢)

3L 5s, named checkertonic in TAMNAMS (also known as anti-oneirotonic), is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 750 ¢ to 800 ¢, or from 400 ¢ to 450 ¢.

Name

TAMNAMS suggests the temperament-agnostic name checkertonic for this scale.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-checkstep Perfect 0-checkstep P0chks 0 0.0 ¢
1-checkstep Minor 1-checkstep m1chks s 0.0 ¢ to 150.0 ¢
Major 1-checkstep M1chks L 150.0 ¢ to 400.0 ¢
2-checkstep Minor 2-checkstep m2chks 2s 0.0 ¢ to 300.0 ¢
Major 2-checkstep M2chks L + s 300.0 ¢ to 400.0 ¢
3-checkstep Perfect 3-checkstep P3chks L + 2s 400.0 ¢ to 450.0 ¢
Augmented 3-checkstep A3chks 2L + s 450.0 ¢ to 800.0 ¢
4-checkstep Minor 4-checkstep m4chks L + 3s 400.0 ¢ to 600.0 ¢
Major 4-checkstep M4chks 2L + 2s 600.0 ¢ to 800.0 ¢
5-checkstep Diminished 5-checkstep d5chks L + 4s 400.0 ¢ to 750.0 ¢
Perfect 5-checkstep P5chks 2L + 3s 750.0 ¢ to 800.0 ¢
6-checkstep Minor 6-checkstep m6chks 2L + 4s 800.0 ¢ to 900.0 ¢
Major 6-checkstep M6chks 3L + 3s 900.0 ¢ to 1200.0 ¢
7-checkstep Minor 7-checkstep m7chks 2L + 5s 800.0 ¢ to 1050.0 ¢
Major 7-checkstep M7chks 3L + 4s 1050.0 ¢ to 1200.0 ¢
8-checkstep Perfect 8-checkstep P8chks 3L + 5s 1200.0 ¢

Generator chain

Generator chain of 3L 5s
Bright gens Scale degree Abbrev.
10 Augmented 2-checkdegree A2chkd
9 Augmented 5-checkdegree A5chkd
8 Augmented 0-checkdegree A0chkd
7 Augmented 3-checkdegree A3chkd
6 Major 6-checkdegree M6chkd
5 Major 1-checkdegree M1chkd
4 Major 4-checkdegree M4chkd
3 Major 7-checkdegree M7chkd
2 Major 2-checkdegree M2chkd
1 Perfect 5-checkdegree P5chkd
0 Perfect 0-checkdegree
Perfect 8-checkdegree		 P0chkd
P8chkd
−1 Perfect 3-checkdegree P3chkd
−2 Minor 6-checkdegree m6chkd
−3 Minor 1-checkdegree m1chkd
−4 Minor 4-checkdegree m4chkd
−5 Minor 7-checkdegree m7chkd
−6 Minor 2-checkdegree m2chkd
−7 Diminished 5-checkdegree d5chkd
−8 Diminished 8-checkdegree d8chkd
−9 Diminished 3-checkdegree d3chkd
−10 Diminished 6-checkdegree d6chkd

Modes

Scale degrees of the modes of 3L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (checkdegree)
0 1 2 3 4 5 6 7 8
7|0 1 LsLssLss Perf. Maj. Maj. Aug. Maj. Perf. Maj. Maj. Perf.
6|1 6 LssLsLss Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 3 LssLssLs Perf. Maj. Maj. Perf. Maj. Perf. Min. Maj. Perf.
4|3 8 sLsLssLs Perf. Min. Maj. Perf. Maj. Perf. Min. Maj. Perf.
3|4 5 sLssLsLs Perf. Min. Maj. Perf. Min. Perf. Min. Maj. Perf.
2|5 2 sLssLssL Perf. Min. Maj. Perf. Min. Perf. Min. Min. Perf.
1|6 7 ssLsLssL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 4 ssLssLsL Perf. Min. Min. Perf. Min. Dim. Min. Min. Perf.

Proposed mode names

The modes of checkertonic can be named after its sister mos 5L 3s (oneirotonic). R-4981 has also proposed names based on grand chess pieces.

Modes of 3L 5s
UDP Cyclic
order		 Step
pattern		 Anti-modes of 5L 3s Grand chess names[proposed]
7|0 1 LsLssLss Anti-Sarnathian (sar-NA(H)TH-iən) King
6|1 6 LssLsLss Anti-Hlanithian (lə-NITH-iən) Queen
5|2 3 LssLssLs Anti-Kadathian (kə-DA(H)TH-iən) Marshall
4|3 8 sLsLssLs Anti-Mnarian (mə-NA(I)R-iən) Cardinal
3|4 5 sLssLsLs Anti-Ultharian (ul-THA(I)R-iən) Rook
2|5 2 sLssLssL Anti-Celephaïsian (kel-ə-FAY-zhən) Bishop
1|6 7 ssLsLssL Anti-Illarnekian (ill-ar-NEK-iən) Knight
0|7 4 ssLssLsL Anti-Dylathian (də-LA(H)TH-iən) Pawn

The order of modes on the white keys JKLMNOPQ are:

  • J Anti-Ultharian, Rook
  • K Anti-Hlanithian, Queen
  • L Anti-Illarnekian, Knight
  • M Anti-Mnarian, Cardinal
  • N Anti-Sarnathian, King
  • O Anti-Celephaïsian, Bishop
  • P Anti-Kadathian, Marshall
  • Q Anti-Dylathian, Pawn
Scale degrees (on J, sLssLsLs = JKLMNOPQ)
UDP Anti-modes of 5L 3s Chess-based names Step pattern 1 2 3 4 5 6 7 8 (9)
7|0 Anti-Sarnathian King LsLssLss J K& L M& N& O P& Q (J)
6|1 Anti-Hlanithian Queen LssLsLss J K& L M N& O P& Q (J)
5|2 Anti-Kadathian Marshall LssLssLs J K& L M N& O P Q (J)
4|3 Anti-Mnarian Cardinal sLsLssLs J K L M N& O P Q (J)
3|4 Anti-Ultharian Rook sLssLsLs J K L M N O P Q (J)
2|5 Anti-Celephaïsian Bishop sLssLssL J K L M N O P Q@ (J)
1|6 Anti-Illarnekian Knight ssLsLssL J K L@ M N O P Q@ (J)
0|7 Anti-Dylathian Pawn ssLssLsL J K L@ M N O@ P Q@ (J)

Notation

The TAMNAMS system is used in this article to refer to 3L 5s step size ratios and step ratio ranges.

The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.

Theory

In contrast to oneirotonic (5L 3s), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth (3/2, usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.

Low harmonic entropy scales

There are two significant harmonic entropy minima with this MOS pattern:

  • Sensi, in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
  • Squares, in which the generator is also a 9/7, but two of them make an 18/11 and four of them make a 4/3, which is improper.

Tuning ranges

Simple tunings

Simple Tunings of 3L 5s
Scale degree Abbrev. Basic (2:1)
11edo 		Hard (3:1)
14edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\11 0.0 0\14 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 1\11 109.1 1\14 85.7 2\19 126.3
Major 1-checkdegree M1chkd 2\11 218.2 3\14 257.1 3\19 189.5
Minor 2-checkdegree m2chkd 2\11 218.2 2\14 171.4 4\19 252.6
Major 2-checkdegree M2chkd 3\11 327.3 4\14 342.9 5\19 315.8
Perfect 3-checkdegree P3chkd 4\11 436.4 5\14 428.6 7\19 442.1
Augmented 3-checkdegree A3chkd 5\11 545.5 7\14 600.0 8\19 505.3
Minor 4-checkdegree m4chkd 5\11 545.5 6\14 514.3 9\19 568.4
Major 4-checkdegree M4chkd 6\11 654.5 8\14 685.7 10\19 631.6
Diminished 5-checkdegree d5chkd 6\11 654.5 7\14 600.0 11\19 694.7
Perfect 5-checkdegree P5chkd 7\11 763.6 9\14 771.4 12\19 757.9
Minor 6-checkdegree m6chkd 8\11 872.7 10\14 857.1 14\19 884.2
Major 6-checkdegree M6chkd 9\11 981.8 12\14 1028.6 15\19 947.4
Minor 7-checkdegree m7chkd 9\11 981.8 11\14 942.9 16\19 1010.5
Major 7-checkdegree M7chkd 10\11 1090.9 13\14 1114.3 17\19 1073.7
Perfect 8-checkdegree P8chkd 11\11 1200.0 14\14 1200.0 19\19 1200.0

Parasoft tunings

Parasoft tunings (step ratios 4:3 to 3:2) are associated with sensi tempermament.

Parasoft Tunings of 3L 5s
Scale degree Abbrev. Supersoft (4:3)
27edo 		7:5
46edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\27 0.0 0\46 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 3\27 133.3 5\46 130.4 2\19 126.3
Major 1-checkdegree M1chkd 4\27 177.8 7\46 182.6 3\19 189.5
Minor 2-checkdegree m2chkd 6\27 266.7 10\46 260.9 4\19 252.6
Major 2-checkdegree M2chkd 7\27 311.1 12\46 313.0 5\19 315.8
Perfect 3-checkdegree P3chkd 10\27 444.4 17\46 443.5 7\19 442.1
Augmented 3-checkdegree A3chkd 11\27 488.9 19\46 495.7 8\19 505.3
Minor 4-checkdegree m4chkd 13\27 577.8 22\46 573.9 9\19 568.4
Major 4-checkdegree M4chkd 14\27 622.2 24\46 626.1 10\19 631.6
Diminished 5-checkdegree d5chkd 16\27 711.1 27\46 704.3 11\19 694.7
Perfect 5-checkdegree P5chkd 17\27 755.6 29\46 756.5 12\19 757.9
Minor 6-checkdegree m6chkd 20\27 888.9 34\46 887.0 14\19 884.2
Major 6-checkdegree M6chkd 21\27 933.3 36\46 939.1 15\19 947.4
Minor 7-checkdegree m7chkd 23\27 1022.2 39\46 1017.4 16\19 1010.5
Major 7-checkdegree M7chkd 24\27 1066.7 41\46 1069.6 17\19 1073.7
Perfect 8-checkdegree P8chkd 27\27 1200.0 46\46 1200.0 19\19 1200.0

Temperaments

The major temperaments in this area are:

  • Sensi (Parasoft checkertonic)
  • Squares (Parahard checkertonic)

Music

Uncreative Name

Scale tree

Generator ranges:

  • Chroma-positive generator: 750 ¢ (5\8) to 800 ¢ (2\3)
  • Chroma-negative generator: 400 ¢ (1\3) to 450 ¢ (3\8)
Scale tree and tuning spectrum of 3L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\8 750.000 450.000 1:1 1.000 Equalized 3L 5s
27\43 753.488 446.512 6:5 1.200
22\35 754.286 445.714 5:4 1.250
39\62 754.839 445.161 9:7 1.286
17\27 755.556 444.444 4:3 1.333 Supersoft 3L 5s
46\73 756.164 443.836 11:8 1.375
29\46 756.522 443.478 7:5 1.400 Sensi (optimal around here)
41\65 756.923 443.077 10:7 1.429
12\19 757.895 442.105 3:2 1.500 Soft 3L 5s
43\68 758.824 441.176 11:7 1.571 Clyde
31\49 759.184 440.816 8:5 1.600
50\79 759.494 440.506 13:8 1.625 Golden sentry (759.4078 ¢)
19\30 760.000 440.000 5:3 1.667 Semisoft 3L 5s
45\71 760.563 439.437 12:7 1.714
26\41 760.976 439.024 7:4 1.750
33\52 761.538 438.462 9:5 1.800
7\11 763.636 436.364 2:1 2.000 Basic 3L 5s
Scales with tunings softer than this are proper
30\47 765.957 434.043 9:4 2.250
23\36 766.667 433.333 7:3 2.333
39\61 767.213 432.787 12:5 2.400
16\25 768.000 432.000 5:2 2.500 Semihard 3L 5s
41\64 768.750 431.250 13:5 2.600 Unnamed golden tuning (768.8815 ¢)
25\39 769.231 430.769 8:3 2.667
34\53 769.811 430.189 11:4 2.750 Hamity
9\14 771.429 428.571 3:1 3.000 Hard 3L 5s
29\45 773.333 426.667 10:3 3.333
20\31 774.194 425.806 7:2 3.500 Squares (optimal around here)
31\48 775.000 425.000 11:3 3.667
11\17 776.471 423.529 4:1 4.000 Superhard 3L 5s
24\37 778.378 421.622 9:2 4.500
13\20 780.000 420.000 5:1 5.000
15\23 782.609 417.391 6:1 6.000 Roman↓, hocus↓
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 5s
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