595/594

595/594, the dakotisma, is a small 17-limit comma measuring about 2.91 cents. It is one of the simplest commas tempered out in 311et, a highly notable general-purpose equal temperament.

Interval information

Ratio	595/594
Factorization	2-1 × 3-3 × 5 × 7 × 11-1 × 17
Monzo	[-1 -3 1 1 -1 0 1⟩
Size in cents	2.912085¢
Name	dakotisma
Color name	17o1uzy2, soluzoyo 2nd
FJS name	[math]\displaystyle{ \text{m2}^{5,7,17}_{11} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	18.4311
Weil norm (log2 max(n, d))	18.4335
Wilson norm (sopfr(nd))	51
Comma size	unnoticeable
S-expression	S34⋅S35
Open this interval in xen-calc

Commatic relations

This comma identifies itself as the difference between the following superparticular pairs:

• 34/33 and 36/35
• 85/84 and 99/98
• 100/99 and 120/119
• 136/135 and 176/175
• 221/220 and 351/350
• 289/288 and 561/560
• 325/324 and 715/714
• 364/363 and 936/935
• 385/384 and 1089/1088
• 441/440 and 1701/1700
• 442/441 and 1716/1715
• 540/539 and 5832/5831
• 561/560 and 9801/9800

not to mention some nonsuperparticular but useful relations:

• 154/153 and 245/243

It factors into the following superparticular pairs:

• 1156/1155 and 1225/1224
• 833/832 and 2080/2079
• 625/624 and 12376/12375

not to mention some nonsuperparticular but useful relations:

• 1575/1573 and 2431/2430
• 1377/1375 and 4375/4374
• 1275/1274 and 4459/4455

Temperaments

Tempering out this comma in the full 17-limit gives the rank-6 dakotismic temperament, or in the 2.3.5.7.11.17 subgroup, the rank-5 dakotic temperament, enabling dakotismic chords. You may find a list of good equal temperaments that support these temperaments below.

Dakotismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

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

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.2950 ¢, ~5/4 = 386.0705 ¢, ~7/4 = 968.4704 ¢, ~11/8 = 551.8578 ¢, ~13/8 = 840.5277 ¢

Optimal ET sequence: 12f, 14cf, 15g, 19eg, 22, 26, 27eg, 34d, 38df, 41, 46, 58, 72, 121, 140, 171, 190g, 212g, 217, 289, 311, 668, 694g, 740g, 1051dg*

* optimal patent val: 861

Dakotic

Subgroup: 2.3.5.7.11.17

Subgroup val mapping:

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

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.2950 ¢, ~5/4 = 386.0705 ¢, ~7/4 = 968.4704 ¢, ~11/8 = 551.8578 ¢

Optimal ET sequence: 12, 14c, 19eg, 22, 27eg, 31g, 34d, 38d, 41, 46, 58, 68, 72, 118, 171, 190g, 193, 212g, 217, 239, 311, 1051dg

Etymology

The dakotisma was named by Praveen Venkataramana in 2022 in honor of Scott Dakota's pioneering study of this comma and of its role in 311et.

See also

• List of superparticular intervals