3L 5s

↖ 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.

MOS data is deprecated. Please use the following templates individually: MOS intervals, MOS genchain, and MOS mode degrees

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
7|0	1	LsLssLss
6|1	6	LssLsLss
5|2	3	LssLssLs
4|3	8	sLsLssLs
3|4	5	sLssLsLs
2|5	2	sLssLssL
1|6	7	ssLsLssL
0|7	4	ssLssLsL

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

• The Nachtlandian Somersault (19edo)

Scale tree

Generator ranges:

• Chroma-positive generator: 750 ¢ (5\8) to 800 ¢ (2\3)
• Chroma-negative generator: 400 ¢ (1\3) to 450 ¢ (3\8)