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| Pattern = LsLssLss
| Pattern = LsLssLss
}}
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{{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 checkertonic scales contain approximations to a perfect fifth (usually as a dim. chk6th or maj. chk5th), 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]]
Retrieved from "https://en.xen.wiki/w/3L_5s"