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¢)
Step size
Large	1\8 to 1\3 (150.0¢ to 400.0¢)
Small	0\3 to 1\8 (0.0¢ to 150.0¢)
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
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.

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 (with relation to root)		Size		Abbrev.
Generic	Specific	L's and s's	Range in cents
0-checkstep (root)	Perfect 0-checkstep	0	0.0¢	P0ms
1-checkstep	Minor 1-checkstep	s	0.0¢ to 150.0¢	m1ms
	Major 1-checkstep	L	150.0¢ to 400.0¢	M1ms
2-checkstep	Minor 2-checkstep	2s	0.0¢ to 300.0¢	m2ms
	Major 2-checkstep	L + s	300.0¢ to 400.0¢	M2ms
3-checkstep	Perfect 3-checkstep	L + 2s	400.0¢ to 450.0¢	P3ms
	Augmented 3-checkstep	2L + s	450.0¢ to 800.0¢	A3ms
4-checkstep	Minor 4-checkstep	L + 3s	400.0¢ to 600.0¢	m4ms
	Major 4-checkstep	2L + 2s	600.0¢ to 800.0¢	M4ms
5-checkstep	Diminished 5-checkstep	L + 4s	400.0¢ to 750.0¢	d5ms
	Perfect 5-checkstep	2L + 3s	750.0¢ to 800.0¢	P5ms
6-checkstep	Minor 6-checkstep	2L + 4s	800.0¢ to 900.0¢	m6ms
	Major 6-checkstep	3L + 3s	900.0¢ to 1200.0¢	M6ms
7-checkstep	Minor 7-checkstep	2L + 5s	800.0¢ to 1050.0¢	m7ms
	Major 7-checkstep	3L + 4s	1050.0¢ to 1200.0¢	M7ms
8-checkstep (octave)	Perfect 8-checkstep	3L + 5s	1200.0¢	P8ms

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