Polishookisma

Interval information

Ratio	8464/8463
Factorization	24 × 3-1 × 7-1 × 13-1 × 232 × 31-1
Monzo	[4 -1 0 -1 0 -1 0 0 2 0 -1⟩
Size in cents	0.204553¢
Name	Polishookisma
Color name	31u23oo3ur2 thiwubitwetho-athuru 2nd
FJS name	[math]\text{d}{-2}^{23,23}_{7,13,31}[/math]
Special properties	square superparticular, reduced
Tenney height (log2 nd)	26.0941
Weil height (log2 max(n, d))	26.0942
Wilson height (sopfr(nd))	108
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.19861 bits
Comma size	unnoticeable
S-expression	S92
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8464/8463, the polishookisma, is a 31-limit superparticular comma of about 0.20 cents.

Commatic relationships

This comma is the difference between the following superparticular pairs:

• 92/91 and 93/92
• 736/735 and 806/805
• 1288/1287 and 1519/1518
• 2016/2015 and 2646/2645
• 2185/2184 and 2945/2944
• 4096/4095 and 7936/7935
• 4186/4185 and 8281/8280

It can be factored into the following superparticular ratios:

• 8625/8624 and 453376/453375
• 8960/8959 and 152881/152880
• 10557/10556 and 42688/42687
• 10648/10647 and 41262/41261
• 13300/13299 and 23276/23275

Temperaments

Tempering out this comma in the full 31-limit leads to the rank-10 temperament polishookismic. Using the 2.3.7.13.23.31 subgroup leads to the rank-5 temperament polishookic.

Polishookismic

Subgroup: 2.3.5.7.11.13.17.23.29.31

Comma list: 8464/8463

Mapping:

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.959, ~5/4 = 386.314, ~7/4 = 968.838, ~11/8 = 551.318, ~13/8 = 840.549, ~17/16 = 104.955, ~19/16 = 297.513, ~23/16 = 628.210, ~29/16 = 1029.577

Polishookic

Subgroup: 2.3.7.13.23.31

Comma list: 8464/8463

Mapping:

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.959, ~7/4 = 968.838, ~13/8 = 840.549, ~23/16 = 628.210

Optimal ET sequence: 5fik, 5, 7, 9ik, 10k, 12, 12fik, 14, 17, 29, 31k, 36k, 45fi, 46, 53, 63, 77, 113, 130, 311, 436, 441, 571, 1335, 1470, 1525, 1600, 1776, 1853, 1983, 2424, 2554, 3125, 5679, 6884, 7455, 10009

Etymology

This comma was named by Francium in 2024. It refers to 8464 Polishook, the asteroid, which in turn was named after David Polishook, an Israeli astronomer.

See also

• Unnoticeable comma
• List of superparticular intervals