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 refers to the structure of octave-equivalent MOS scales with generators ranging from 1\3 (one degrees 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.

There are two significant harmonic entropy minima with this MOS pattern. Sensi, in which the generator is 9/7, two of them make 5/3, and seven of them make 6/1, is the proper one. Squares, in which the generator is also 9/7, but two of them make 18/11 and four of them make 8/3, is improper.

Standing assumptions

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.

Names

The TAMNAMS name is sensoid (named after the regular temperament sensi).

Tuning ranges

Simple tunings

Degree	Size in 11edo (basic)	Size in 14edo (hard)	Size in 19edo (soft)	Note name on J	Approximate ratios	#Gens up
min. sen2nd	1\11, 109.1	1\14, 85.7	2\19, 126.3	K		+3
maj. sen2nd	2\11, 218.2	3\14, 257.1	3\19, 189.5	K&		-5
min. sen3rd	2\11, 218.2	2\14, 171.4	4\19, 252.6	L@		+6
maj. sen3rd	3\11, 327.3	4\14, 342.9	5\19, 316.8	L		-2
perf. sen4th	4\11, 436.4	5\14, 428.6	7\19, 442.1	M		+1
aug. sen4th	5\11, 545.5	7\14, 600.0	8\19, 505.3	M&		-7
min. sen5th	5\11, 545.5	6\14, 514.3	9\19, 568.4	N		+4
maj. sen5th	6\11, 656.6	8\14, 685.7	10\19, 631.6	N&		-4
dim. sen6th	6\11, 656.6	7\14, 600.0	11\19, 694.7	O@		+7
perf. sen6th	7\11, 763.6	8\14, 771.4	12\19, 757.9	O		-1
min. sen7th	8\11, 872.7			P		+2
maj. sen7th	9\11, 981.8			P&		-6
min. sen8th	9\11, 981.8			Q		+5
maj. sen8th	10\11, 1090.9			Q&		-3

Parasoft

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

Sortable table of major and minor intervals in parasoft sensoid 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. sen2nd	2\19, 126.3	3\27, 133.3	5\46, 130.4	K	14/13	+3
maj. sen2nd	3\19, 189.5	4\27, 177.8	7\46, 182.6	K&	10/9	-5
min. sen3rd	4\19, 252.6	6\27, 266.7	10\46, 260.9	L@	7/6	+6
maj. sen3rd	5\19, 315.8	7\27, 311.1	12\46, 313.0	L	6/5	-2
perf. sen4th	7\19, 442.1	10\27, 444.4	17\46, 443.5	M	9/7, 13/10	+1
aug. sen4th	8\19, 505.3	11\27, 488.9	19\46, 495.7	M&	4/3	-7
min. sen5th	9\19, 568.4	13\27, 577.8	22\46, 573.9	N	7/5, 18/13	+4
maj. sen5th	10\19, 631.6	14\27, 622.2	24\46, 626.1	N&	10/7, 13/9	-4
dim. sen6th	11\19, 694.7	16\27, 711.1	27\46, 704.3	O@	3/2	+7
perf. sen6th	12\19, 757.9	17\27, 755.6	20\46, 756.5	O	14/9, 20/13	-1
min. sen7th	14\19, 884.2	20\27, 888.9	34\46, 887.0	P	5/3	+2
maj. sen7th	15\19, 947.4	21\27, 933.3	36\46, 939.1	P&	12/7	-6
min. sen8th	16\19, 1010.5	23\27, 1022.2	39\46, 1017.4	Q	9/5	+5
maj. sen8th	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

Sensoid 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 sensoid)
• Squares (Parahard sensoid)

Scale tree

Generator						Cents	Comments
1\3						400
					8\23	417.39
				7\20		420
			6\17			423.53	L/s = 4
					17\48	425
				11\31		425.81
		5\14				428.57	L/s = 3
				14\39		430.77
					23\64	431.25
			9\25			432
	4\11					436.36	Boundary of propriety: generators larger than this are proper
						439.23
			11\30			440
					29\79	440.51	Golden sensoid
				18\49		440.82
						441.185
		7\19				442.11	Optimum rank range (L/s=3/2) sensoid
				17\46		443.48
			10\27			444.44
				13\35		445.71
					16\43	446.51
3\8						450