User:FloraC/Sandbox

Temperament pages

Databoxes has been canceled, but the cleanup will continue

Note:

1. Order: subgroup, comma list, (sval) mapping, (sval) mapping generators, gencom mapping, gencom, map to lattice, lattice basis, wedgie, POTE generator(s), minimax tuning, tuning ranges, algebraic generator, vals, 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 tuning are based on tonality diamond, it should explicitly state the odd limit, or a diamond function of ratios.
5. Replace "EDOs" with Template:Val list.
6. 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
• Parakleismic (2 different 7-limit extensions) canceled, not strong extensions
• Superpyth (2 different 7-limit extensions) canceled, not strong extensions

Who's next?

• Meantone family
• Archytas clan
• Father family
• Trienstonic clan
• Septisemi temperaments
• Archytas family
• Slendro clan
• Semiphore family
• Marvel temperaments
• Marvel family
• Mint temperaments
• Mint family
• Gamelismic clan
• Gamelismic family
• Jubilismic clan
• Jubilismic family
• Didymus rank three family
• Kleismic family
• Kleismic rank three family
• Shibboleth family
• Keemic temperaments
• Keemic family
• Schismatic family
• Hemimean clan
• Hemimean family
• Luna family
• Canousmic temperaments
• Canou family
• Starling temperaments
• Starling family
• Sensipent family
• Sensamagic clan
• Sensamagic family
• Magic family
• Unicorn family
• Trisedodge family
• Sycamore family
• Rastmic temperaments
• Semicomma family
• Orwellismic temperaments
• Hemimage temperaments
• Hemimage family
• Porwell temperaments
• Porwell family
• Hemifamity temperaments
• Hemifamity family
• Porcupine family
• Tetracot family
• Misty family
• Diaschismic family
• Amity family
• Lehmerismic temperaments
• Kalismic temperaments
• Ragismic family
• Landscape family
• Dimcomp family
• Mirkwai family
• Mirkwai clan
• Breed family
• Breedsmic temperaments
• Ragismic microtemperaments
• Escapade family
• Gravity family
• Cataharry family
• Cataharry temperaments
• Varunismic temperaments
• Werckismic temperaments
• Valinorsmic temperaments

As a regular temperament

3-limit

Subgroup: 2.3

Comma list: [-49 31⟩

Mapping: ⟨31 49]

TE octave stretch: +1.63¢

TE absolute error (¢): 1.64

TE relative error (%): 4.22

5-limit

Subgroup: 2.3.5

Comma list: 81/80, 393216/390625

Mapping: ⟨31 49 72]

TE octave stretch: +0.98¢

TE absolute error (¢): 1.63

TE relative error (%): 4.20

7-limit

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125, 1029/1024

Mapping: ⟨31 49 72 87]

TE octave stretch: +0.83¢

TE absolute error (¢): 1.43

TE relative error (%): 3.70

Septimal meantone

Deutsch

Main article: Meantone

See also: Wikipedia: Septimal meantone temperament

The 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, 7/5, C-F#, the tritone, and 21/16, C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas 126/125 and 225/224 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

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 ]]

POTE generator: ~3/2 = 696.495

Minimax tuning:

• 7- and 9-odd-limit

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]

Eigenmonzos: 2, 5

Tuning ranges:

• valid range: [694.737, 700.000] (11\19 to 7\12)
• nice range: [694.786, 701.955]
• strict range: [694.786, 700.000]

Algebraic generator: Cybozem, the real root of 15x³ - 10x² - 18, 503.4257 cents. The recurrence converges quickly.

Badness: 0.0137

Scales: meantone5, meantone7, meantone12

Archytas

Main article: Archytas

Subgroup: 2.3.5.7

Comma list: 64/63

Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]

Lattice basis:

3/2 length = 1.0508, 5/4 length = 2.3219

Angle (3/2, 5/4) = 90 degrees

POTE generators: ~3/2 = 709.3213, ~5/4 = 393.3747

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]

Eigenmonzos: 2, 6/5, 7/5

• 9-odd-limit

[[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]

Eigenmonzos: 2, 6/5, 9/7

Scales: archytas12, archytas12synch

Scale tree

(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