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Notes for temperament pages

Comment: the example below looks great to me by the very formatting.

Note:

1. Order: subgroup, comma list, mapping, mapping generators (or simply "generators", this is the same thing), wedgie, minimax tuning, tuning ranges, algebraic generator, complexity spectrum, vals, badness.
2. Comma list shows 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. Minimax tuning are based on tonality diamond, so it should explicitly state the odd limit, or a diamond function of ratios.
5. Use Template:Val list.

Who's next?

• Meantone family
• Archytas clan
• Father family
• Trienstonic clan
• Septisemi temperaments
• Archytas family
• Marvel temperaments
• Marvel family
• Schismatic family

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 also tempers out the common 7-limit comma 225/224 and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.495

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)

#	Cents	Approximate ratios
0	0.0	1/1
1	696.5	3/2
2	193.0	9/8, 10/9
3	889.5	5/3
4	386.0	5/4
5	1082.5	15/8, 28/15
6	579.0	7/5
7	75.5	21/20, 25/24, 28/27
8	772.0	14/9, 25/16
9	268.5	7/6
10	965.0	7/4
11	461.4	21/16

Technical 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]]

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] (19 to 12)
• nice range: [694.786, 701.955]
• strict range: [694.786, 700.000]

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

Badness: 0.0137

Archytas

Period: 1\1

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

EDO generators: (12, 5)\15, (7, 13)\22, (9, 16)\27

Scales: archytas12, archytas12synch

Technical data

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

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

Commas

41edo tempers out the following commas using its patent val, ⟨41 65 95 115 142 152 168 174 185 199 203].

Prime limit	Ratio	Name(s)
3	[65 -41⟩	41-comma
5	[-5 -10 9⟩	Shibboleth
5	[-25 7 6⟩	Ampersand
5	3125/3072	Magic comma
5	[5 -9 4⟩	Tetracot comma
5	[20 -17 3⟩	Roda
5	[-15 8 1⟩	Schisma
7	[0 -7 6 -1⟩	Great BP diesis
7	[-10 7 8 -7⟩	Blackjackisma
7	875/864	Keema
7	3125/3087	Gariboh
7	[10 -11 2 1⟩	Tolerma
7	[-15 3 2 2⟩	Mirwomo comma
7	245/243	Sensamagic
7	4000/3969	Octagar
7	[-15 0 -2 7⟩	Quince
7	1029/1024	Gamelisma
7	225/224	Marvel comma
7	[0 3 4 -5⟩	Mirkwai
7	[5 -7 -1 3⟩	Hemimage
7	5120/5103	Hemifamity
7	[25 -14 0 -1⟩	Garischisma
7	2401/2400	Breedsma
11	[15 0 1 0 -5⟩	Thuja comma
11	245/242	Cassacot
11	100/99	Ptolemisma
11	1344/1331	Hemimin
11	896/891	Pentacircle
11	[16 0 0 -2 -3⟩	Orgonisma
11	243/242	Rastma
11	385/384	Keenanisma
11	441/440	Werckisma
11	1375/1372	Moctdel
11	540/539	Swetisma
11	3025/3024	Lehmerisma
11	[-1 2 -4 5 -2⟩	Odiheim
13	343/338
13	105/104	Animist comma
13	[12 -7 0 1 0 -1⟩	Secorian
13	275/273	Gassorma
13	144/143	Grossma
13	196/195	Mynucuma
13	640/637	Huntma
13	1188/1183	Kestrel comma
13	325/324	Marveltwin
13	352/351	Minthma
13	364/363	Gentle comma
13	847/845	Cuthbert
13	729/728	Squbema
13	4096/4095	Schismina
13	[3 -2 0 -1 3 -2⟩	Harmonisma
17	2187/2176	Septendecimal schisma
17	256/255	Septendecimal kleisma
17	715/714	Septendecimal bridge comma
19	210/209	Spleen comma
19	361/360	Go comma
19	513/512	Undevicesimal comma
19	1216/1215	Eratosthenes' comma
23	736/729	Vicesimotertial comma
29	145/144	29th-partial chroma