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Template: Temperament data

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

Catalog of 5-limit rank-2 temperaments

Sintel's TE logflat badness.

Temperament	Complexity	Error (¢)	Badness	Mapping	Comma list
	2.63	40.1	.392	[⟨2 3 0], ⟨0 0 1]]	10/9
Antitonic	2.83	41.6	.508	[⟨2 3 0], ⟨0 0 1]]	9/8
Father	3.17	20.4	.349	[⟨1 0 4], ⟨0 1 -1]]	16/15
Dicot	3.73	11.0	.306	[⟨1 1 2], ⟨0 2 1]]	25/24
Bug	4.30	17.9	.769	[⟨1 0 0], ⟨0 2 3]]	27/25
Meantone	5.08	2.44	.173	[⟨1 0 -4], ⟨0 1 4]]	81/80
Mavila	5.68	9.36	.928	[⟨1 0 7], ⟨0 1 -3]]	135/128
Augmented	6.40	3.71	.523	[⟨3 0 7], ⟨0 1 0]]	128/125
Porcupine	6.86	4.14	.722	[⟨1 2 3], ⟨0 3 5]]	250/243
Blackwood	7.29	7.14	1.50	[⟨5 8 0], ⟨0 0 1]]	256/243
Diminished	7.54	4.79	1.11	[⟨4 0 3], ⟨0 1 1]]	648/625
Supersharp	7.58	12.3	2.88	[⟨2 0 -5], ⟨0 1 3]]	800/729
Diaschismic	8.76	1.29	.467	[⟨2 0 11], ⟨0 1 -2]]	2048/2025
Laconic	8.87	10.1	3.80	[⟨1 1 1], ⟨0 3 7]]	2187/2000
Lafayette	9.31	10.8	4.70	[⟨1 1 2], ⟨0 5 3]]	3456/3125
Symbolic	9.38	9.40	4.19	[⟨1 3 2], ⟨0 4 -1]]	2048/1875
Sixix	9.81	7.06	3.59	[⟨1 3 4], ⟨0 5 6]]	3125/2916
Magic	9.98	1.71	.919	[⟨1 0 2], ⟨0 5 1]]	3125/3072
Uncle	10.0	11.6	6.33	[⟨1 0 12], ⟨0 1 -6]]	4096/3645
Whitewood	10.1	6.49	3.63	[⟨7 11 0], ⟨0 0 1]]	2187/2048
Hanson	11.1	.422	.310	[⟨1 0 1], ⟨0 6 5]]	15625/15552
Ripple	11.2	4.35	3.26	[⟨1 2 3], ⟨0 5 8]]	6561/6250
Negri	11.3	2.61	2.04	[⟨1 2 2], ⟨0 4 -3]]	16875/16384
Tetracot	11.5	1.39	1.14	[⟨1 1 1], ⟨0 4 9]]	20000/19683
	11.9	6.38	5.77	[⟨1 0 5], ⟨0 3 -5]]	32768/30375
Superpyth	12.2	3.26	3.17	[⟨1 0 -12], ⟨0 1 9]]	20480/19683
Helmholtz	12.8	.0881	.0999	[⟨1 0 15], ⟨0 1 -8]]	32805/32768
Wesley	13.6	4.25	5.81	[⟨1 4 3], ⟨0 7 2]]	78125/73728
Sensipent	14.1	.550	.826	[⟨1 6 8], ⟨0 7 9]]	78732/78125
Stump	14.4	2.91	4.71	[⟨1 0 6], ⟨0 3 -7]]	273375/262144
Passion	14.5	2.42	3.95	[⟨1 2 2], ⟨0 5 -4]]	262144/253125
Doublewide	14.7	3.09	5.32	[⟨2 1 3], ⟨0 4 3]]	390625/373248
Würschmidt	16.3	.404	.952	[⟨1 7 3], ⟨0 8 1]]	393216/390625
Amity	16.4	.217	.515	[⟨1 3 6], ⟨0 5 13]]	1600000/1594323
Valentine	16.7	1.14	2.88	[⟨1 1 2], ⟨0 9 5]]	1990656/1953125
Immunity	17.2	1.59	4.33	[⟨1 0 -8], ⟨0 2 13]]	1638400/1594323
Shibboleth	17.3	1.91	5.34	[⟨1 4 5], ⟨0 9 10]]	1953125/1889568
Compton	17.4	.778	2.22	[⟨12 19 0], ⟨0 0 1]]	531441/524288
Orson	17.5	.332	.957	[⟨1 0 3], ⟨0 7 -3]]	2109375/2097152
Unicorn	18.0	1.12	3.53	[⟨1 2 3], ⟨0 8 13]]	1594323/1562500
Mynic	18.6	1.69	5.86	[⟨1 9 9], ⟨0 10 9]]	10077696/9765625
Ampersand	19.9	.917	3.89	[⟨1 1 3], ⟨0 6 -7]]	34171875/33554432
Fifive	20.8	.992	4.83	[⟨2 2 3], ⟨0 5 7]]	9765625/9565938
Misty	21.3	.476	2.50	[⟨3 0 26], ⟨0 1 -4]]	67108864/66430125
Gravity	21.4	.415	2.19	[⟨1 5 12], ⟨0 6 17]]	129140163/128000000
Rodan	22.2	.669	3.95	[⟨1 1 -1], ⟨0 3 17]]	[20 -17 3⟩
Python	22.2	1.17	6.92	[⟨1 0 -23], ⟨0 1 16]]	[-23 16 -1⟩
Mabila	23.8	.753	5.45	[⟨1 6 1], ⟨0 10 -3]]	[28 -3 -10⟩
Parakleismic	24.8	.123	1.02	[⟨1 5 6], ⟨0 13 14]]	[8 14 -13⟩
Quartonic	24.9	.331	2.75	[⟨1 2 3], ⟨0 11 18]]	[3 -18 11⟩
Escapade	25.8	.213	1.97	[⟨1 2 2], ⟨0 9 -7]]	[32 -7 -9⟩
Ditonic	26.3	.399	3.92	[⟨1 6 3], ⟨0 13 2]]	[-27 -2 13⟩
Vishnuzmic	26.5	.0727	.731	[⟨2 4 5], ⟨0 7 3]]	[23 6 -14⟩
Vulture	27.3	.0890	.972	[⟨1 0 -6], ⟨0 4 21]]	[24 -21 4⟩
Trisedodge	27.7	.518	5.92	[⟨5 1 7], ⟨0 3 2]]	[19 10 -15⟩
	28.6	.243	3.06	[⟨1 4 -1], ⟨0 8 -11]]	[-36 11 8⟩

5-limit rank-2 temperaments by TE simple badness

Breed's TE 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
Diaschismic	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

5-limit norm-based tunings

	Manhattan	Euclidean		Chebyshevian
		Constrained	Constrained & skewed
Equilateral	CEOP: ~3/2 = 697.654 ¢ (1/5-comma)	CEE: ~3/2 = 696.895 ¢ (4/17-comma)	CSEE: ~3/2 = 696.453 ¢ (11/43-comma)	CEC: ~3/2 = 696.578 ¢ (1/4-comma)
Tenney	CTOP: ~3/2 = 698.020 ¢	CTE: ~3/2 = 697.214 ¢	CWE: ~3/2 = 696.651 ¢	CTC: ~3/2 = 696.578 ¢ (1/4-comma)
Benedetti, Wilson	CBOP: ~3/2 = 698.160 ¢ (3/17-comma)	CBE: ~3/2 = 697.374 ¢ (36/169-comma)	CSBE: ~3/2 = 696.787 ¢ (31/129-comma)	CBC: ~3/2 = 696.578 ¢ (1/4-comma)

7-limit norm-based tunings

	Manhattan	Euclidean		Chebyshevian
		Constrained	Constrained & skewed
Equilateral	CEOP: ~3/2 = 697.344 ¢	CEE: ~3/2 = 696.884 ¢	CSEE: ~3/2 = 696.725 ¢	CEC: ~3/2 = 696.883 ¢
Tenney	CTOP: ~3/2 = 696.646 ¢	CTE: ~3/2 = 696.952 ¢	CWE: ~3/2 = 696.656 ¢	CTC: ~3/2 = 696.883 ¢
Benedetti, Wilson	CBOP: ~3/2 = 697.842 ¢	CBE: ~3/2 = 697.015 ¢	CSBE: ~3/2 = 696.631 ¢	CBC: ~3/2 = 696.883 ¢

Temperament pages

Note:

1. Order: subgroup, comma list, mapping, mapping generators, (subgroup-val mapping, gencom mapping), lattice basis, optimal tunings (WE and CWE), minimax tuning, tuning ranges, algebraic generator (?), optimal ET sequence, badness, complexity spectrum, and others.
2. Comma list should be the Tenney-minimal commas sufficient to define the temperament, stated in Normal forms #Normal forms for commas.
3. Mapping generators should show all the ratios as used in the mapping, including the period.
4. For subgroup temperaments, "mapping" becomes "subgroup-val mapping" and "gencom mapping"".

• If the subgroup is trivial, simply show "WE"/"CWE", otherwise show "subgroup/inharmonic WE/CWE" instead.

1. Cent symbol in tunings.
2. Since minimax tunings are based on tonality diamonds, it should explicitly state the odd limit, or a diamond function of ratios.
3. Algebraic generators are to be discussed later.

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-3 family (perhaps)

Progress:

• Meantone family
• Didymus rank-3 family
• Archytas clan
• Archytas family
• Father family
• Trienstonic clan
• Septisemi temperaments
• Semaphoresmic clan
• Semaphoresmic family
• Jubilismic clan
• Jubilismic family
• Mint temperaments
• Mint family
• Augmented family
• Diminished family
• Passion family
• Ripple family
• Dicot family
• Bug family
• Mavila family
• Smate family
• Gamelismic clan
• Gamelismic family
• Marvel temperaments
• Marvel 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-3 family
• Würschmidt family
• Unicorn family
• Immunity family
• Fifive family
• Trisedodge family
• Quintosec family
• Quintile family
• Sycamore family
• Semicomma family
• Orwellismic temperaments
• Orwellismic family
• Hemimean clan
• Hemimean family
• Hemifamity temperaments
• Hemifamity family
• Porwell temperaments
• Porwell family
• Hemimage temperaments
• Hemimage family
• Porcupine family
• Porcupine rank-3 family
• Tetracot family
• Diaschismic family
• Diaschismic rank-3 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
• Quince clan
• Breedsmic temperaments
• Breed family
• Escapade family
• Gravity family
• Buzzardsmic clan
• Cataharry temperaments
• Cataharry family
• Varunismic temperaments
• Rastmic rank-3 clan
• Biyatismic clan
• Ptolemismic clan
• Valinorsmic clan
• Pentacircle clan
• Keenanismic temperaments
• Werckismic temperaments
• Swetismic temperaments
• Wizmic microtemperaments
• Metric microtemperaments
• Horwell temperaments
• Horwell family
• Luna family
• Vulture family
• Alphatricot family
• Minortonic family
• Gammic family
• Vishnuzmic family
• Garischismic clan
• Canousmic temperaments
• Canou family
• Semicanousmic clan
• Semiporwellismic clan
• Olympic clan
• Alphaxenic rank-3 clan
• Mirwomo temperaments
• Gariboh clan
• Gariboh family
• Octagar temperaments
• Octagar family
• Whitewood family
• Wesley family
• Mabila family
• Maquila family
• Maja family
• Ditonmic family
• Greenwoodmic temperaments
• Avicennmic temperaments
• Very low accuracy temperaments
• Very high accuracy temperaments
• Equivalence continua
• Fractional-octave temperaments
• Subgroup temperaments

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