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{{MOS intro}}
{{MOS intro}}
In contrast to oneirotonic (5L 3s) scales, 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.


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.
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.


== Standing assumptions ==
==Intervals==
:''This article assumes [[TAMNAMS]] for naming step ratios, intevrvals, and scale degrees.''
 
Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.
{{MOS intervals}}
 
== Notation ==
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
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), &/@ = up/down by chroma.
The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.


== Names ==
== Theory ==
The [[TAMNAMS]] name for 3L 5s is '''checkertonic'''.
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.


== Intervals ==
=== Low harmonic entropy scales ===
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.
There are two significant harmonic entropy minima with this MOS pattern:


== Tuning ranges ==
* [[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.
=== Simple tunings ===
* [[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===
{| class="wikitable right-2 right-3 right-4 sortable "
{| class="wikitable right-2 right-3 right-4 sortable "
|-
|-
! class="unsortable"|Degree
! class="unsortable" |Degree
! Size in [[11edo]] (basic)
! Size in [[11edo]] (basic)
! Size in [[14edo]] (hard)
!Size in [[14edo]] (hard)
! Size in [[19edo]] (soft)
!Size in [[19edo]] (soft)
! class="unsortable"| Note name on J
! class="unsortable" |Note name on J
! #Gens up
!#Gens up
|-
|-
| min. chk2nd
|min. chk2nd
| 1\11, 109.1
|1\11, 109.1
| 1\14, 85.7
| 1\14, 85.7
| 2\19, 126.3
|2\19, 126.3
| K
|K
| +3
| +3
|-
|-
| maj. chk2nd
|maj. chk2nd
| 2\11, 218.2
|2\11, 218.2
| 3\14, 257.1
|3\14, 257.1
| 3\19, 189.5
|3\19, 189.5
| K&
|K&
| -5
| -5
|-
|-
| min. chk3rd
|min. chk3rd
| 2\11, 218.2
|2\11, 218.2
| 2\14, 171.4
|2\14, 171.4
| 4\19, 252.6
|4\19, 252.6
| L@
|L@
| +6
| +6
|-
|-
| maj. chk3rd
| maj. chk3rd
| 3\11, 327.3
|3\11, 327.3
| 4\14, 342.9
| 4\14, 342.9
| 5\19, 315.8
| 5\19, 315.8
| L
|L
| -2
| -2
|-
|-
| perf. chk4th
|perf. chk4th
| 4\11, 436.4
|4\11, 436.4
| 5\14, 428.6
|5\14, 428.6
| 7\19, 442.1
| 7\19, 442.1
| M
| M
| +1
| +1
|-
|-
| aug. chk4th
|aug. chk4th
| 5\11, 545.5
|5\11, 545.5
| 7\14, 600.0
|7\14, 600.0
| 8\19, 505.3
|8\19, 505.3
| M&
|M&
| -7
| -7
|-
|-
| min. chk5th
|min. chk5th
| 5\11, 545.5
|5\11, 545.5
| 6\14, 514.3
|6\14, 514.3
| 9\19, 568.4
|9\19, 568.4
| N
|N
| +4
| +4
|-
|-
| maj. chk5th
|maj. chk5th
| 6\11, 656.6
|6\11, 656.6
| 8\14, 685.7
|8\14, 685.7
| 10\19, 631.6
|10\19, 631.6
| N&
|N&
| -4
| -4
|-
|-
| dim. chk6th
|dim. chk6th
| 6\11, 656.6
|6\11, 656.6
| 7\14, 600.0
| 7\14, 600.0
| 11\19, 694.7
|11\19, 694.7
| O@
|O@
| +7
| +7
|-
|-
| perf. chk6th
|perf. chk6th
| 7\11, 763.6
|7\11, 763.6
| 8\14, 771.4
|8\14, 771.4
| 12\19, 757.9
|12\19, 757.9
| O
| O
| -1
| -1
|-
|-
| min. chk7th
|min. chk7th
| 8\11, 872.7
|8\11, 872.7
| 10\14, 857.1
|10\14, 857.1
| 14\19, 884.2
|14\19, 884.2
| P
| P
| +2
| +2
|-
|-
| maj. chk7th
|maj. chk7th
| 9\11, 981.8
| 9\11, 981.8
| 12\14, 1028.6
|12\14, 1028.6
| 15\19, 947.4
|15\19, 947.4
| P&
| P&
| -6
| -6
|-
|-
| min. chk8th
|min. chk8th
| 9\11, 981.8
|9\11, 981.8
| 11\14, 942.9
|11\14, 942.9
| 16\19, 1010.5
|16\19, 1010.5
| Q@
|Q@
| +5
| +5
|-
|-
| maj. chk8th
|maj. chk8th
| 10\11, 1090.9
|10\11, 1090.9
| 13\14, 1114.3
|13\14, 1114.3
| 17\19, 1073.7
|17\19, 1073.7
| Q
| Q
| -3
| -3
|}
|}


=== Parasoft ===
===Parasoft===
Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).
Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).


Line 142: Line 151:
{| class="wikitable right-2 right-3 right-4 sortable "
{| class="wikitable right-2 right-3 right-4 sortable "
|-
|-
! class="unsortable"|Degree
! class="unsortable" |Degree
! Size in [[19edo]] (soft)
!Size in [[19edo]] (soft)
! Size in [[27edo]] (supersoft)
!Size in [[27edo]] (supersoft)
! Size in [[46edo]]
!Size in [[46edo]]
! class="unsortable"| Note name on J
! class="unsortable" | Note name on J
! class="unsortable"| Approximate ratios
! class="unsortable" |Approximate ratios
! #Gens up
!#Gens up
|-
|-
| unison
|unison
| 0\19, 0.00
|0\19, 0.00
| 0\27, 0.00
|0\27, 0.00
| 0\46, 0.00  
|0\46, 0.00
| J
|J
| 1/1
|1/1
| 0
|0
|-
|-
| min. chk2nd
|min. chk2nd
| 2\19, 126.3
|2\19, 126.3
| 3\27, 133.3
|3\27, 133.3
| 5\46, 130.4
|5\46, 130.4
| K
|K
| 14/13
|14/13
| +3
| +3
|-
|-
| maj. chk2nd
| maj. chk2nd
| 3\19, 189.5
|3\19, 189.5
| 4\27, 177.8
|4\27, 177.8
| 7\46, 182.6
| 7\46, 182.6
| K&
|K&
| 10/9
|10/9
| -5
| -5
|-
|-
| min. chk3rd
|min. chk3rd
| 4\19, 252.6
|4\19, 252.6
| 6\27, 266.7
|6\27, 266.7
| 10\46, 260.9
| 10\46, 260.9
| L@
|L@
| 7/6
|7/6
| +6
| +6
|-
|-
| maj. chk3rd
|maj. chk3rd
| 5\19, 315.8
|5\19, 315.8
| 7\27, 311.1
|7\27, 311.1
| 12\46, 313.0
| 12\46, 313.0
| L
|L
| 6/5
|6/5
| -2
| -2
|-
|-
| perf. chk4th
|perf. chk4th
| 7\19, 442.1
|7\19, 442.1
| 10\27, 444.4
|10\27, 444.4
| 17\46, 443.5
|17\46, 443.5
| M
| M
| 9/7, 13/10
|9/7, 13/10
| +1
| +1
|-
|-
| aug. chk4th
|aug. chk4th
| 8\19, 505.3
|8\19, 505.3
| 11\27, 488.9
|11\27, 488.9
| 19\46, 495.7
|19\46, 495.7
| M&
|M&
| 4/3
|4/3
| -7
| -7
|-
|-
| min. chk5th
|min. chk5th
| 9\19, 568.4
|9\19, 568.4
| 13\27, 577.8
|13\27, 577.8
| 22\46, 573.9
| 22\46, 573.9
| N
|N
| 7/5, 18/13
|7/5, 18/13
| +4
| +4
|-
|-
| maj. chk5th
|maj. chk5th
| 10\19, 631.6
|10\19, 631.6
| 14\27, 622.2
| 14\27, 622.2
| 24\46, 626.1  
|24\46, 626.1
| N&
|N&
| 10/7, 13/9
|10/7, 13/9
| -4
| -4
|-
|-
| dim. chk6th
|dim. chk6th
| 11\19, 694.7
|11\19, 694.7
| 16\27, 711.1
|16\27, 711.1
| 27\46, 704.3
|27\46, 704.3
| O@
| O@
| 3/2
|3/2
| +7
| +7
|-
|-
| perf. chk6th
|perf. chk6th
| 12\19, 757.9
|12\19, 757.9
| 17\27, 755.6
| 17\27, 755.6
| 20\46, 756.5
|20\46, 756.5
| O
|O
| 14/9, 20/13
| 14/9, 20/13
| -1
| -1
|-
|-
| min. chk7th
|min. chk7th
| 14\19, 884.2
|14\19, 884.2
| 20\27, 888.9
| 20\27, 888.9
| 34\46, 887.0
|34\46, 887.0
| P
|P
| 5/3
|5/3
| +2
| +2
|-
|-
| maj. chk7th
|maj. chk7th
| 15\19, 947.4
|15\19, 947.4
| 21\27, 933.3
|21\27, 933.3
| 36\46, 939.1
|36\46, 939.1
| P&
|P&
| 12/7
|12/7
| -6
| -6
|-
|-
| min. chk8th
|min. chk8th
| 16\19, 1010.5
|16\19, 1010.5
| 23\27, 1022.2
|23\27, 1022.2
| 39\46, 1017.4
|39\46, 1017.4
| Q@
|Q@
| 9/5
|9/5
| +5
| +5
|-
|-
| maj. chk8th
|maj. chk8th
| 17\19, 1073.7
| 17\19, 1073.7
| 24\27, 1066.7
|24\27, 1066.7
| 41\46, 1069.6
|41\46, 1069.6
| Q
|Q
| 13/7
|13/7
| -3
| -3
|}
|}
Line 273: Line 282:
Tunings in this region have a regular temperament interpretation called [[sensi]].
Tunings in this region have a regular temperament interpretation called [[sensi]].


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


# Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss  
#Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss
# Anti-Hlanithian (lə-NITH-iən): LssLsLss  
#Anti-Hlanithian (lə-NITH-iən): LssLsLss
# Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs  
#Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs
# Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
# Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
# Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
#Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
# Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
#Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
# Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
#Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
# Anti-Dylathian (də-LA(H)TH-iən): ssLssLsL
#Anti-Dylathian (də-LA(H)TH-iən): ssLssLsL


The modes on the white keys JKLMNOPQJ are:
The modes on the white keys JKLMNOPQJ are:
* J Anti-Ultharian
*J Anti-Ultharian
* K Anti-Hlanithian
* K Anti-Hlanithian
* L Anti-Illarnekian
* L Anti-Illarnekian
* M Anti-Mnarian
*M Anti-Mnarian
* N Anti-Sarnathian
*N Anti-Sarnathian
* O Anti-Celephaïsian
*O Anti-Celephaïsian
* P Anti-Kadathian
*P Anti-Kadathian
* Q Anti-Dylathian
*Q Anti-Dylathian


{| class="wikitable"
{| class="wikitable"
|-
|-
|+ Table of modes (based on J, from brightest to darkest)
|+Table of modes (based on J, from brightest to darkest)
|-
|-
! Mode
!Mode
! 1
! 1
! 2
!2
! 3
!3
! 4
! 4
! 5
!5
! 6
!6
! 7
!7
! 8
!8
! (9)
!(9)
|-
|-
| Anti-Sarnathian
|Anti-Sarnathian
| J
| J
| K&
|K&
| L
|L
| M&
|M&
| N&
|N&
| O
|O
| P&
| P&
| Q
|Q
| (J)
|(J)
|-
|-
| Anti-Hlanithian
|Anti-Hlanithian
| J
|J
| K&
|K&
| L
| L
| M
| M
| N&
| N&
| O
|O
| P&
|P&
| Q
| Q
| (J)
|(J)
|-
|-
| Anti-Kadathian
|Anti-Kadathian
| J
|J
| K&
| K&
| L
|L
| M
|M
| N&
|N&
| O
|O
| P
|P
| Q
|Q
| (J)
|(J)
|-
|-
| Anti-Mnarian
|Anti-Mnarian
| J
|J
| K
|K
| L
|L
| M
|M
| N&
|N&
| O
|O
| P
|P
| Q
|Q
| (J)
|(J)
|-
|-
| Anti-Ultharian
| Anti-Ultharian
| J
|J
| K
|K
| L
|L
| M
|M
| N
| N
| O
|O
| P
|P
| Q
|Q
| (J)
|(J)
|-
|-
| Anti-Celephaïsian
|Anti-Celephaïsian
| J
|J
| K
|K
| L
|L
| M
|M
| N
|N
| O
| O
| P
|P
| Q@
|Q@
| (J)
|(J)
|-
|-
| Anti-Illarnekian
|Anti-Illarnekian
| J
|J
| K
|K
| L@
|L@
| M
|M
| N
|N
| O
|O
| P
| P
| Q@
|Q@
| (J)
|(J)
|-
|-
| Anti-Dylathian
|Anti-Dylathian
| J
|J
| K
|K
| L@
|L@
| M
|M
| N
|N
| O@
|O@
| P
|P
| Q@
|Q@
| (J)
|(J)
|}
|}


== Temperaments ==
==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)


== Scale tree ==
==Scale tree==
Generator ranges:
Generator ranges:
* Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
*Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
* Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)
* Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)


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


[[Category:8-tone scales]]
[[Category:8-tone scales]]
[[Category:checkertonic]]
[[Category:checkertonic]]
Retrieved from "https://en.xen.wiki/w/3L_5s"