3L 5s: Difference between revisions

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| Pattern = LsLssLss
| Pattern = LsLssLss
}}
}}
{{MOS intro|Other Names=anti-oneirotonic}}


'''3L 5s''' refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 1\3 (one degree of [[3edo]] = 400¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 3edo, s becomes so small it disappears (and all that remains are the three equal L's). The pattern is also named ''antioneirotonic'' because it is the [[oneirotonic]] (5L 3s) MOS pattern with large and small steps switched. In contrast to oneirotonic scales, which often require the usage of completely new chords to have consonant-sounding music, some hard checkertonic scales contain approximations to a perfect fifth, and thus can be used for traditional root-3rd-P5 harmony.
== Name ==
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.


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, is the proper one. [[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, is improper.
== Scale properties ==
{{TAMNAMS use}}


== Standing assumptions ==
=== Intervals ===
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
{{MOS intervals}}
 
The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.
 
== Names ==
The [[TAMNAMS]] name for 3L 5s is '''checkertonic'''.
 
== Intervals ==
Note: In TAMNAMS, a k-step interval class in checkertonic may be called a "k-step", "k-mosstep", or "k-checkstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
 
== Tuning ranges ==
=== Simple tunings ===
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! Size in [[11edo]] (basic)
! Size in [[14edo]] (hard)
! Size in [[19edo]] (soft)
! class="unsortable"| Note name on J
! #Gens up
|-
| min. chk2nd
| 1\11, 109.1
| 1\14, 85.7
| 2\19, 126.3
| K
| +3
|-
| 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]].
=== Generator chain ===
{{MOS genchain}}


== Modes ==
=== Modes ===
Checkertonic modes can be named by prefixing ''anti-'' to their counterpart modes in the MOS sister [[oneirotonic]].
{{MOS mode degrees}}


# Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss
==== Proposed mode names ====
# Anti-Hlanithian (lə-NITH-iən): LssLsLss
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.
# Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs
{{MOS modes
# Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
| Table Headers=
# Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
Anti-modes of 5L 3s $
# Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
Grand chess names<sup>[proposed]</sup>
# Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
| Table Entries=
# Anti-Dylathian (də-LA(H)TH-iən): ssLssLsL
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 modes on the white keys JKLMNOPQJ are:
* J Anti-Ultharian, Rook
* J Anti-Ultharian
* K Anti-Hlanithian, Queen
* K Anti-Hlanithian
* L Anti-Illarnekian, Knight
* L Anti-Illarnekian
* M Anti-Mnarian, Cardinal
* M Anti-Mnarian
* N Anti-Sarnathian, King
* N Anti-Sarnathian
* O Anti-Celephaïsian, Bishop
* O Anti-Celephaïsian
* P Anti-Kadathian, Marshall
* P Anti-Kadathian
* Q Anti-Dylathian, Pawn
* Q Anti-Dylathian


{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Scale degrees (on J, {{nowrap|sLssLsLs {{=}} JKLMNOPQ}})
|-
|-
|+ Table of modes (based on J, from brightest to darkest)
! [[UDP]]
|-
! Anti-modes of 5L 3s
! Mode
! Chess-based names
! Step pattern
! 1
! 1
! 2
! 2
Line 309: Line 77:
! (9)
! (9)
|-
|-
| 7{{pipe}}0
| Anti-Sarnathian
| Anti-Sarnathian
| King
| LsLssLss
| J
| J
| K&
| K&amp;
| L
| L
| M&
| M&amp;
| N&
| N&amp;
| O
| O
| P&
| P&amp;
| Q
| Q
| (J)
| (J)
|-
|-
| 6{{pipe}}1
| Anti-Hlanithian
| Anti-Hlanithian
| Queen
| LssLsLss
| J
| J
| K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P&
| P&amp;
| Q
| Q
| (J)
| (J)
|-
|-
| 5{{pipe}}2
| Anti-Kadathian
| Anti-Kadathian
| Marshall
| LssLssLs
| J
| J
| K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P
| P
Line 342: Line 119:
| (J)
| (J)
|-
|-
| 4{{pipe}}3
| Anti-Mnarian
| Anti-Mnarian
| Cardinal
| sLsLssLs
| J
| J
| K
| K
| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P
| P
Line 353: Line 133:
| (J)
| (J)
|-
|-
| 3{{pipe}}4
| Anti-Ultharian
| Anti-Ultharian
| Rook
| sLssLsLs
| J
| J
| K
| K
Line 364: Line 147:
| (J)
| (J)
|-
|-
| 2{{pipe}}5
| Anti-Celephaïsian
| Anti-Celephaïsian
| Bishop
| sLssLssL
| J
| J
| K
| K
Line 375: Line 161:
| (J)
| (J)
|-
|-
| 1{{pipe}}6
| Anti-Illarnekian
| Anti-Illarnekian
| Knight
| ssLsLssL
| J
| J
| K
| K
Line 386: Line 175:
| (J)
| (J)
|-
|-
| 0{{pipe}}7
| Anti-Dylathian
| Anti-Dylathian
| Pawn
| ssLssLsL
| J
| J
| K
| K
Line 397: Line 189:
| (J)
| (J)
|}
|}
== 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 ==
== Temperaments ==
Line 402: Line 216:
* [[Sensi]] (Parasoft checkertonic)
* [[Sensi]] (Parasoft checkertonic)
* [[Squares]] (Parahard checkertonic)
* [[Squares]] (Parahard checkertonic)
== Music ==
; [[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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