2058/2057

Interval information

Ratio	2058/2057
Factorization	2 × 3 × 73 × 11-2 × 17-1
Monzo	[1 1 0 3 -2 0 -1⟩
Size in cents	0.84142604¢
Name	xenisma
Color name	17u1uuz32, sululutrizo 2nd
FJS name	[math]\text{M2}^{7,7,7}_{11,11,17}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	22.0134
Weil height (log2 max(n, d))	22.0141
Wilson height (sopfr (nd))	65
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.20641 bits
Comma size	unnoticeable
open this interval in xen-calc

2058/2057, the xenisma, is an unnoticeable 17-limit (also 2.3.7.11.17 subgroup) comma measuring about 0.84 cents. It is the amount by which a stack of two 14/11's exceed 34/21.

Commatic relations

This comma is the difference between the following superparticular pairs:

• 364/363 and 442/441
• 441/440 and 561/560
• 936/935 and 1716/1715
• 1225/1224 and 3025/3024
• 1701/1700 and 9801/9800

Not to mention some nonsuperparticular but useful relations:

• 273/272 and 1573/1568
• 1575/1573 and 1275/1274

It factors into the following superparticular pairs:

• 2401/2400 and 14400/14399
• 2080/2079 and 194481/194480

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 xenismic temperament, or in the 2.3.7.11.17 subgroup, the rank-4 xenic temperament. In either case it enables xenismic chords.

Xenic

Subgroup: 2.3.7.11.17

Comma list: 2058/2057

Sval mapping: [⟨1 0 0 0 1], ⟨0 1 0 0 1], ⟨0 0 1 0 3], ⟨0 0 0 1 -2]]

sval mapping generators: ~2, ~3, ~7, ~11

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.9397, ~7/4 = 968.6818, ~11/8 = 551.4639
• CWE: ~2 = 1\1, ~3/2 = 701.9446, ~7/4 = 968.6908, ~11/8 = 551.4738

Optimal ET sequence: 46, 58, 63, 72, 89, 118, 135, 207, 342, 400, 535, 742, 1395, 1930, 2672

Xenismic

Subgroup: 2.3.5.7.11.13.17

Comma list: 2058/2057

Mapping:

[⟨	1	0	0	0	0	0	1	],
⟨	0	1	0	0	0	0	1	],
⟨	0	0	1	0	0	0	0	],
⟨	0	0	0	1	0	0	3	],
⟨	0	0	0	0	1	0	-2	],
⟨	0	0	0	0	0	1	0	]]

mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.9397, ~5/4 = 386.3137, ~7/4 = 968.6818, ~11/8 = 551.4639, ~13/8 = 840.5277
• CWE: ~2 = 1\1, ~3/2 = 701.9445, ~5/4 = 386.3208, ~7/4 = 968.6908, ~11/8 = 551.4738, ~13/8 = 840.5389

Optimal ET sequence: 43, 46, 58, 72, 103, 121, 149, 161, 190g, 224, 270, 311, 354, 400, 460, 581, 742, 814, 935, 1084, 1323, 1395, 2137, 3072e, 3342eg, 4156eg*

* optimal patent val: 3756

Etymology

The xenisma was named by Margo Schulter in 2000.