User:FloraC/Sandbox

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 4 10 4 13 12 ]]

Optimal tuning (CTE): ~3/2 = 696.9521

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.0137

5-limit rank-2 temperaments by TE simple badness

Breed's simple badness.

High-badness temperaments

Temperament	Complexity	Error (¢)	Badness (moct)	Mapping	Comma list
Yo	.367	26.0	7.95	[⟨1 0 -1], ⟨0 1 -2]]	10/9
Antitonic	.395	27.0	8.89	[⟨2 3 0], ⟨0 0 1]]	9/8
Father	.443	13.2	4.87	[⟨1 0 4], ⟨0 1 -1]]	16/15
Bug	.602	11.6	5.81	[⟨1 0 0], ⟨0 2 3]]	27/25
Supersharp	1.06	7.94	7.01	[⟨2 0 -5], ⟨0 1 3]]	800/729
Laconic	1.24	6.52	6.74	[⟨1 1 1], ⟨0 3 7]]	2187/2000
Lafayette	1.30	7.00	7.60	[⟨1 1 2], ⟨0 5 3]]	3456/3125
Symbolic	1.31	6.09	6.66	[⟨1 3 4], ⟨0 -5 -6]]	2048/1875
Sixix	1.37	4.57	5.22	[⟨1 3 2], ⟨0 -4 1]]	3125/2916
Uncle	1.40	7.53	8.80	[⟨1 0 12], ⟨0 1 -6]]	4096/3645
Whitewood	1.42	4.20	4.95	[⟨7 11 0], ⟨0 0 1]]	2187/2048
	1.66	4.13	5.72	[⟨1 0 5], ⟨0 3 -5]]	32768/30375

Main sequence

Temperament	Complexity	Error (¢)	Badness (moct)	Mapping	Comma list
Dicot	.521	7.09	3.08	[⟨1 1 2], ⟨0 2 1]]	25/24
Meantone	.711	1.58	.937	[⟨1 0 -4], ⟨0 1 4]]	81/80
Mavila	.795	6.06	4.02	[⟨1 0 7], ⟨0 1 -3]]	135/128
Augmented	.894	2.40	1.79	[⟨3 0 7], ⟨0 1 0]]	128/125
Porcupine	.960	2.68	2.14	[⟨1 2 3], ⟨0 -3 -5]]	250/243
Blackwood	1.02	4.63	3.93	[⟨5 8 0], ⟨0 0 1]]	256/243
Diminished	1.05	3.10	2.73	[⟨4 0 3], ⟨0 1 1]]	648/625
Srutal	1.22	.835	.852	[⟨2 0 11], ⟨0 1 -2]]	2048/2025
Magic	1.40	1.11	1.29	[⟨1 0 2], ⟨0 5 1]]	3125/3072
Hanson	1.55	.274	.353	[⟨1 0 1], ⟨0 6 5]]	15625/15552
Ripple	1.56	2.82	3.66	[⟨1 2 3], ⟨0 -5 -8]]	6561/6250
Negri	1.58	1.69	2.23	[⟨1 2 2], ⟨0 -4 3]]	16875/16384
Tetracot	1.61	.900	1.21	[⟨1 1 1], ⟨0 4 9]]	20000/19683
Superpyth	1.70	2.11	2.99	[⟨1 0 -12], ⟨0 1 9]]	20480/19683
Helmholtz	1.79	.0570	.0851	[⟨1 0 15], ⟨0 1 -8]]	32805/32768
Wesley	1.91	2.75	4.37	[⟨1 4 3], ⟨0 -7 -2]]	78125/73728
Sensipent	1.97	.356	.584	[⟨1 6 8], ⟨0 7 9]]	78732/78125
Stump	2.02	1.88	3.16	[⟨1 0 6], ⟨0 3 -7]]	273375/262144
Passion	2.02	1.57	2.64	[⟨1 2 2], ⟨0 -5 4]]	262144/253125
Doublewide	2.06	2.00	3.43	[⟨2 1 3], ⟨0 4 3]]	390625/373248
Würschmidt	2.29	.262	.499	[⟨1 7 3], ⟨0 8 1]]	393216/390625
Amity	2.29	.140	.268	[⟨1 3 6], ⟨0 -5 -13]]	1600000/1594323
Valentine	2.34	.736	1.44	[⟨1 1 2], ⟨0 9 5]]	1990656/1953125
Immunity	2.40	1.03	2.06	[⟨1 0 -8], ⟨0 2 13]]	1638400/1594323
Shibboleth	2.42	1.24	2.50	[⟨1 4 5], ⟨0 -9 -10]]	1953125/1889568
Compton	2.44	.504	1.02	[⟨12 19 0], ⟨0 0 1]]	531441/524288
Orson	2.44	.215	.438	[⟨1 0 3], ⟨0 7 -3]]	2109375/2097152
Unicorn	2.52	.725	1.52	[⟨1 2 3], ⟨0 -8 -13]]	1594323/1562500
Mynic	2.60	1.10	2.37	[⟨1 9 9], ⟨0 -10 -9]]	10077696/9765625
Ampersand	2.78	.594	1.38	[⟨1 1 3], ⟨0 6 -7]]	34171875/33554432
Fifive	2.91	.643	1.56	[⟨2 2 3], ⟨0 5 7]]	9765625/9565938
Misty	2.99	.308	.767	[⟨3 0 26], ⟨0 1 -4]]	67108864/66430125
Gravity	2.99	.269	.669	[⟨1 5 12], ⟨0 -6 -17]]	129140163/128000000
Rodan	3.10	.433	1.12	[⟨1 1 -1], ⟨0 3 17]]	[20 -17 3⟩
	3.11	.757	1.96	[⟨1 0 -23], ⟨0 1 16]]	[-23 16 -1⟩
Mabila	3.32	.488	1.35	[⟨1 6 1], ⟨0 -10 3]]	[28 -3 -10⟩
Parakleismic	3.47	.0798	.231	[⟨1 5 6], ⟨0 -13 -14]]	[8 14 -13⟩
Quartonic	3.48	.214	.621	[⟨1 2 3], ⟨0 -11 -18]]	[3 -18 11⟩
Escapade	3.60	.138	.414	[⟨1 2 2], ⟨0 -9 7]]	[32 -7 -9⟩
Ditonic	3.68	.258	.792	[⟨1 6 3], ⟨0 -13 -2]]	[-27 -2 13⟩
Vishnuzmic	3.71	.0471	.145	[⟨2 4 5], ⟨0 -7 -3]]	[23 6 -14⟩
Vulture	3.81	.0576	.183	[⟨1 0 -6], ⟨0 4 21]]	[24 -21 4⟩
Trisedodge	3.87	.336	1.08	[⟨5 1 7], ⟨0 3 2]]	[19 10 -15⟩
	4.00	.157	.524	[⟨1 4 -1], ⟨0 -8 11]]	[-36 11 8⟩

Tunings to be reviewed

5-limit Prime-Optimized Tunings

Weight-skew\Order	Chebyshevian	Euclidean	Manhattan
Tenney	CTC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CTE ~3/2 = 697.2143¢	CTOP ~3/2 = 698.0197¢
Tenney-Weil	CTWC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CTWE ~3/2 = 696.6512¢	CTWOP ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)
Equilateral	CEC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CEE ~3/2 = 696.8947¢ Eigenmonzo basis: 2.1875 (4/17-comma tuning)	CEOP ~3/2 = 697.6537¢ Eigenmonzo basis: 2.15 (1/5-comma tuning)
Equilateral-Weil	CEWC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CEWE ~3/2 = 696.4534¢ Eigenmonzo basis: 2.48828125/3 (11/43-comma tuning)	CEWOP ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)
Benedetti	CBC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CBE ~3/2 = 697.3738¢ Eigenmonzo basis: 2.[0 25 36⟩ (36/169-comma tuning)	CBOP ~3/2 = 698.1598¢ Eigenmonzo basis: 2.30375 (3/17-comma tuning)
Benedetti-Weil	CBWC ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)	CBWE ~3/2 = 696.7868¢ Eigenmonzo basis: 2.[0 5 31⟩ (31/129-comma tuning)	CBWOP ~3/2 = 696.5784¢ Eigenmonzo basis: 2.5 (1/4-comma tuning)

7-limit Prime-Optimized Tunings

Weight-skew\Order	Chebyshevian	Euclidean	Manhattan
Tenney	CTC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CTE ~3/2 = 696.9521¢	CTOP ~3/2 = 696.6458¢
Tenney-Weil	CTWC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CTWE ~3/2 = 696.6562¢	CTWOP ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7
Equilateral	CEC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CEE ~3/2 = 696.8843¢ Eigenmonzo basis: 2.[0 1 4 10⟩	CEOP ~3/2 = 697.3437¢ Eigenmonzo basis: 2.21
Equilateral-Weil	CEWC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CEWE ~3/2 = 696.7248¢ Eigenmonzo basis: 2.4117715/9	CEWOP ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7
Benedetti	CBC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CBE ~3/2 = 697.0147¢ Eigenmonzo basis: 2.[0 1225 1764 2250⟩	CBOP ~3/2 = 697.8422¢ Eigenmonzo basis: 2.750141
Benedetti-Weil	CBWC ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7	CBWE ~3/2 = 696.6306¢ Eigenmonzo basis: 2.[0 -3290 3171 7215⟩	CBWOP ~3/2 = 696.8826¢ Eigenmonzo basis: 2.7

Temperament pages

Note:

1. Order: subgroup, comma list, (sval) mapping, (sval) mapping generators, gencom mapping, gencom, lattice basis, wedgie, optimal tuning (CTE), minimax tuning, tuning ranges, algebraic generator, Optimal ET sequence, badness, complexity spectrum, and others.
2. Comma list should show the simplest commas sufficient to define the temperament, stated in Normal lists #Normal interval list.
3. Mapping generators should show all the ratios as used in the mapping, including the period.
4. Since minimax tunings are based on tonality diamond, it should explicitly state the odd limit, or a diamond function of ratios.
5. For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (sic), simply show "POTE", otherwise show "POL2" or "POT2" instead of "POTE".

Get a family for:

• Ripple (3 different 7-limit extensions) done
• Smate (2 different 7-limit extensions) done
• Passion (4 different 7-limit extensions, 3 strong and 1 weak) done
• Undim done
• Quintaleap done
• Quindromeda done
• Parakleismic (many reasonable but unnamed 7-limit extensions)
• Schismatic rank three family (perhaps)

Who's next?

• Meantone family
• Didymus rank three family
• Archytas clan
• Archytas family
• Father family
• Trienstonic clan
• Septisemi temperaments
• Slendro clan
• Semiphore family
• Jubilismic clan
• Jubilismic family
• Mint temperaments
• Mint family
• Augmented family
• Dimipent family
• Dicot family
• Bug family
• Pelogic family
• Marvel temperaments
• Marvel family
• Gamelismic clan
• Gamelismic family
• Starling temperaments
• Starling family
• Sensamagic clan
• Sensamagic family
• Magic family
• Sensipent family
• Keemic temperaments
• Keemic family
• Sengic family
• Schismatic family
• Kleismic family
• Kleismic rank three family
• Hemimean clan
• Hemimean family
• Unicorn family
• Shibboleth family
• Immunity family
• Trisedodge family
• Qintosec family
• Pental family
• Sycamore family
• Rastmic rank three clan
• Semicomma family
• Orwellismic temperaments
• Orwellismic family
• Hemimage temperaments
• Hemimage family
• Porwell temperaments
• Porwell family
• Hemifamity temperaments
• Hemifamity family
• Porcupine family
• Porcupine rank three family
• Tetracot family
• Diaschismic family
• Diaschismic rank three family
• Compton family
• Amity family
• Misty family
• Undim family
• Lehmerismic temperaments
• Kalismic temperaments
• Ragismic microtemperaments
• Ragismic family
• Landscape microtemperaments
• Landscape family
• Dimcomp family
• Mirkwai clan
• Mirkwai family
• Breedsmic temperaments
• Breed family
• Escapade family
• Gravity family
• Cataharry temperaments
• Cataharry family
• Varunismic temperaments
• Keenanismic temperaments
• Valinorsmic clan
• Pentacircle clan
• Werckismic temperaments
• Swetismic temperaments
• Wizmic microtemperaments
• Metric microtemperaments
• Luna family
• Vulture family
• Tricot family
• Minortonic family
• Gammic family
• Garischismic clan
• Canousmic temperaments
• Canou family
• Semicanousmic clan
• Semiporwellismic clan
• Olympic clan
• Alphaxenic rank three clan
• Mirwomo temperaments
• Horwell temperaments
• Horwell family
• Gariboh clan
• Gariboh family
• Octagar temperaments
• Octagar family

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-49 31⟩	[⟨31 49]]	+1.63	1.64	4.22
2.3.5	81/80, 393216/390625	[⟨31 49 72]]	+0.98	1.63	4.20
2.3.5.7	81/80, 126/125, 1029/1024	[⟨31 49 72 87]]	+0.83	1.43	3.70
2.3.5.7.11	81/80, 99/98, 121/120, 126/125	[⟨31 49 72 87 107]]	+1.21	1.49	3.84

Scale tree

6-tone

1L 5s, 2L 4s, 3L 3s, 4L 2s, 5L 1s

7-tone

1L 6s, 2L 5s, 3L 4s, 4L 3s, 5L 2s, 6L 1s

8-tone

1L 7s, 2L 6s, 3L 5s, 4L 4s, 5L 3s, 6L 2s, 7L 1s

9-tone

1L 8s, 2L 7s, 3L 6s, 4L 5s, 5L 4s, 6L 3s, 7L 2s, 8L 1s

10-tone

1L 9s, 2L 8s, 3L 7s, 4L 6s, 5L 5s, 6L 4s, 7L 3s, 8L 2s, 9L 1s

11-tone

1L 10s, 2L 9s, 3L 8s, 4L 7s, 5L 6s, 6L 5s, 7L 4s, 8L 3s, 9L 2s, 10L 1s

12-tone

1L 11s, 2L 10s, 3L 9s, 4L 8s, 5L 7s, 6L 6s, 7L 5s, 8L 4s, 9L 3s, 10L 2s, 11L 1s

(Bounded by branch depth = 7)

Generator							Cents	L	s	L/s	Comments
4\7							685.714	1	1	1.000
						27\47	689.362	7	6	1.167
					23\40		690.000	6	5	1.200
						42\73	690.411	11	9	1.222
				19\33			690.909	5	4	1.250
						53\92	691.304	14	11	1.273
					34\59		691.525	9	7	1.286
						49\85	691.765	13	10	1.300
			15\26				692.308	4	3	1.333
						56\97	692.784	15	11	1.364
					41\71		692.958	11	8	1.375
						67\116	693.103	18	13	1.385
				26\45			693.333	7	5	1.400
						63\109	693.578	17	12	1.417
					37\64		693.750	10	7	1.429
						48\83	693.976	13	9	1.444
		11\19					694.737	3	2	1.500	L/s = 3/2
						51\88	695.455	14	9	1.556
					40\69		695.652	11	7	1.571
						69\119	695.798	19	12	1.583
				29\50			696.000	8	5	1.600
						66\131	696.183	21	13	1.615	Golden meantone
					47\81		696.296	13	8	1.625
						65\112	696.429	18	11	1.636
			18\31				696.774	5	3	1.667	Meantone is in this region
						61\105	697.143	17	10	1.700
					43\74		697.297	12	7	1.714
						68\117	697.436	19	11	1.727
				25\43			697.674	7	4	1.750
						57\98	697.959	16	9	1.778
					32\55		698.182	9	5	1.800
						39\67	698.507	11	6	1.833
	7\12						700.000	2	1	2.000	Basic diatonic (Generators smaller than this are proper)
						38\65	701.539	11	5	2.200
					31\53		701.887	9	4	2.250
						55\94	702.128	16	7	2.286
				24\41			702.409	7	3	2.333
						65\111	702.703	19	8	2.375
					41\70		702.857	12	5	2.400
						58\99	703.030	17	7	2.428
			17\29				703.448	5	2	2.500
						61\104	703.846	18	7	2.571
					44\75		704.000	13	5	2.600
						71\121	704.132	21	8	2.625	Golden neogothic
				27\46			704.348	8	3	2.667
						64\109	704.587	19	7	2.714
					37\63		704.762	11	4	2.750
						47\80	705.000	14	5	2.800
		10\17					705.882	3	1	3.000	L/s = 3/1
						43\73	706.849	13	4	3.250
					33\56		707.143	10	3	3.333
						56\95	707.368	17	5	3.400
				23\39			707.692	7	2	3.500
						59\100	708.000	18	5	3.600
					36\61		708.197	11	3	3.667
						49\83	708.434	15	4	3.750
			13\22				709.091	4	1	4.000	Archy is in this region
						42\71	709.859	13	3	4.333
					29\49		710.204	9	2	4.500
						45\76	710.526	14	3	4.667
				16\27			711.111	5	1	5.000
						35\59	711.864	11	2	5.500
					19\32		712.500	6	1	6.000
						22\37	713.514	7	1	7.000
3\5							720.000	1	0	→ inf