3L 5s: Difference between revisions

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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]]

Revision as of 21:01, 8 October 2023

↖ 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, 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.

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.

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 ¢

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

Degree Size in 11edo (basic) Size in 14edo (hard) Size in 19edo (soft) 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:

Degree Size in 19edo (soft) Size in 27edo (supersoft) Size in 46edo Note name on J 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

Checkertonic modes can be named by prefixing anti- to their counterpart modes in the MOS sister oneirotonic.

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

The modes on the white keys JKLMNOPQJ are:

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

Temperaments

The major temperaments in this area are:

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

Scale tree

Generator ranges:

  • Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
  • Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)
Generator Cents L s L/s Comments
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
(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
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