Gabisma

Interval information

Ratio	1665/1664
Subgroup monzo	2.3.5.13.37 [-7 2 1 -1 1⟩
Size in cents	1.040093¢
Name	gabisma
Color name	37o3uy1 thisothuyo unison
FJS name	[math]\text{d}{-2}^{5,37}_{13}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	21.4017
Weil height (log2 max(n, d))	21.4026
Wilson height (sopfr(nd))	75
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.20921 bits
Comma size	unnoticeable
open this interval in xen-calc

1665/1664, the gabisma, is a 37-limit superparticular comma of about 1.0 cents.

Commatic relationships

This comma is the difference between the following superparticular pairs:

• 105/104 and 112/111
• 225/224 and 260/259
• 481/480 and 676/675
• 495/494 and 704/703
• 729/728 and 1296/1295

It can be factored into the following superparticular intervals:

• 1683/1682 and 155585/155584
• 1701/1700 and 78625/78624
• 1702/1701 and 76545/76544
• 1729/1728 and 44955/44954
• 1860/1859 and 15873/15872
• 1925/1924 and 12321/12320
• 2025/2024 and 9361/9360
• 2109/2108 and 7905/7904
• 2146/2145 and 7425/7424
• 2185/2184 and 6993/6992
• 2553/2552 and 4785/4784
• 2601/2600 and 4625/4624

Temperaments

Tempering out this comma in the full 37-limit leads to the rank-11 temperament gabismic. Using the 2.3.5.13.37 subgroup leads to the rank-4 temperament gabic.

Gabismic

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list. 1665/1664

Mapping:

[⟨	1	1	2	2	3	3	4	4	4	4	4	6	],
⟨	0	1	0	0	0	0	0	0	0	0	0	-2	],
⟨	0	0	1	0	0	0	0	0	0	0	0	-1	],
⟨	0	0	0	1	0	0	0	0	0	0	0	0	],
⟨	0	0	0	0	1	0	0	0	0	0	0	0	],
⟨	0	0	0	0	0	1	0	0	0	0	0	1	],
⟨	0	0	0	0	0	0	1	0	0	0	0	0	],
⟨	0	0	0	0	0	0	0	1	0	0	0	0	],
⟨	0	0	0	0	0	0	0	0	1	0	0	0	],
⟨	0	0	0	0	0	0	0	0	0	1	0	0	],
⟨	0	0	0	0	0	0	0	0	0	0	1	0	]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.862, ~5/4 = 386.214, ~7/4 = 968.826, ~11/8 = 551.318, ~13/8 = 840.781, ~17/16 = 104.955, ~19/16 = 297.513, ~23/16 = 628.274, ~29/16 = 1029.577, ~31/16 = 1145.036

Gabic

Subgroup: 2.3.5.13.37

Comma list: 1665/1664

Mapping:

[⟨	1	1	2	3	6	],
⟨	0	1	0	0	-2	],
⟨	0	0	1	0	-1	],
⟨	0	0	0	1	1	]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.862, ~5/4 = 386.214, ~13/8 = 840.781

Optimal ET sequence: 5, 7f, 7l, 9, 10, 12l, 12f, 15, 19, 34, 53, 171, 224, 277, 354, 407, 631, 684, 1038, 1091l, 1532l, 1756l, 1809l, 2163ll, 2216ll

Etymology

This comma was named by Francium in 2025. It refers to 1665 Gaby, the asteroid, which in turn was named after Gaby Reinmuth, the daughter-in-law of discoverer Karl Wilhelm Reinmuth.

See also

• Unnoticeable comma
• List of superparticular intervals