7L 2s

7L 2s, named armotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 2 small steps, repeating every octave. Generators that produce this scale range from 666.7 ¢ to 685.7 ¢, or from 514.3 ¢ to 533.3 ¢. Scales of this form are strongly associated with Armodue theory, as applied to septimal mavila and Hornbostel temperaments.

↖ 6L 1s	↑ 7L 1s	8L 1s ↗
← 6L 2s	7L 2s	8L 2s →
↙ 6L 3s	↓ 7L 3s	8L 3s ↘

Scale structure
Step pattern	LLLLsLLLs sLLLsLLLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	5\9 to 4\7 (666.7 ¢ to 685.7 ¢)
Dark	3\7 to 4\9 (514.3 ¢ to 533.3 ¢)
TAMNAMS information
Name	armotonic
Prefix	arm-
Abbrev.	arm
Related MOS scales
Parent	2L 5s
Sister	2L 7s
Daughters	9L 7s, 7L 9s
Neutralized	5L 4s
2-Flought	16L 2s, 7L 11s
Equal tunings
Equalized (L:s = 1:1)	5\9 (666.7 ¢)
Supersoft (L:s = 4:3)	19\34 (670.6 ¢)
Soft (L:s = 3:2)	14\25 (672.0 ¢)
Semisoft (L:s = 5:3)	23\41 (673.2 ¢)
Basic (L:s = 2:1)	9\16 (675.0 ¢)
Semihard (L:s = 5:2)	22\39 (676.9 ¢)
Hard (L:s = 3:1)	13\23 (678.3 ¢)
Superhard (L:s = 4:1)	17\30 (680.0 ¢)
Collapsed (L:s = 1:0)	4\7 (685.7 ¢)
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Name

The TAMNAMS name for this pattern is armotonic, in reference to Armodue theory. Superdiatonic is also in use.

Intervals

This article assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.

Intervals of 7L 2s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-armstep	Perfect 0-armstep	P0arms	0	0.0 ¢
1-armstep	Minor 1-armstep	m1arms	s	0.0 ¢ to 133.3 ¢
	Major 1-armstep	M1arms	L	133.3 ¢ to 171.4 ¢
2-armstep	Minor 2-armstep	m2arms	L + s	171.4 ¢ to 266.7 ¢
	Major 2-armstep	M2arms	2L	266.7 ¢ to 342.9 ¢
3-armstep	Minor 3-armstep	m3arms	2L + s	342.9 ¢ to 400.0 ¢
	Major 3-armstep	M3arms	3L	400.0 ¢ to 514.3 ¢
4-armstep	Perfect 4-armstep	P4arms	3L + s	514.3 ¢ to 533.3 ¢
	Augmented 4-armstep	A4arms	4L	533.3 ¢ to 685.7 ¢
5-armstep	Diminished 5-armstep	d5arms	3L + 2s	514.3 ¢ to 666.7 ¢
	Perfect 5-armstep	P5arms	4L + s	666.7 ¢ to 685.7 ¢
6-armstep	Minor 6-armstep	m6arms	4L + 2s	685.7 ¢ to 800.0 ¢
	Major 6-armstep	M6arms	5L + s	800.0 ¢ to 857.1 ¢
7-armstep	Minor 7-armstep	m7arms	5L + 2s	857.1 ¢ to 933.3 ¢
	Major 7-armstep	M7arms	6L + s	933.3 ¢ to 1028.6 ¢
8-armstep	Minor 8-armstep	m8arms	6L + 2s	1028.6 ¢ to 1066.7 ¢
	Major 8-armstep	M8arms	7L + s	1066.7 ¢ to 1200.0 ¢
9-armstep	Perfect 9-armstep	P9arms	7L + 2s	1200.0 ¢

Note names

7L 2s, when viewed under Armodue theory, can be notated using Armodue notation.

Theory

Temperament interpretations

Mavila is an important harmonic entropy minimum here, insofar as 678¢ can be considered a fifth. Other temperaments include septimal mavila and Hornbostel.

Modes

Modes of 7L 2s

UDP	Cyclic order	Step pattern
8|0	1	LLLLsLLLs
7|1	6	LLLsLLLLs
6|2	2	LLLsLLLsL
5|3	7	LLsLLLLsL
4|4	3	LLsLLLsLL
3|5	8	LsLLLLsLL
2|6	4	LsLLLsLLL
1|7	9	sLLLLsLLL
0|8	5	sLLLsLLLL

Scale tree

Todo: cleanup Clean up scale tree

Optional types of 'JI Blown Fifth' Generators: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119 & 250/169.

Generator ranges:

• Chroma-positive generator: 666.6667 cents (5\9) to 685.7143 cents (4\7)
• Chroma-negative generator: 514.2857 cents (3\7) to 533.3333 cents (4\9)

Generator			Generator size (cents)	Pentachord steps	Comments
4\7			685.714	1 1 1 0
53\93			683.871	13 13 13 1
	102\179		683.798	25 25 25 2	Approximately 0.03 cents away from 95/64
49\86			683.721	12 12 12 1
	94\165		683.636	23 23 23 2
45\79			683.544	11 11 11 1
	86\151		683.444	21 21 21 2
41\72			683.333	10 10 10 1
	78\137		683.212	19 19 19 2
37\65			683.077	9 9 9 1
	70\123		682.927	17 17 17 2
33\58			682.758	8 8 8 1	2 generators equal 11/10, 6 equal 4/3, creating a hybrid Mavila/Porcupine scale with three perfect 5ths as well as the flat ones.
	62\109		682.569	15 15 15 2
29\51			682.353	7 7 7 1
	54\95		682.105	13 13 13 2
25\44			681.818	6 6 6 1
	46\81		681.4815	11 11 11 2
21\37			681.081	5 5 5 1
	59\104		680.769	14 14 14 3
	38\67		680.597	9 9 9 2
	55\97		680.412	13 13 13 3
17\30			680	4 4 4 1	L/s = 4
	47\83		679.518	11 11 11 3
	30\53		679.245	7 7 7 2
	43\76		678.947	10 10 10 3
	56\99		678.788	13 13 13 4
	69\122		678.6885	16 16 16 5
	82\145		678.621	19 19 19 6
	95\168		678.571	22 22 22 7
			678.569	π π π 1	L/s = π
	108\191		678.534	25 25 25 8
	121\214		678.505	28 28 28 9	28;9 Superdiatonic 1/28-tone (a slight exceeded representation of the ratio 262144/177147, the Pythagorean wolf Fifth)
	134\237		678.481	31 31 31 10	HORNBOSTEL TEMPERAMENT (1/31-tone; Optimum high size of Hornbostel '6th')
13\23			678.261	3 3 3 1	HORNBOSTEL TEMPERAMENT (Armodue 1/3-tone)
	126\223		678.027	29 29 29 10	HORNBOSTEL TEMPERAMENT (Armodue 1/29-tone)
	113\200		678	26 26 26 9	HORNBOSTEL (& OGOLEVETS) TEMPERAMENT (Armodue 1/26-tone; Best equillibrium between 6/5, Phi (833.1 Cent) and Square root of Pi (990.9 Cent), the notes '3', '7' & '8')
	100\177		677.966	23 23 23 8
	87\154		677.922	20 20 20 7
	74\131		677.863	17 17 17 6	Armodue-Hornbostel 1/17-tone (the Golden Tone System of Thorvald Kornerup and a temperament of the Alexei Ogolevets's list of temperaments)
	61\108		677.778	14 14 14 5	Armodue-Hornbostel 1/14-tone
		109\193	677.720	25 25 25 9	Armodue-Hornbostel 1/25-tone
	48\85		677.647	11 11 11 4	Armodue-Hornbostel 1/11-tone (Optimum accuracy of Phi interval, the note '7')
			677.562	e e e 1	L/s = e
	35\62		677.419	8 8 8 3	Armodue-Hornbostel 1/8-tone
		92\163	677.301	21 21 21 8	21;8 Superdiatonic 1/21-tone
			677.28	φ+1 φ+1 φ+1 1	Split φ superdiatonic relation (34;13 - 55;21 - 89;34 - 144;55 - 233;89 - 377;144 - 610;233..)
	57\101		677.228	13 13 13 5	13;5 Superdiatonic 1/13-tone
22\39			676.923	5 5 5 2	Armodue-Hornbostel 1/5-tone (Optimum low size of Hornbostel '6th')
	75\133		676.692	17 17 17 7	17;7 Superdiatonic 1/17-tone (Note the very accuracy of the step 75 with the ratio 34/23 with an error of +0.011 Cents)
	53\94		676.596	12 12 12 5
	31\55		676.364	7 7 7 3	7;3 Superdiatonic 1/7-tone
	40\71		676.056	9 9 9 4	9;4 Superdiatonic 1/9-tone
	49\87		675.862	11 11 11 5	11;5 Superdiatonic 1/11-tone
	58\103		675.728	13 13 13 6	13;6 Superdiatonic 1/13-tone
9\16			675	2 2 2 1	[BOUNDARY OF PROPRIETY: smaller generators are strictly proper]ARMODUE ESADECAFONIA (or Goldsmith Temperament)
	59\105		674.286	13 13 13 7	Armodue-Mavila 1/13-tone
	50\89		674.157	11 11 11 6	Armodue-Mavila 1/11-tone
	41\73		673.973	9 9 9 5	Armodue-Mavila 1/9-tone (with an approximation of the Perfect Fifth + 1/5 Pyth.Comma [706.65 Cents]: 43\73 is 706.85 Cents)
	32\57		673.684	7 7 7 4	Armodue-Mavila 1/7-tone (the 'Commatic' version of Armodue, because its high accuracy of the 7/4 interval, the note '8')
			673.577	√3 √3 √3 1
	55\98		673.469	12 12 12 7
	78\139		673.381	17 17 17 10	Armodue-Mavila 1/17-tone
	101\180		673.333	22 22 22 13
23\41			673.171	5 5 5 3	5;3 Golden Armodue-Mavila 1/5-tone
	60\107		672.897	13 13 13 8	13;8 Golden Mavila 1/13-tone
			672.85	φ φ φ 1	GOLDEN MAVILA (L/s = φ)
		97\173	672.832	21 21 21 13	21;13 Golden Mavila 1/21-tone (Phi is the step 120\173)
	37\66		672.727	8 8 8 5	8;5 Golden Mavila 1/8-tone
	51\91		672.527	11 11 11 7	11;7 Superdiatonic 1/11-tone
			672.523	π π π 2
		116\207	672.464	25 25 25 16	25;16 Superdiatonic 1/25-tone
	65\116		672.414	14 14 14 9	14;9 Superdiatonic 1/14-tone
	79\141		672.340	17 17 17 11	17;11 Superdiatonic 1/17-tone
	93\166		672.289	20 20 20 13
	107\191		672.251	23 23 23 15
	121\216		672.222	26 26 26 17	26;17 Superdiatonic 1/26-tone
	135\241		672.199	29 29 29 19	29;19 Superdiatonic 1/29-tone
14\25			672	3 3 3 2	3;2 Golden Armodue-Mavila 1/3-tone
	145\259		671.815	31 31 31 21	31;21 Superdiatonic 1/31-tone
	131\234		671.795	28 28 28 19	28;19 Superdiatonic 1/28-tone
	117\209		671.770	25 25 25 17
	103\184		671.739	22 22 22 15
	89\159		671.698	19 19 19 13
	75\134		671.642	16 16 16 11
	61\109		671.560	13 13 13 9
	47\84		671.429	10 10 10 7
		80\143	671.329	17 17 17 12
	33\59		671.186	7 7 7 5
	52\93		670.968	11 11 11 8
19\34			670.588	4 4 4 3
	43\77		670.13	9 9 9 7
24\43			669.767	5 5 5 4
	53\95		669.474	11 11 11 9
29\52			669.231	6 6 6 5
	63\113		669.0265	13 13 13 11
34\61			668.8525	7 7 7 6
	73\131		668.702	15 15 15 13
39\70			668.571	8 8 8 7
	83\149		668.456	17 17 17 15
44\79			668.354	9 9 9 8
	93\167		668.2365	19 19 19 17
49\88			668.182	10 10 10 9
	103\185		668.108	21 21 21 9
54\97			668.041	11 11 11 10
	113\203		667.98	23 23 23 21
59\106			667.925	12 12 12 11
	123\221		667.873	25 25 25 23
64\115			667.826	13 13 13 12
5\9			666.667	1 1 1 1